## 1. Life and Achievements




Hermann Weyl was born on 9 November 1885 in the small town of 
Elmshorn near Hamburg. In 1904 he entered Gottingen University, 
where his teachers included Hilbert, Klein and Minkowski. Weyl was 
particularly impressed with Hilbert's lectures on number theory and 
resolved to study everything he had written. Hilbert's work on 
integral equations became the focus of Weyl's (1908) doctoral 
dissertation, written under Hilbert's direction. In this and in 
subsequent papers Weyl made important contributions to the theory of 
self-adjoint operators. Virtually all of Weyl's many publications 
during his stay in Gottingen until 1913 dealt with integral 
equations and their applications.




After Weyl's (1910b) habilitation, he became a Privatdozent and was 
thereby entitled to give lectures at the University of 
Gottingen. Weyl chose to lecture on Riemann's theory of 
algebraic functions during the winter semester of 1911-12. 
These lectures became the basis of Weyl's (1913) first book *Die 
Idee der Riemannschen Flache* (*The Concept of a Riemann 
Surface*). This work, in which function theory, geometry and 
topology are unified, constitutes the first modern and comprehensive 
treatment of Riemann surfaces. The work also contains the first 
construction of an abstract manifold. Emphasizing that the `points of
the manifold' can be quite arbitrary, Weyl based his definition of a 
general two-dimensional manifold or surface on an extension of the 
neighbourhood axioms that Hilbert (1902) had proposed for the 
definition of a plane. The work is indicative of Weyl's exceptional 
gift for harmoniously uniting into a coherent whole a patchwork of 
distinct mathematical fields.




In 1913 Weyl was offered, and accepted, a professorship at the 
Eidgenossische Technische Hochschule--ETH (Swiss Federal 
Institute of Technology)--in Zurich. Weyl's years in 
Zurich were extraordinarily productive and resulted in some of 
his finest work, especially in the foundations of mathematics and 
physics. When he arrived in Zurich in the fall of 1913, Einstein
and Grossmann were struggling to overcome a difficulty in their 
effort to provide a coherent mathematical formulation of the general 
theory of relativity. Like Hilbert, Weyl appreciated the importance 
of a close relationship between mathematics and physics. It was 
therefore only natural that Weyl should become interested in 
Einstein's theory and the potential mathematical challenges it might 
offer. Following the outbreak of the First World War, however, in May
1915 Weyl was called up for military service. But Weyl's academic 
career was interrupted only briefly, since in 1916 he was exempted 
from military duties for reasons of health. In the meantime Einstein 
had accepted an offer from Berlin and had left Zurich in 1914. 
Einstein's departure had weakened the theoretical physics program at 
the ETH and (as reported by Frei and Stammbach (1992, 26) the 
administration hoped that Weyl's presence would alleviate the 
situation. But Weyl needed no external prompting to work in, and to 
teach, theoretical physics: his interest in the subject in general 
and, above all, in the theory of relativity, gave him more than 
sufficient motivation in that regard. Weyl decided to lecture on the 
general theory of relativity in the summer semester of 1917, and 
these lectures became the basis of his famous book 
*Raum-Zeit-Materie* (*Space-Time-Matter*) of 1918.




During 1917-24, Weyl directed his energies equally to the 
development of the mathematical and philosophical foundations of 
relativity theory, and to the broader foundations of mathematics. It 
is in these two areas that his philosophical erudition, nourished 
from his youth, manifests itself most clearly. The year 1918, the 
same year in which *Space-Time-Matter* appeared, also saw the 
publication of *Das Kontinuum* (*The Continuum*), a 
work in which Weyl constructs a new foundation for mathematical 
analysis free of what he had come to see as fatal flaws in the 
set-theoretic formulation of Cantor and Dedekind. Soon afterwards 
Weyl embraced Brouwer's mathematical intuitionism; in the early 1920s
he published a number of papers elaborating on and defending the 
intuitionistic standpoint in the foundations of mathematics.




It was also during the first years of the 1920s that Weyl came to 
appreciate the power and utility of group theory, initially in 
connection with his work on the solution to the Riemann-Helmholtz-Lie
problem of space. Weyl analyzed this problem, the 
*Raumproblem*, in a series of articles and lectures during the
period 1921-23. Weyl (1949b, 400) noted that his interest in 
the philosophical foundations of the general theory of relativity had
motivated his analysis of the representations and invariants of the 
continuous groups:



> I can say that the wish to understand what really
> is the mathematical substance behind the formal apparatus of
> relativity theory led me to the study of the representations and
> invariants of groups; and my experience in this regard is probably not
> unique.




This newly acquired appreciation of group theory led Weyl to what he 
himself considered his greatest work in mathematics, a general theory
of the representations and invariants of the classical Lie groups 
(Weyl 1924a, 1924f, 1925, 1926a, 1926b, 1926c). Later Weyl (1939) 
wrote a book, *The Classical Groups: Their Invariants and 
Representations*, in which he returned to the theory of 
invariants and representations of semisimple Lie groups. In this work
he realized his ambition "to derive the decisive results for 
the most important of these groups by direct algebraic construction, 
in particular for the full group of all non-singular linear 
transformations and for the orthogonal group."




Weyl applied his work in group theory and his earlier work in 
analysis and spectral theory to the new theory of quantum mechanics. 
Weyl's mathematical analysis of the foundations of quantum mechanics 
showed that regularities in a physical theory are most fruitfully 
understood in terms of symmetry groups. Weyl's (1928) book 
*Gruppentheorie und Quantenmechanik* (*Group Theory and 
Quantum Mechanics*) deals not only with the theory of quantum 
mechanics but also with relativistic quantum electrodynamics. In this
work Weyl also presented a very early analysis of discrete symmetries
which later stimulated Dirac to predict the existence of the positron
and the antiproton.




During his years in Zurich Weyl received, and turned down, 
numerous offers of professorships by other 
universities--including an invitation in 1923 to become Felix 
Klein's successor at Gottingen. It was only in 1930 that he 
finally accepted the call to become Hilbert's successor there. His 
second stay in Gottingen was to be brief. Repelled by Nazism, 
"deeply revolted," as he later wrote, "by the 
shame which this regime had brought to the German name," he 
left Germany in 1933 to accept an offer of permanent membership of 
the newly founded Institute for Advanced Study in Princeton. Before 
his departure for Princeton he published *The Open World* 
(1932); his tenure there saw the publication of *Mind and 
Nature* (1934), the aforementioned *The Classical Groups* 
(1939), *Algebraic Theory of Numbers* (1940), *Meromorphic 
Functions and Analytic Curves* (1943), *Philosophy of 
Mathematics and Natural Science* (1949; an enlarged English 
version of a 1927 work *Philosophie der Mathematik und 
Naturwissenschaften*), and *Symmetry* (1952). In 1951 he 
formally retired from the Institute, remaining as an emeritus member 
until his death, spending half his time there and half in 
Zurich. He died in Zurich suddenly, of a heart attack, on 9
December 1955.


## 2. Metaphysics




Weyl was first and foremost a mathematician, and certainly not a 
"professional" philosopher. But as a German intellectual 
of his time it was natural for him to regard philosophy as a pursuit 
to be taken seriously. In Weyl's case, unusually even for a German 
mathematician, it was idealist philosophy that from the beginning 
played a significant role in his thought. Kant, Husserl, Fichte, and,
later, Leibniz, were at various stages major influences on Weyl's 
philosophical thinking. As a schoolboy Weyl had been impressed by 
Kant's "Critique of Pure Reason." He was especially taken
with Kant's doctrine that space and time are not inherent in the 
objects of the world, existing as such and independently of our 
awareness, but are, rather, forms of our intuition As he reports in 
Insight and Reflection, (Weyl 1955), his youthful enthusiasm for Kant
crumbled soon after he entered Gottingen University in 1904. 
There he read Hilbert's Foundations of Geometry, a tour-de-force of 
the axiomatic method, in comparison to which Kant's "bondage to
Euclidean geometry" now appeared to him naive. After this 
philosophical reverse he lapsed into an indifferent positivism for a 
while. But in 1912 he found a new and exciting source of 
philosophical enlightenment in Husserl's 
 phenomenology.[1]
 It was also at about this time that Fichte's metaphysical idealism 
came to "capture his imagination." Although Weyl later 
questioned idealist philosophy, and became dissatisfied with 
phenomenology, he remained faithful throughout his life to the 
primacy of intuition that he had first learned from Kant, and to the 
irreducibility of individual consciousness that had been confirmed in
his view by Fichte and Husserl.




Weyl never provided a systematic account of his philosophical views,
and sorting out his overall philosophical position is no easy matter.
Despite the importance of intuition and individual consciousness in
Weyl's philosophical outlook, it would nevertheless be inexact to
describe his outlook as being that of a "pure" idealist,
since certain "realist" touches seem also to be present,
in his approach to physics, at least. His metaphysics appears to rest
on three elements, the first two of which may be considered
"idealist", and the third "realist": these
are, respectively, the Ego or "I", the (Conscious) Other
or "Thou", and the external or "objective"
world. 




It is the first of these constituents, the Ego, to which Weyl 
ascribes primacy. Indeed, in Weyl's words,



> 
> 
> The world exists only as met with by an ego, as one appearing to
> a consciousness; the consciousness in this function does not
> belong to the world, but stands out against the being as the 
> sphere of vision, of meaning, of image, or however else one may call
> it. (Weyl 1934, 1)




The Ego alone has direct access to the given, that is, to the raw 
materials of the existent which are presented to consciousness with 
an immediacy at once inescapable and irreducible. The Ego is singular
in that, from its own standpoint, it is unique. But in an act of 
self-reflection, through grasping (in Weyl's words) "what I am 
for myself", the Ego comes to recognize that it has a 
*function*, namely as "conscious-existing carrier of the
world of phenomena." It is then but a short step for the Ego to
transcend its singularity through the act of defining an 
"ego" to be an entity performing that same function 
*for itself*. That is, an ego is precisely what I am for 
myself (in other words, what the Ego is for itself)--again a 
"conscious-existing carrier of the world of 
phenomena"--and yet *other* than myself. 
"Thou" is the term the Ego uses to address, and so to 
identify, an ego in this sense. "Thou" is thus the Ego 
generalized, the Ego refracted through itself. The Ego grasps that it
exists within a world of Thous, that is, within a world of other Egos
similar to itself. While the Ego has, of necessity, no direct access 
to any Thou, it can, through analogy and empathy, grasp what it is to
be Thou, a conscious being like oneself. By that very fact the Ego 
recognizes in the Thou the same luminosity it sees in itself.




The relationship of the Ego with the external world, the realm of 
"objective" reality, is of an entirely different nature. 
There is no analogy that the Ego can draw--as it can with the 
Thou--between itself and the external world, since that world 
(presumably) lacks consciousness. The external world is radically 
other, and opaque to the 
 Ego[2].
 Like Kant's noumenal realm, the external world is outside the 
immediacy of consciousness; it is, in a word, *transcendent*. 
Since this transcendent world is not directly accessible to the Ego, 
as far as the latter is concerned the existence of that world must 
arise through *postulation*, "a matter of metaphysics, 
not a judgment but an act of acknowledgment or 
 belief."[3]
 Indeed, according to Weyl, it is not strictly necessary for the Ego 
to postulate the existence of such a world, even given the existence 
of a world of Thous:




For as long as I do not proceed beyond what is given, or, more 
exactly, what is given at the moment, there is no need for the 
substructure of an objective world. Even if I include memory and in 
principle acknowledge it as valid testimony, if I furthermore accept 
as data the contents of the consciousness of others on equal terms 
with my own, thus opening myself to the mystery of intersubjective 
communication, I would still not have to proceed as we actually do, 
but might ask instead for the 'transformations' which 
mediate between the images of the several consciousnesses. Such a 
presentation would fit in with Leibniz's monadology. (Weyl 1949, 
117.)




But once the existence of the transcendent world is postulated, its 
opacity to the Ego can be partly overcome by constructing a 
representation of it through the use of *symbols*, the 
procedure called by Weyl *symbolic construction*, (or 
*constructive 
 cognition)*[4]
 and which he regarded as the cornerstone of scientific explanation. 
He outlines the process as follows (Weyl 1934, 53):


1. Upon that which is given, certain reactions are performed by
which the given is in general brought together with other elements
capable of being varied arbitrarily. If the results to be read from
these reactions are found to be independent of the variable auxiliary
elements they are then introduced as attributes inherent in the things
themselves (even if we do not actually perform those reactions on
which their meaning rests, but only believe in the possibility of
their being performed).
2. By the introduction of symbols, the judgements are split up and a
part of the manipulations is made independent of the given and its
duration by being shifted onto the representing symbols which are time
resisting and simultaneously serve the purpose of preservation and
communication. Thereby the unrestricted handling of notions arises in
counterpoint to their application, ideas in a relatively independent
manner confront reality.
3. Symbols are not produced simply "according to demand"
wherever they correspond to actual occurrences, but they are embedded
into an ordered manifold of possibilities created by free construction
and open towards infinity. Only in this way may we contrive to predict
the future, for the future is not given actually.




Weyl's procedure thus amounts to the following. In step 1, a given 
configuration is subjected to variation. One then identifies those 
features of the configuration that remain unchanged under the 
variation--the *invariant* features; these are in turn, 
through a process of reification, deemed to be properties of an 
unchanging substrate--the "things themselves". It is
precisely the invariance of such features that renders them (as well 
as the "things themselves") capable of being represented 
by the "time resisting" symbols Weyl introduces in step 
2. As (written) symbols these are communicable without temporal 
distortion and can be subjected to unrestricted manipulation without 
degradation. It is the flexibility conferred thereby which enables 
the use of symbols to be conformable with reality. Nevertheless (step
3) symbols are not haphazardly created in response to immediate 
stimuli; they are introduced, rather, in a structured, yet freely 
chosen manner which reflects the idea of an underlying 
order--the "one real world"--about which not 
everything is, or can be, known--it is, like the future, 
"open towards infinity". Weyl observes that the 
reification implicit in the procedure of symbolic construction leads 
inevitably to its iteration, for "the transition from step to 
step is made necessary by the fact that the objects at one step 
reveal themselves as manifestations of a higher reality, the reality 
of the next step" (Weyl (1934), 32-33). But in the end 
"systematic scientific explanation will finally reverse the 
order: first it will erect its symbolical world by itself, without 
any reference, then, skipping all intermediate steps, try to describe
which symbolical configurations lead to which data of 
consciousness" (*ibid*.). In this way the symbolic world
becomes (mistakenly) identified with the transcendent world.




It is symbolic construction which, in Weyl's vision, allows us access
to the "objective" world presumed to underpin our 
immediate perceptions; indeed, Weyl holds that the objective world, 
being beyond the grasp (the "lighted circle") of 
intuition, can *only* be presented to us in symbolic 
 form[5].
 We can see a double dependence on the Ego in Weyl's idea of symbolic 
construction to get hold of an objective world beyond the mental. For
not only is that world "constructed" by the Ego, but the 
materials of construction, the symbols themselves, as signs intended 
to convey meaning, have no independent existence beyond their 
graspability by a consciousness. By their very nature these symbols 
cannot point directly to an external world (even given an unshakable 
belief in the existence of that world) lying beyond consciousness. 
Weyl's metaphysical triad thus reduces to what might be called a 
*polarized dualism*, with the mental (I, Thou) as the primary,
independent pole and objective reality as a secondary, dependent 
 pole[6].
 





In Weyl's view mathematics simply lies - as it did for Brouwer
- within the Ego's "lighted circle of intuition" and
so is, in principle at least, completely presentable to that
intuition. But the nature of physics is more complicated. To the
extent that physics is linked to the transcendent world of objective
reality, it cannot arise as the direct object of intuition, but must,
like the transcendent world itself, be presented in symbolic form;
more exactly, as the result of a process of symbolic construction. It
is this which, in Weyl's vision, allows us access to to the
"objective" world presumed to underpin our immediate perceptions. 




Weyl's conviction that the objective world can only be presented to us
through symbolic construction may serve to explain his apparently
untroubled attitude towards the highly counterintuitive nature of
quantum theory. Indeed, the claims of numerous physicists that the
quantum microworld is accessible to us only through abstract
mathematical description provides a vindication of Weyl's thesis that
objective reality cannot be grasped directly, but only through the
mediation of symbols. 


In his later years Weyl attempted to enlarge his metaphysical triad 
(I, Thou, objective world) to a tetrad, by a process of completion, 
as it were, to embrace the "godhead that lives in impenetrable 
silence", the objective counterpart of the Ego, which had been 
suggested to him by his study of Eckhart. But this effort was to remain
uncompleted.






During his long philosophical voyage Weyl stopped at a number of 
ports of call: in his youth, Kantianism and positivism; then 
Husserlian phenomenological idealism; later Brouwerian intuitionism 
and finally a kind of theological existentialism. But apart from his 
brief flirtation with positivism (itself, as he says, the result of a
disenchantment with Kant's "bondage to Euclidean 
geometry"), Weyl's philosophical orientation remained in its 
essence idealist (even granting the significant realist elements 
mentioned above). Nevertheless, while he continued to acknowledge the
importance of phenomenology, his remarks in *Insight and 
Reflection* indicate that he came to regard Husserl's doctrine as
lacking in two essential respects: first, it failed to give due 
recognition to the (construction of) transcendent external world, 
with which Weyl, in his capacity as a natural scientist, was 
concerned; secondly, and perhaps in Weyl's view even more seriously, 
it failed to engage with the enigma of selfhood: the fact that I am 
the person I am. Grappling with the first problem led Weyl to 
identify symbolic construction as providing sole access to objective 
reality, a position which brought him close to Cassirer in certain 
respects; while the second problem seems to have led him to 
existentialism and even, through his reading of Eckhart, to a kind of
religious mysticism.


## 3. Work in the foundations and philosophy of mathematics




Towards the end of his *Address on the Unity of Knowledge*, 
delivered at the 1954 Columbia University bicentennial celebrations, 
Weyl enumerates what he considers to be the essential constituents of
knowledge. At the top of his 
 list[7]
 comes



> 
> 
> ...intuition, mind's ordinary act of seeing what is
> given to it. (Weyl 1954, 629)




In particular Weyl held to the view that intuition, or 
*insight*--rather than *proof*--furnishes the
ultimate foundation of *mathematical* knowledge. Thus in his 
*Das Kontinuum* of 1918 he says:



> 
> In the Preface to Dedekind (1888) we read that "In science,
> whatever is provable must not be believed without proof." This
> remark is certainly characteristic of the way most mathematicians
> think. Nevertheless, it is a preposterous principle. As if such an
> indirect concatenation of grounds, call it a proof though we may, can
> awaken any "belief" apart from assuring ourselves through
> immediate insight that each individual step is correct. In all cases,
> this process of confirmation--and not the proof--remains
> the ultimate source from which knowledge derives its authority; it is
> the "experience of truth". (Weyl 1987, 119)




Weyl's idealism naturally inclined him to the view that the ultimate 
basis of his own subject, mathematics, must be found in the 
intuitively given as opposed to the transcendent. Nevertheless, he 
recognized that it would be unreasonable to require all mathematical 
knowledge to possess intuitive immediacy. In *Das Kontinuum*, 
for example, he says:



> 
> The states of affairs with which mathematics deals are, apart from the
> very simplest ones, so complicated that it is practically impossible
> to bring them into full givenness in consciousness and in this way to
> grasp them completely. (*Ibid*., 17)




Nevertheless, Weyl felt that this fact, inescapable as it might be, 
could not justify extending the bounds of mathematics to embrace 
notions, such as the actual infinite, which cannot be given fully in 
intuition even in principle. He held, rather, that such extensions of
mathematics into the transcendent are warranted only by the fact that
mathematics plays an indispensable role in the physical sciences, in 
which intuitive evidence is necessarily transcended. As he says in 
The Open 
 World[8]:
 



> ... if mathematics is taken by itself, one should
> restrict oneself with Brouwer to the intuitively cognizable truths
> ... nothing compels us to go farther. But in the natural sciences
> we are in contact with a sphere which is impervious to intuitive
> evidence; here cognition necessarily becomes symbolical construction.
> Hence we need no longer demand that when mathematics is taken into the
> process of theoretical construction in physics it should be possible to
> set apart the mathematical element as a special domain in which all
> judgments are intuitively certain; from this higher standpoint which
> makes the whole of science appear as one unit, I consider Hilbert to be
> right. (Weyl 1932, 82).




In *Consistency in Mathematics* (1929), Weyl characterized the
mathematical method as



> the a priori construction of the possible in opposition to
> the a posteriori description of what is actually
>  given.[9]
> 




The problem of identifying the limits on constructing "the 
possible" in this sense occupied Weyl a great deal. He was 
particularly concerned with the concept of the mathematical 
*infinite*, which he believed to elude 
"construction" in the naive set-theoretical sense 
 [10].
 Again to quote a passage from *Das Kontinuum:*



> 
> No one can describe an infinite set other than by indicating
> properties characteristic of the elements of the set.... The
> notion that a set is a "gathering" brought together by
> infinitely many individual arbitrary acts of selection, assembled and
> then surveyed as a whole by consciousness, is nonsensical;
> "inexhaustibility" is essential to the infinite. (Weyl
> 1987, 23)




But still, as Weyl attests towards the end of *The Open 
World*, "the demand for totality and the metaphysical 
belief in reality inevitably compel the mind to represent the 
infinite as closed being by symbolical construction". The 
conception of the completed infinite, even if nonsensical, is 
inescapable.


### 3.1 *Das Kontinuum*




Another mathematical "possible" to which Weyl gave a 
great deal of thought is the *continuum*. During the period 
1918-1921 he wrestled with the problem of providing the 
mathematical continuum--the real number line--with a 
logically sound formulation. Weyl had become increasingly critical of
the principles underlying the set-theoretic construction of the 
mathematical continuum. He had come to believe that the whole 
set-theoretical approach involved vicious 
 circles[11]
 to such an extent that, as he says, "every cell (so to speak) 
of this mighty organism is permeated by contradiction." In 
*Das Kontinuum* he tries to overcome this by providing 
analysis with a *predicative* formulation--not, as 
Russell and Whitehead had attempted, by introducing a hierarchy of 
logically ramified types, which Weyl seems to have regarded as 
excessively complicated--but rather by confining the 
comprehension principle to formulas whose bound variables range over 
just the initial given entities (numbers). Accordingly he restricts 
analysis to what can be done in terms of natural numbers with the aid
of three basic logical operations, together with the operation of 
substitution and the process of "iteration", i.e., 
primitive recursion. Weyl recognized that the effect of this 
restriction would be to render unprovable many of the central results
of classical analysis--e.g., Dirichlet's principle that any 
bounded set of real numbers has a least upper 
 bound[12]--but
 he was prepared to accept this as part of the price that must be paid
for the security of mathematics.




As Weyl saw it, there is an unbridgeable gap between intuitively 
given continua (e.g. those of space, time and motion) on the one 
hand, and the "discrete" exact concepts of mathematics 
(e.g. that of natural 
 number[13])
 on the other. The presence of this chasm meant that the construction 
of the mathematical continuum could not simply be "read 
off" from intuition. It followed, in Weyl's view, that the 
mathematical continuum must be treated as if it were an element of 
the transcendent realm, and so, in the end, justified in the same way
as a physical theory. It was not enough that the mathematical theory 
be *consistent*; it must also be *reasonable*. 



*Das Kontinuum* embodies Weyl's attempt at formulating a 
theory of the continuum which satisfies the first, and, as far as 
possible, the second, of these requirements. In the following 
passages from this work he acknowledges the difficulty of the task:



> 
> 
> 
> ... the conceptual world of mathematics is so foreign to what 
> the intuitive continuum presents to us that the demand for 
> coincidence between the two must be dismissed as absurd. (Weyl 1987, 
> 108)
> 
> 
> 
> 
> ... the continuity given to us immediately by intuition (in the 
> flow of time and of motion) has yet to be grasped mathematically as a
> totality of discrete "stages" in accordance with that 
> part of its content which can be conceptualized in an exact way. 
> (*Ibid*., 
>  24)[14]
> 
> 
> 
> 
> 
> Exact time- or space-points are not the ultimate, underlying atomic 
> elements of the duration or extension given to us in experience. On 
> the contrary, only reason, which thoroughly penetrates what is 
> experientially given, is able to grasp these exact ideas. And only in
> the arithmetico- analytic concept of the real number belonging to the
> purely formal sphere do these ideas crystallize into full 
> definiteness. (*Ibid*., 94)
> 
> 
> 
> 
> When our experience has turned into a real process in a real world 
> and our phenomenal time has spread itself out over this world and 
> assumed a cosmic dimension, we are not satisfied with replacing the 
> continuum by the exact concept of the real number, in spite of the 
> essential and undeniable inexactness arising from what is given. 
> (*Ibid*., 93) 
> 
> 
> 




As these quotations show, Weyl had come to accept that it was in 
principle impossible to furnish the continuum as presented to 
intuition with an exact mathematical formulation : so, with 
reluctance, he lowered his sights. In *Das Kontinuum* his goal
was, first and foremost, to establish the *consistency* of the
mathematical theory of the continuum by putting the 
*arithmetical* notion of real number on a firm logical basis. 
Once this had been achieved, he would then proceed to show that this 
theory is *reasonable* by employing it as the foundation for a
plausible account of continuous process in the objective physical 
 world.[15]





In SS6 of *Das Kontinuum* Weyl presents his conclusions as
to the relationship between the intuitive and mathematical 
continua. He poses the question: Does the mathematical 
framework he has erected provide an adequate representation of 
physical or temporal continuity as it is *actually 
experienced*? In posing this question we can see the continuing 
influence of Husserl and phenomenological doctrine. Weyl begins his 
investigation by noting that, according to his theory, if one asks 
whether a given function is continuous, the answer is not fixed once 
and for all, but is, rather, dependent on the extent of the domain of
real numbers which have been defined up to the point at which the 
question is posed. Thus the continuity of a function must always 
remain *provisional*; the possibility always exists that a 
function deemed continuous *now* may, with the emergence of 
"new" real numbers, turn out to be discontinuous *in 
the future*. 
 [16]





To reveal the discrepancy between this formal account of continuity 
based on real numbers and the properties of an intuitively given 
continuum, Weyl next considers the experience of seeing a pencil 
lying on a table before him throughout a certain time interval. The 
position of the pencil during this interval may be taken as a 
function of the time, and Weyl takes it as a fact of observation that
during the time interval in question this function is continuous and 
that its values fall within a definite range. And so, he says,



> 
> This observation entitles me to assert that during a certain period
> this pencil was on the table; and even if my right to do so is not
> absolute, it is nevertheless reasonable and well-grounded. It is
> obviously absurd to suppose that this right can be undermined by
> "an expansion of our principles of definition"--as if
> new moments of time, overlooked by my intuition could be added to this
> interval, moments in which the pencil was, perhaps, in the vicinity of
> Sirius or who knows where. If the temporal continuum can be
> represented by a variable which "ranges over" the real
> numbers, then it appears to be determined thereby how narrowly or
> widely we must understand the concept "real number" and
> the decision about this must not be entrusted to logical deliberations
> over principles of definition and the like. (Weyl 1987, 88)




To drive the point home, Weyl focuses attention on the fundamental 
continuum of *immediately given phenomenal time*, that is, as 
he characterizes it,



> 
> ... to that constant form of my experiences of consciousness by
> virtue of which they appear to me to flow by successively. (By
> "experiences" I mean what I experience, exactly as I
> experience it. I do not mean real psychical or even physical processes
> which occur in a definite psychic-somatic individual, belong to a real
> world, and, perhaps, correspond to the direct experiences.)
> (*Ibid*., 88)




In order to correlate mathematical concepts with phenomenal time in 
this sense Weyl grants the possibility of introducing a rigidly 
punctate "now" and of identifying and exhibiting the 
resulting temporal points. On the collection of these temporal points
is defined the relation of *earlier than* as well as a 
congruence relation of *equality of temporal intervals*, the 
basic constituents of a simple mathematical theory of time. Now Weyl 
observes that the discrepancy between phenomenal time and the concept
of real number would vanish if the following pair of conditions could
be shown to be satisfied:


1. The immediate expression of the intuitive finding that during
a certain period I saw the pencil lying there were construed in such a
way that the phrase "during a certain period" was replaced
by "in every temporal point which falls within a certain time
span OE". [Weyl goes on to say parenthetically here that he admits
"that this no longer reproduces what is intuitively present, but
one will have to let it pass, *if it is really legitimate to
dissolve a period into temporal points*.")
2. If \(P\) is a temporal point, then the domain of rational
numbers to which \(l\) belongs if and only if there is a time
point \(L\) earlier than \(P\) such that

\[
OL = l{.}OE
\]



can be constructed arithmetically in pure number theory on the basis 
of our principles of definition, and is therefore a real number in 
our sense. (*Ibid*., 89)




Condition 2 means that, if we take the time span \(OE\) as a 
unit, then each temporal point \(P\) is correlated with a 
definite real number. In an addendum Weyl also stipulates the 
converse.




But can temporal intuition itself provide evidence for the truth or 
falsity of these two conditions? Weyl thinks not. In fact, he states 
quite categorically that



> 
> ... everything we are demanding here is obvious nonsense: to
> these questions, the intuition of time provides no answer--just
> as a man makes no reply to questions which clearly are addressed to
> him by mistake and, therefore, are unintelligible when addressed to
> him. (*Ibid*., 90)




The grounds for this assertion are by no means immediately evident, 
but one gathers from the passages following it that Weyl regards the 
experienced *continuous flow* of phenomenal time as 
constituting an insuperable barrier to the whole enterprise of 
representing the continuum as experienced in terms of individual 
points, and even to the characterization of "individual 
temporal point" itself. As he says,



> The view of a flow consisting of points and, therefore,
> also dissolving into points turns out to be mistaken: precisely what
> eludes us is the nature of the continuity, the flowing from point to
> point; in other words, the secret of how the continually enduring
> present can continually slip away into the receding past. Each one of
> us, at every moment, directly experiences the true character of this
> temporal continuity. But, because of the genuine primitiveness of
> phenomenal time, we cannot put our experiences into words. So we shall
> content ourselves with the following description. What I am conscious
> of is for me both a being-now and, in its essence, something which,
> with its temporal position, slips away. In this way there arises the
> persisting factual extent, something ever new which endures and changes
> in consciousness. (*Ibid*., 91-92)




Weyl sums up what he thinks can be affirmed about "objectively 
presented time"--by which he presumably means 
"phenomenal time described in an objective 
manner"--in the following two assertions, which he claims 
apply equally, mutatis mutandis, to every intuitively given 
continuum, in particular, to the continuum of spatial extension. 
(*Ibid*., 92):


1. An individual point in it is non-independent, i.e., is pure
nothingness when taken by itself, and exists only as a "point of
transition" (which, of course, can in no way be understood
mathematically);
2. It is due to the essence of time (and not to contingent
imperfections in our medium) that a fixed temporal point cannot be
exhibited in any way, that always only an approximate, never an exact
determination is possible.




The fact that single points in a true continuum "cannot be 
exhibited" arises, Weyl asserts, from the fact that they are 
not genuine individuals and so cannot be characterized by their 
properties. In the physical world they are never defined absolutely, 
but only in terms of a coordinate system, which, in an arresting 
metaphor, Weyl describes as "the unavoidable residue of the 
eradication of the ego." This metaphor, which Weyl was to 
employ more than 
 once[17],
 again reflects the continuing influence of phenomenological doctrine 
in his thinking : here, the thesis that the existent is given in the 
first instance as the contents of a consciousness.


### 3.2 Weyl and Brouwerian Intuitionism




By 1919 Weyl had come to embrace Brouwer's views on the intuitive 
continuum. Given the idealism that always animated Weyl's thought, 
this is not surprising, since Brouwer assigned the thinking subject a
central position in the creation of the mathematical 
 world[18].
 




In his early thinking Brouwer had held that that the continuum is 
presented to intuition as a whole, and that it is impossible to 
construct all its points as individuals. But later he radically 
transformed the concept of "point", endowing points with 
sufficient fluidity to enable them to serve as generators of a 
"true" continuum. This fluidity was achieved by admitting
as "points", not only fully defined discrete numbers such
as 1/9, \(e\), and the like--which have, so to speak, 
already achieved "being"--but also 
"numbers" which are in a perpetual state of 
"becoming" in that the entries in their decimal (or 
dyadic) expansions are the result of free acts of choice by a subject
operating throughout an indefinitely extended time. The resulting 
choice sequences cannot be conceived as finished, completed objects: 
at any moment only an initial segment is known. Thus Brouwer obtained
the mathematical continuum in a manner compatible with his belief in 
the primordial intuition of time--that is, as an unfinished, in 
fact unfinishable entity in a perpetual state of growth, a 
"medium of free development". In Brouwer's vision, the 
mathematical continuum is indeed "constructed", not, 
however, by initially shattering, as did Cantor and Dedekind, an 
intuitive continuum into isolated points, but rather by assembling it
from a complex of continually changing overlapping parts.




Brouwer's impact looms large in Weyl's 1921 paper, On the New 
Foundational Crisis of Mathematics. Here Weyl identifies two distinct
views of the continuum: "atomistic" or 
"discrete"; and "continuous". In the first of
these the continuum is composed of individual real numbers which are 
well-defined and can be sharply distinguished. Weyl describes his 
earlier attempt at reconstructing analysis in Das Kontinuum as 
atomistic in this sense:



> 
> Existential questions concerning real numbers only become
> meaningful if we analyze the concept of real number in this
> extensionally determining and delimiting manner. Through this
> conceptual restriction, an ensemble of individual points is, so to
> speak, picked out from the fluid paste of the continuum. The continuum
> is broken up into isolated elements, and the flowing-into-each other of
> its parts is replaced by certain conceptual relations between these
> elements, based on the "larger-smaller" relationship. This
> is why I speak of the *atomistic* conception of the
> continuum. (Weyl 1921, 91)




By this time Weyl had repudiated atomistic theories of the continuum,
including that of Das 
 Kontinuum.[19]
 While intuitive considerations, together with Brouwer's influence, 
must certainly have fuelled Weyl's rejection of such theories, it 
also had a logical basis. For Weyl had come to regard as meaningless 
the formal procedure--employed in Das Kontinuum--of 
negating universal and existential statements concerning real numbers
conceived as developing sequences or as sets of rationals. This had 
the effect of undermining the whole basis on which his theory had 
been erected, and at the same time rendered impossible the very 
formulation of a "law of excluded middle" for such 
statements. Thus Weyl found himself espousing a 
 position[20]
 considerably more radical than that of Brouwer, for whom negations of
quantified statements had a perfectly clear constructive meaning, 
under which the law of excluded middle is simply not generally 
affirmable.




Of existential statements Weyl says:



> An existential statement--e.g., "there is an
> even number"--is not a judgement in the proper sense at
> all, which asserts a state of affairs; existential states of affairs
> are the empty invention of logicians. (Weyl [1921], 97)




Weyl termed such pseudostatements "judgment abstracts", 
likening them, with typical literary flair, to "a piece of 
paper which announces the presence of a treasure, without divulging 
its location." Universal statements, although possessing 
greater substance than existential ones, are still mere intimations 
of judgments, "judgment instructions", for which Weyl 
provides the following metaphorical description:



> If knowledge be compared to a fruit and the realization of
> that knowledge to the consumption of the fruit, then a universal
> statement is to be compared to a hard shell filled with fruit. It is,
> obviously, of some value, however, not as a shell by itself, but only
> for its content of fruit. It is of no use to me as long as I do not
> open it and actually take out a fruit and eat it. (*Ibid*.,
> 98)




Above and beyond the claims of logic, Weyl welcomed Brouwer's 
construction of the continuum by means of sequences generated by free
acts of choice, thus identifying it as a "medium of free 
Becoming" which "does not dissolve into a set of real 
numbers as finished entities". Weyl felt that Brouwer, through 
his doctrine of 
 Intuitionism[21],
 had come closer than anyone else to bridging that "unbridgeable
chasm" between the intuitive and mathematical continua. In 
particular, he found compelling the fact that the Brouwerian 
continuum is not the union of two disjoint nonempty parts--that 
it is, in a word, indecomposable. "A genuine continuum," 
Weyl says, "cannot be divided into separate 
 fragments."[22]
 In later publications he expresses this more colourfully by quoting 
Anaxagoras to the effect that a continuum "defies the chopping 
off of its parts with a hatchet."




Weyl also agreed with Brouwer that all functions everywhere defined 
on a continuum are continuous, but here certain subtle differences of
viewpoint emerge. Weyl contends that what mathematicians had taken to
be discontinuous functions actually consist of several continuous 
functions defined on separated continua. In Weyl's view, for example, 
the "discontinuous" function defined by 
\(f(x) = 0\) for \(x \lt 0\) and 
\(f(x) = 1\) for \(x \ge 0\) in fact consists of 
the *two* functions with constant values and 1 respectively 
defined on the separated continua \(\{x: x \lt 0\}\) and
\(\{x: x \ge 0\}\). (The union of these two continua 
fails to be the whole of the real continuum because of the failure of
the law of excluded middle: it is not the case that, for be any real 
number \(x\), either \(x \lt 0\) or \(x \ge 0\).) 
Brouwer, on the other hand, had not dismissed the possibility that 
discontinuous functions could be defined on proper parts of a 
continuum, and still seems to have been searching for an appropriate 
way of formulating this 
 idea.[23]
 In particular, at that time Brouwer would probably have been inclined
to regard the above function \(f\) as a genuinely discontinuous 
function defined on a proper part of the real continuum. For Weyl, it
seems to have been a self-evident fact that all functions defined on 
a continuum are continuous, but this is because Weyl confines 
attention to functions which turn out to be continuous by definition.
Brouwer's concept of function is less restrictive than Weyl's and it 
is by no means immediately evident that such functions must always be
continuous. 




Weyl defined real functions as mappings correlating each interval in 
the choice sequence determining the argument with an interval in the 
choice sequence determining the value "interval by 
interval" as it were, the idea being that approximations to the
input of the function should lead effectively to corresponding 
approximations to the input. Such functions are continuous by 
definition. Brouwer, in contrast, considers real functions as 
correlating choice sequences with choice sequences, and the 
continuity of these is by no means obvious. The fact that Weyl 
refused to grant (free) choice sequences--whose identity is in 
no way predetermined--sufficient individuality to admit them as 
arguments of functions betokens a commitment to the conception of the
continuum as a "medium of free Becoming" even deeper, 
perhaps, than that of Brouwer.




There thus being only minor differences between Weyl's and Brouwer's 
accounts of the continuum, Weyl accordingly abandoned his earlier 
attempt at the reconstruction of analysis, and joined Brouwer. He 
explains:



> 
> I tried to find solid ground in the impending state of dissolution of
> the State of analysis (which is in preparation, although still only
> recognized by few)without forsaking the order on which it is founded,
> by carrying out its fundamental principle purely and honestly. And I
> believe I was successful--as far as this is possible. For
> *this order is itself untenable*, as I have now convinced
> myself, and Brouwer--that is the revolution!... It would
> have been wonderful had the old dispute led to the conclusion that the
> atomistic conception as well as the continuous one can be carried
> through. Instead the latter triumphs for good over the former. It is
> Brouwer to whom we owe the new solution of the continuum
> problem. History has destroyed again from within the provisional
> solution of Galilei and the founders of the differential and the
> integral calculus. (Weyl 1921, 98-99)




Weyl's initial enthusiasm for intuitionism seems later to have waned.
This may have been due to a growing belief on his part that the 
mathematical sacrifices demanded by adherence to intuitionistic 
doctrine (e.g., the abandonment of the least upper bound principle, 
and other important results of classical analysis) would prove to be 
intolerable to practicing mathematicians. Witness this passage from 
*Philosophy of Mathematics and Natural Science*:



> 
> 
> Mathematics with Brouwer gains its highest intuitive clarity. He
> succeeds in developing the beginnings of analysis in a natural manner,
> all the time preserving the contact with intuition much more closely
> than had been done before. It cannot be denied, however, that in
> advancing to higher and more general theories the inapplicability of
> the simple laws of classical logic eventually results in an almost
> unbearable awkwardness. And the mathematician watches with pain the
> greater part of his towering edifice which he believed to be built of
> concrete blocks dissolve into mist before his eyes. (Weyl [1949],
> 54)




Nevertheless, it is likely that Weyl remained convinced to the end of
his days that intuitionism, despite its technical 
"awkwardness", came closest, of all mathematical 
approaches, to capturing the essence of the continuum.


### 3.3 Weyl and Hilbert




Weyl's espousal of the intuitionistic standpoint in the foundations 
of mathematics in 1920-21 inevitably led to friction with his 
old mentor Hilbert. Hilbert's conviction had long been that there 
were in principle no limitations on the possibility of a full 
scientific understanding of the natural world, and, analogously, in 
the case of mathematics, that once a problem was posed with the 
required precision, it was, at least in principle, soluble. In 1904 
he was moved to respond to Emil du Bois-Reymond's famous declaration 
concerning the limits of science, ignoramus et ignorabimus ("we
are ignorant and we shall remain ignorant"):



> 
> We hear within us the perpetual call. There is the problem. Seek the
> solution. You can find it by pure reason, for in mathematics there is
> no
>  ignorabimus.[24]
> 




Hilbert was unalterably opposed to any restriction of mathematics 
"by decree", an obstacle he had come up against in the 
early stages of his career in the form of Leopold Kronecker's (the 
influential 19th century German mathematician) 
anathematization of all mathematics venturing beyond the finite. In 
Brouwer's intuitionistic program--with its draconian 
restrictions on what was admissible in mathematical argument, in 
particular, its rejection of the law of excluded middle, 
"pure" existence proofs, and virtually the whole of 
Cantorian set theory--Hilbert saw the return of Kroneckerian 
constaints on mathematics (and also, perhaps, a trace of du 
Bois-Reymond's "ignorabimus") against which he had 
struggled for so long. Small wonder, then, that Hilbert was upset 
when Weyl joined the Brouwerian 
 camp.[25]





Hilbert's response was to develop an entirely new approach to the 
foundations of mathematics with the ultimate goal of establishing 
beyond doubt the consistency of the whole of classical mathematics, 
including arithmetic, analysis, and Cantorian set theory. With the 
attainment of that goal, classical mathematics would be placed 
securely beyond the destructive reach of the intuitionists. The core 
of Hilbert's program was the translation of the whole apparatus of 
classical mathematical demonstration into a simple, finitistic 
framework (which he called "metamathematics") involving 
nothing more, in principle, than the straightforward manipulation of 
symbols, taken in a purely formal sense, and devoid of further 
 meaning.[26]
 Within metamathematics itself, Hilbert imposed a standard of 
demonstrative evidence stricter even than that demanded by the 
intuitionists, a form of finitism rivalling (ironically) that of 
Kronecker. The demonstration of the consistency of classical 
mathematics was then to be achieved by showing, within the 
constraints of strict finitistic evidence insisted on by Hilbert, 
that the formal metamathematical counterpart of a classical proof in 
that system can never lead to an assertion evidently false, such as 
\(0 = 1\).




Hilbert's program rested on the insight that, *au fond*, the 
only part of mathematics whose reliability is entirely beyond 
question is the *finitistic*, or *concrete* part: in 
particular, finite manipulation of surveyable domains of distinct 
objects including mathematical symbols presented as marks on paper. 
Mathematical propositions referring only to concrete objects in this 
sense Hilbert called *real*, *concrete*, or 
*contentual* propositions, and all other mathematical 
propositions he distinguished as possessing an *ideal*, or 
*abstract* character. (Thus, for example, \(2 + 2 = 4\) would 
count as a real proposition, while *there exists an odd perfect 
number* would count as an ideal one.) Hilbert viewed ideal 
propositions as akin to the ideal lines and points "at 
infinity" of projective geometry. Just as the use of these does
not violate any truths of the "concrete" geometry of the 
usual Cartesian plane, so he hoped to show that the use of ideal 
propositions--even those of Cantorian set theory--would 
never lead to falsehoods among the real propositions, that, in other 
words, such use *would never contradict any self-evident fact 
about concrete objects*. Establishing this by strictly concrete, 
and so unimpeachable means was thus the central aim of Hilbert's 
program. Hilbert may be seen to have followed Kant in attempting to 
ground mathematics on the apprehension of spatiotemporal 
configurations; but Hilbert restricted these configurations to 
concrete signs (such as inscriptions on paper). Hilbert regarded 
consistency as the touchstone of existence, and so for him the 
important thing was the fact that no inconsistencies can arise within
the realm of concrete signs, since correct descriptions of concrete 
objects are always mutually compatible. In particular, within the 
realm of concrete signs, actual infinity cannot generate 
inconsistencies since, again along with Kant, he held that this 
concept cannot correspond to any concrete object. Hilbert's view 
seems accordingly to have been that the formal soundness of 
mathematics issues ultimately, not from a *logical* source, 
but from a *concrete* 
 one[27],
 in much the same way as the consistency of truly reported empirical 
statements is guaranteed by the concreteness of the external 
 world[28].
 




Weyl soon grasped the significance of Hilbert's program, and came to 
acknowledge its "immense significance and 
 scope"[29].
 Whether that program could be successfully carried out was, of 
course, still an open question. But independently of this issue Weyl 
was concerned about what he saw as the loss of content resulting from
Hilbert's thoroughgoing formalization of mathematics. "Without 
doubt," Weyl warns, "if mathematics is to remain a 
serious cultural concern, then some *sense* must be attached 
to Hilbert's game of formulae." Weyl thought that this sense 
could only be supplied by "fusing" mathematics and 
physics so that "the mathematical concepts of number, function,
etc. (or Hilbert's symbols) generally partake in the theoretical 
construction of reality in the same way as the concepts of energy, 
gravitation, electron, 
 etc."[30]
 Indeed, in Weyl's view, "it is the function of mathematics to 
be at the service of the natural sciences". But still:



> 
> The propositions of theoretical physics... lack that feature
> which Brouwer demands of the propositions of mathematics, namely, that
> each should carry within itself its own intuitively comprehensible
> meaning.... Rather, what is tested by confronting theoretical
> physics with experience is the system as a whole. It seems that we
> have to differentiate between phenomenal knowledge or
> insight--such as is expressed in the statement: "This leaf
> (given to me in a present act of perception) has this green color
> (given to me in this same perception)"--and theoretical
> construction. Knowledge furnishes truth, its organ is
> "seeing" in the widest sense. Though subject to error, it
> is essentially definitive and unalterable. Theoretical construction
> seems to be bound only to one strictly formulable rational principle,
> concordance, which in mathematics, where the domain of sense data
> remains untouched, reduces to consistency; its organ is creative
> imagination. (Weyl 1949, 61-62)




Weyl points out that, just as in theoretical physics, Hilbert's 
account of mathematics "already... goes beyond the bounds 
of intuitively ascertainable states of affairs through... ideal 
assumptions." (Weyl 1927, 484.) If Hilbert's realm of 
contentual or "real" propositions--the domain of 
metamathematics--corresponds to that part of the world directly 
accessible to what Weyl terms "insight" or 
"phenomenal knowledge", then "serious" 
mathematics--the mathematics that practicing mathematicians are 
actually engaged in doing--corresponds to Hilbert's realm of 
"ideal" propositions. Weyl regarded this realm as the 
counterpart of the domain generated by "symbolic 
construction", the transcendent world focussed on by 
theoretical physics. Hence his memorable characterization:



> 
> The set-theoretical approach is the stage of naive realism which is
> unaware of the transition from the given to the transcendent. Brouwer
> represents idealism, by demanding the reduction of all truth to the
> intuitively given. In [Hilbert's] formalism, finally, consciousness
> makes the attempt to "jump over its own shadow", to leave
> behind the stuff of the given, to represent the
> transcendent--but, how could it be otherwise?, only through the
> symbol. (Weyl 1949, 65-66)




In Weyl's eyes, Hilbert's approach embodied the "symbolic 
representation of the transcendent, which demands to be 
satisfied", and so he regarded its emergence as a natural 
development. But by 1927 Weyl saw Hilbert's doctrine as beginning to 
prevail over intuitionism, and in this an adumbration of *"a
decisive defeat of the philosophical attitude of pure 
phenomenology*, which thus proves to be insufficient for the 
understanding of creative science even in the area of cognition that 
is most primal and most readily open to 
 evidence--mathematics."[31]
 Since by this time Weyl had become convinced that "creative 
science" must *necessarily* transcend what is 
phenomenologically given, he had presumably already accepted that 
pure phenomenology is incapable of accounting for theoretical 
physics, let alone the whole of existence. But it must have been 
painful for him to concede the analogous claim in the case of 
*mathematics*. In 1932, he asserts: "If mathematics is 
taken by itself, one should restrict oneself with Brouwer to the 
intuitively cognizable truths ... nothing compels us to go 
farther." If mathematics could be "taken by 
itself", then there would be no need for it to justify its 
practices by resorting to "symbolic construction", to 
employ symbols which in themselves "signify 
nothing"--nothing, at least, accessible to intuition. But,
unlike Brouwer, Weyl seems finally to have come to terms with the 
idea that mathematics could not simply be "taken by 
itself", that it has a larger role to play in the world beyond 
its service as a paradigm, however pure, of subjective certainty.




The later impact of Godel's incompleteness theorems on Hilbert's
program led Weyl to remark in 
 1949:[32]




> 
> The ultimate foundations and the ultimate meaning of mathematics
> remain an open problem; we do not know in what direction it will find
> its solution, nor even whether a final objective answer can be
> expected at all. "Mathematizing" may well be a creative
> activity of man, like music, the products of which not only in form
> but also in substance defy complete objective rationalization. The
> undecisive outcome of Hilbert's bold enterprise cannot fail to affect
> the philosophical interpretation. (Weyl 1949, 219)




The fact that "Godel has left us little hope that a 
formalism wide enough to encompass classical mathematics will be 
supported by a proof of consistency" seems to have led Weyl to 
take a renewed interest in "axiomatic systems developed before 
Hilbert without such ambitious dreams", for example Zermelo's 
set theory, Russell's and Whitehead's ramified type theory and 
Hilbert's own axiom systems for geometry (as well, possibly, as 
Weyl's own system in *Das Kontinuum*, which he modestly fails 
to mention). In one of his last papers, *Axiomatic Versus 
Constructive Procedures in Mathematics*, written sometime after 
1953, he saw the battle between Hilbertian formalism and Brouwerian 
intuitionism in which he had participated in the 1920s as having 
given way to a "dextrous blending" of the axiomatic 
approach to mathematics championed by Bourbaki and the algebraists 
(themselves mathematical descendants of Hilbert) with constructive 
procedures associated with geometry and topology.




It seems appropriate to conclude this account of Weyl's work in the 
foundations and philosophy of mathematics by allowing the man himself
to have the last word:



> 
> This history should make one thing clear: we are less certain than
> ever about the ultimate foundations of (logic and) mathematics; like
> everybody and everything in the world today, we have our
> "crisis". We have had it for nearly fifty years. Outwardly
> it does not seem to hamper our daily work, and yet I for one confess
> that it has had a considerable practical influence on my mathematical
> life: it directed my interests to fields I considered relatively
> "safe", and it has been a constant drain on my enthusiasm
> and determination with which I pursued my research work. The
> experience is probably shared by other mathematicians who are not
> indifferent to what their scientific endeavours mean in the contexts
> of man's whole caring and knowing, suffering and creative existence in
> the world. (Weyl 1946, 13)


## 4. Contributions to the Foundations of Physics


### 4.1 Spacetime Geometries and Weyl's Unified Field Theory




Weyl's clarification of the role of coordinates, invariance or 
symmetry principles, his important concept of gauge invariance, his 
group-theoretic results concerning the uniqueness of the Pythagorean 
form of the metric, his generalization of Levi-Civita's concept of 
parallelism, his development of the geometry of paths, his discovery 
of the causal-inertial method which prepared the way to empirically 
determine the spacetime metric in a non-circular, non-conventional 
manner, his deep analysis of the concept of motion and the role of 
Mach's Principle, are but a few examples of his important 
contributions to the philosophical and mathematical foundations of 
modern spacetime theory. 




Weyl's book, *Raum-Zeit-Materie*, beautifully exemplifies the 
fruitful and harmonious interplay of mathematics, physics and 
philosophy. Here Weyl aims at a mathematical and philosophical 
elucidation of the problem of space and time in general. In the 
preface to the great classical work of 1923, the fifth German 
edition, after mentioning the importance of mathematics to his work, 
Weyl says: 



>  
> 
> Despite this, the book does not disavow its basic, philosophical
> orientation: its central focus is *conceptual analysis*;
> physics provides the experiential basis, mathematics the sharp
> tools. In this new edition, this tendency has been further
> strengthened; although the growth of speculation was trimmed, the
> supporting foundational ideas were more intuitively, more carefully
> and more completely developed and analyzed.
> 
> 


#### 4.1.1 Weyl's metric-independent construction of the symmetric linear connection




Extending and abstracting from Gauss's treatment of curved surfaces 
in Euclidian space, Riemann constructed an infinitesimal geometry of 
\(n\)-dimensional manifolds. The coordinate assignments 
\(x^{k}(p)\ [k \in \{1, \ldots ,n\}\)] of the points \(p\) in such an \(n\)-dimensional 
Riemannian manifold are quite arbitrary, subject only to the 
requirement of arbitrary differential coordinate 
 transformations.[33]
 Riemann's assumption that in an infinitesimal neighbourhood of a 
point, Euclidean geometry and hence Pythagoras's theorem holds, finds
its formal expression in Riemann's equation 



\[\tag{1}
ds^2 = \sum\_{i,j} g\_{ij}(x^{k}(p))dx^{i}dx^{j}\ [\text{where } g\_{ij}(x^{k}(p)) = g\_{ji}(x^{k}(p))]
\]



for the square of the length \(ds\) of an infinitesimal line 
element that leads from the point \(p = x(p) = 
(x^{1}(p), \ldots ,x^{n}(p))\) to an arbitrary infinitely 
near point \(p' = x(p') = (x^{1}(p) + dx^{1}(p), \ldots ,x^{n}(p) + dx^{n}(p))\). 




The assumption that Euclidean geometry holds in the infinitesimally 
small means that the \(dx^{i}(p)\) 
transform *linearly* under arbitrary coordinate 
transformations. Using the *Einstein summation*
*convention*[34],
 equation (1) can simply be written as 



\[\tag{2}
ds^{2} = g\_{ij}(x^{k}(p))dx^{i}dx^{j}.
\]



Riemann assumed the validity of the Pythagorean metric only in the 
infinitely small. Riemannian geometry is essentially a geometry of 
infinitely near points and conforms to the requirement that all laws 
are to be formulated as *field* laws. Field laws are 
*close-action-laws* which relate the field magnitudes only to 
infinitesimally neighbouring points in 
 space.[35]
 The value of some field magnitude at each point depends only on the 
values of other field magnitudes in the infinitesimal neighbourhoods 
of the corresponding points. The field magnitudes consist of partial 
derivatives of position functions at some point, and this requires 
the knowledge of the behavior of the position functions only with 
respect to the neighbourhood of that point. To construct a field law,
only the behavior of the world in the infinitesimally small is 
 required.[36]





Riemann's ideas were brought to a concrete realization fifty years 
later in Einstein's general theory of relativity. The basic idea 
underlying the general theory of relativity was Einstein's 
recognition that the metric field, which has such powerful real 
effects on matter, cannot be a rigid once and for all given geometric
structure of the spacetime, but must itself be something real, that 
not only has effects on matter, but is in turn also affected by 
matter. Riemann had already suggested that analogous to the 
electromagnetic field, the metric field reciprocally interacts with 
matter. Einstein came to this idea of reciprocity between matter and 
field independently of Riemann, and in the context of his theory of 
general relativity, applied this principle of reciprocity to four 
dimensional spacetime. Thus Einstein could adopt Riemann's 
infinitesimal geometry with the important difference: given the 
causal requirements of Einstein's theory of special relativity, 
Riemann's quadratic form is not positive definite but indefinite; it 
has signature 
 1.[37]
 Weyl (1922a) says: 



>  
> 
> All our considerations until now were based on the assumption, that
> the metric structure of space is something that is fixed and
> given. Riemann already pointed to another possibility which was
> realized through General Relativity. The metrical structure of the
> extensive medium of the external world is a field of physical reality,
> which is causally dependent on the state of matter.
> 
> 




And in another place Weyl (1918b) remarks: 



>  
> 
> The metric is not a property of the world [spacetime] in itself,
> rather spacetime as a form of appearance is a completely formless
> four-dimensional continuum in the sense of analysis situs, but the
> metric expresses something real, something which exists in the world,
> which exerts through centrifugal and gravitational forces physical
> effects on matter, and whose state is conversely conditioned through
> the distribution and nature of matter.
> 
> 




After Einstein applied Riemannian geometry to his theory of general 
relativity, Riemannian geometry became the focus of intense research.
In particular, G. Ricci and T. Levi-Civita's so-called *Absolute 
Differential Calculus* developed and clarified the Riemannian 
notions of an *affine connection* and *covariant 
differentiation*. The decisive step in this development, however,
was T. Levi-Civita's discovery in 1917 of the concept of 
*infinitesimal parallel vector displacement*, and the
fact that such parallel vector displacement is uniquely determined by
the metric field of Riemannian geometry. Levi-Civita's construction 
of infinitesimal parallel transport on a manifold required the 
process of embedding the manifold into a flat higher-dimensional 
metric space. In 1918, Weyl generalized Levi-Civita's concept of 
parallel transport by means of an *intrinsic* construction 
that does not require the process of such an embedding, and is 
therefore independent of a metric. Weyl's intrinsic construction 
results in a *metric-independent, symmetric linear 
connection*. Weyl simply referred to the latter as 
*affine*
*connection*.[38]





Weyl defines what he means by an *affine connection* in the 
following way: A point \(p\) on the manifold \(M\) is affinely 
connected with its immediate neighborhood, if and only of for every 
tangent vector \(v\_{p}\) at \(p\), a tangent 
vector \(v\_{q}\) at \(q\) is determined to which
the tangent vector \(v\_{p}\) gives rise under 
parallel displacement from \(p\) to the infinitesimally 
neighboring point \(q\). This definition merely says that a 
manifold is affinely connected if it admits the process of 
infinitesimal parallel displacement of a vector. 




Weyl's next definition characterizes the *essential nature* of
infinitesimal parallel displacement. The definition says that at any 
arbitrary point of the manifold there exists a *geodesic 
coordinate system* such that the components of any vector at that
point are not altered by an infinitesimal parallel displacement with 
respect to it. This is a geometrical way of expressing Einstein's 
requirement that the gravitational field can always be made to vanish
locally. According to Weyl (1923b, 115), it characterizes the 
*nature* of an affine connection on the manifold. A manifold 
which is an affine manifold is homogeneous in this sense. Moreover, 
manifolds do not exist whose affine structure is of a different 
*nature*. 




The transport of a tangent vector \(v\_{p}\) at 
\(p\) to an infinitesimally nearby point \(q\) results in the 
tangent vector \(v\_{q}\) at \(q\), namely, 



\[\tag{3}
v\_{q} = v\_{p} + dv\_{p}.
\]



This infinitesimal tangent vector transport Weyl defines as 
*infinitesimal parallel displacement* if and only if there 
exists a coordinate system \(\overline{x}\), called a *geodesic coordinate system*
for the neighborhood of \(p\), relative to which the transported 
tangent vector \(\overline{v}\_{q}\) at \(q\) 
possesses the same components as the original tangent vector \(\overline{v}\_{p}\) at \(p\); that is, 



\[\tag{4}
\overline{v}^{\,i}\_q - \overline{v}^{\,i}\_p = d\overline{v}^{\,i}\_p = 0.
\]


![figure](fig1.png)
Figure 1: Parallel transport in a geodesic coordinate system \(\overline{x}\)





For an arbitrary coordinate system \(x\) the components \(dv^{\,i}\_p\)
vanish whenever \(v^{\,i}\_p\) or \(dx^{\,i}\_p\) vanish. Consequently,
\(dv^{\,i}\_p\) is *bi-linear* in \(v^{\,i}\_p\) and
\(dx^{\,i}\_p\); that is, 



\[\tag{5}
dv^{\,i}\_p = -\Gamma^{\,i}\_{jk}(x^i(p))v^{\,i}\_p dx^{\,k}\_p,
\]



where, in the case of four dimensions, the \(4^{3} = 64\)
coefficients \(\Gamma^{\,i}\_{jk} (x^{i}(p))\)
are coordinate functions, that is, functions of 
\(x^{i}(p)\) \((i = 1, \ldots, 4)\), 
and the minus sign is introduced to agree with convention. 



![figure](fig2.png)
Figure 2: Parallel transport in an arbitrary coordinate system \(x\)





It is important to understand that there is no intrinsic notion of 
infinitesimal parallel displacement on a differentiable manifold. A 
notion of "parallelism" is not something that a manifold 
would possess merely by virtue of being a smooth manifold; additional
structure has to be introduced which resides on the manifold and 
which permits the notion of infinitesimal parallelism. A manifold is 
an "affine manifold" \((M, \Gamma)\) if in addition to its 
manifold structure (differential topological structure) it is also 
endowed with an affine structure \(\Gamma\) that assigns to each of its 
points 64 coefficients \(\Gamma^{i}\_{jk} (x^{i}(p))\)
satisfying the symmetry condition \(\Gamma^{i}\_{jk} (x^{i}(p))
= \Gamma^{i}\_{kj} (x^{i}(p))\).





An \(n\)-dimensional manifold \(M\), which is an affinely 
connected manifold, Weyl (1918b) interprets physically as an 
\(n\)-dimensional world (spacetime) filled with a gravitational 
field. Weyl says, "...the affine connection appears in 
physics as the *gravitational field*..." Since 
there exists at each spacetime point a geodesic coordinate system in 
which the components \(\Gamma^{i}\_{jk}\) of the symmetric linear 
connection vanish, the gravitational field can be made to vanish at 
each point of the manifold. 




The *classical theory of physical geometry*, developed by 
Helmholtz, Poincare and Hilbert, regarded the concept of 
"metric congruence" as the only basic relation of 
geometry, and constructed physical geometry from this one notion 
alone in terms of the relative positions and displacements of 
physical congruence standards. Although Einstein's general theory of 
relativity championed a *dynamical view of spacetime 
geometry* that is very different from the *classical theory of
physical geometry*, Einstein initially approached the problem of 
the structure of spacetime from the metrical point of view. It was 
Weyl (1923b) who emphasized and developed the metric-independent 
construction of the symmetric linear connection and who pointed out 
the rationale for doing so. In both the non-relativistic and 
relativistic contexts, it is the symmetric linear connection, and not
the metric, which plays the essential role in the formulation of all 
physical laws that are expressed in terms of differential equations. 
It is the symmetric linear connection that relates the state of a 
system at a spacetime point to the states at neighboring spacetime 
events and enters into the differentials of the corresponding 
magnitudes. In both Newtonian physics and the theory of general 
relativity, all dynamical laws presuppose the projective and affine 
structure and hence the Law of Inertia. In fact, the whole of tensor 
analysis with its covariant derivatives is based on the affine 
concept of infinitesimal *parallel displacement* and *not*on the metric. 




Weyl's metric independent construction not only led to a deeper 
understanding of the mathematical characterization of gravity, it 
also prepared the way for new constructions and generalizations in 
differential geometry and the general theory of relativity. In 
particular, it led to 


1. The development of the *geometry of paths*, first introduced by Weyl in 1918.
2. Weyl's discovery of the *causal-inertial method* which prepared the way to empirically determine the spacetime metric in a non-circular, non-conventional manner.
3. Weyl's generalization of Riemannian geometry in his attempt to unify gravity and electromagnetism.
4. Weyl's introduction of the concept of gauge in the context of his attempt to unify gravity and electromagnetism.




For more detail on Weyl's metric independent construction of the 
affine connection (linear symmetric connection), see 
 the supplement.
 


#### 4.1.2 Projective Geometry or the Geometry of Paths




Weyl's metric-independent construction of the affine structure led to
the development of differential *projective* geometries or the
*geometries of paths.* The interest in projective geometry is 
in the *paths*, that is, in the continuous set of points of 
the *image set of curves*, rather than in the possible 
parameter descriptions of curves. A curve has one degree of freedom; 
it depends on one parameter, and its image set or path is a 
one-dimensional continuous set of points of the manifold. One 
represents a *curve* on a manifold \(M\) as a smooth map 
(i.e., \(C^{\infty})\) \(\gamma\) from some open interval 
\(I = (-\varepsilon, \varepsilon)\) of the real line \(\mathbb{R}\) into
\(M\). 



![figure](fig3.png)
Figure 3:
A curve on the manifold \(M\) is the smooth map 
\(\gamma : I \subset \mathbb{R} \rightarrow M\)





It is important to understand that what one means by a 
"curve" is the map (the parametric description) itself, 
and *not* the set of its image points, the path. Consequently,
two curves are mathematically considered to be different curves if 
they are given by different maps (different parameter descriptions), 
*even if their image set, that is, their path, is the 
same*. If we change a curve's parameter description we change the
curve but not its image set (its path), the points it passes through.
A path is therefore sometimes defined as an equivalence class of 
curves under arbitrary parameter transformations. Hence, projective 
geometry may be defined as an equivalence class of affine geometries.





A geodesic curve in *flat space* is a straight line. Its 
tangent at one point is parallel to the tangent at previous or 
subsequent points. A straight line in Euclidean space is the only 
curve that parallel-transports its own tangent vector. This notion of
parallel transport of the tangent vector also characterizes geodesic 
curves in *curved space*. That is, a curve \(\gamma\) in curved 
space, which parallel-transports its own tangent vector along all of 
its points, is called a *geodesic* curve. Given a manifold 
with an affine structure and some arbitrary local coordinate system, 
the coordinate functions (components) \(\gamma^{i}\) of a
geodesic curve \(\gamma\) satisfy the second-order non-linear 
differential equations 



\[\tag{6}
\frac{d^2\gamma^i}{ds^2} + \Gamma^{i}\_{jk} \frac{d\gamma^j}{ds} \frac{d\gamma^k}{ds} = 0.
\]



One may characterize the *projective geometry* \(\Pi\) on an 
affine manifold either in terms of an equivalence class of geodesic 
curves under arbitrary parameter 
 diffeomorphisms[39],
 thereby eliminating all the parameter descriptions and hence all 
possible notions of distance along the curves satisfying 
 (6),[40]
 or one may take the process of *autoparallelism of directions*as fundamental in defining a projective structure. 



![figure](fig4.png)
Figure 4: 
A path \(\xi\) is an equivalence class [\(\gamma\)] of curves 
under all parameter diffeomorphisms \(\mu: \mathbb{R} \rightarrow \mathbb{R};
\lambda \mapsto \mu(\lambda)\)





Weyl took the latter approach. According to Weyl, the infinitesimal 
process of parallel displacements of vectors contains, as a special 
case, the infinitesimal displacement of a *direction* into its
own *direction*. Such an infinitesimal autoparallelism of 
*directions* is characteristic of the projective structure of 
an affinely connected manifold. 



**Infinitesimal Autoparallelism of a Direction:**  

An *infinitesimal autoparallelism of a direction* \(R\) at an
arbitrary point \(p\) consists in the parallel displacement of \(R\)
at \(p\) to a neighbouring point \(p'\) which lies in the direction
\(R\) at \(p\).




A curve is geodesic if and only if its *tangent direction*
\(R\) experiences infinitesimal autoparallelism when moved 
along all the points of the curve. This characterization of a 
geodesic curve constitutes an abstraction from affine geometry. 
Through this abstraction, a geodesic curve is definable exclusively 
in terms of autoparallelism of tangent *directions*, and not 
tangent *vectors*. Roughly speaking, an affine geometry is 
essentially a projective geometry with the notion of distance defined
along the curves. By eliminating all possible notions of distance 
along curves, or equivalently, all the parameter descriptions of the 
curves, one abstracts the projective geometry form affine geometry. 




As mentioned above, a projective geometry \(\Pi\) may be defined as an 
equivalence class of affine geometries, that is, an equivalence class
of *projectively related* affine connections [\(\Gamma\)]. Weyl 
presented the details of his approach to projective geometry, which 
uses the notion of *autoparallelism of direction*, in a set of
lectures delivered in Barcelona and Madrid in the spring of 1922 
(Weyl (1923a); see also Weyl (1921c)). Weyl began with the following 
necessary and sufficient condition for the invariance of the 
projective structure \(\Pi\) under a transformation 
\(\Gamma \rightarrow \overline{\Gamma}\) of the affine 
structure: 



**Projective Transformation:**  
 
A transformation \(\Gamma \rightarrow \overline{\Gamma}\)
preserves the projective structure \(\Pi\) of a 
manifold with an affine structure \(\Gamma\), and is called a 
*projective transformation*, if and only if



\[\tag{7}
(\overline{\Gamma} - \Gamma)^i\_{jk} v^{\,j}v^{\,k} \propto v^{\,i},
\]

where \(v^{\,i}\) is an arbitrary vector.




Weyl's definition says that a change in the affine structure of
the manifold \(M\) preserves the projective structure \(\Pi\) of \(M\)
if the vectors \(v^{i}\_{q}\) and \(\overline{v}^{i}\_{q}\) at \(q\)
that result from the vector \(v^{i}\_{p}\) at \(p\) by parallel
transport under \(\Gamma\) and \(\overline{\Gamma}\) respectively,
differ at most in length but not in
 direction.[41]





A spacetime manifold \(M\) is a "projective manifold" 
(M, \(\Pi)\), if in addition to its manifold structure (differential 
topological structure), it is also endowed with a projective structure
\(\Pi\) that assigns to each of its manifold points 64 coefficients 
\(\Pi^{\,i}\_{jk} x^{i}(p))\)
satisfying certain symmetry 
 conditions.[42]
 These projective coefficients characterize the equivalence class 
[\(\Gamma\)] of projectively equivalent connections, that is, 
connections equivalent under the *projective transformation* 
(7). 




In physical spacetime the projective structure has an immediate 
intuitive significance according to Weyl. The real world is a 
non-empty spacetime filled with an inertial-gravitational field, 
which Weyl calls the *guiding field* 
 \((F\)*u**hrungsfeld*)[43].
 It is an indubitable fact, according to Weyl (1923a, 13), that a body
which is let free in a certain spacetime direction (time-like 
direction) carries out a uniquely determined natural motion from 
which it can only be diverted through an external force. The process 
of autoparallelism of direction appears, thus, as the tendency of 
persistence of the spacetime direction of a free particle whose 
motion is governed by, what Weyl calls, the *guiding field*\((F\)*u**hrungsfeld*). This natural 
motion occurs on the basis of an effective infinitesimal tendency of 
persistence, that parallelly displaces the spacetime direction 
\(R\) of a body at an arbitrary point \(p\) on its trajectory 
to a neighbouring point \(p'\) that lies in the direction 
\(R\) at \(p\). 




If external forces exert themselves on a body, then a motion results 
which is determined through the conflict between the tendency of 
persistence due to the guiding field and the force. The tendency of 
persistence of the guiding field is a type of constraining guidance, 
that the inertial-gravitational field exerts on every body. Weyl 
(1923b, 219) says: 



>  
> 
> Galilei's inertial law shows, that there exists a type of 
> constraining guidance in the *world* [spacetime] that imposes
> on a body that is let free in some definite world direction a unique 
> natural motion from which it can only be diverted through external 
> forces; this occurs on the basis of an effective infinitesimal 
> tendency of persistence from point to point, that 
> *auto-parallelly* transfers the world direction \(r\) of 
> the body at an arbitrary point \(P\) to an infinitesimally close 
> neighboring point \(P'\), that lies in the direction 
> \(r\) at \(P\).
> 
> 
> 


#### 4.1.3 Conformal Geometry, Weyl Geometry, and Weyl's Unified Field Theory




Shortly after the completion of the general theory of relativity in 
1915, Einstein, Weyl, and others began to work on a unified field 
theory. It was natural to assume at that 
 time[44]
 that this task would only involve the unification of gravity and 
electromagnetism. In Einstein's geometrization of gravity, the 
Newtonian gravitational potential, and the Newtonian gravitational 
force, are respectively replaced by the components of the metric 
tensor \(g\_{ij}(x)\), and the components of 
the symmetric linear connection \(\Gamma^{i}\_{jk}(x)\). In the general 
theory of relativity the gravitational field is thus accounted for in
terms of the curvature of spacetime, but the electromagnetic field 
remains completely unrelated to the spacetime geometry. Einstein's 
mathematical formulation of his theory of general relativity does 
not, however, provide room for the geometrization of the other long 
range force field, the electromagnetic 
 field.[45]
 It was therefore natural to ask whether nature's only two long range 
fields of force have a common origin. Consequently, it was quite 
natural to suggest that the electromagnetic field might also be 
ascribed to some property of spacetime, instead of being merely 
something embedded in spacetime. Since, however, the components 
\(g\_{ij}(x)\) of the metric tensor are 
already sufficiently determined by Einstein's field equations, this 
would require setting up a more general differential geometry than 
the one which underlies Einstein's theory, in order to make room for 
incorporating electromagnetism into spacetime geometry. Such a 
generalized differential geometry would describe both long range 
forces, and a new theory based on this geometry would constitute a 
unified field theory of electromagnetism and gravitation. 




In 1918, Weyl proposed such a theory. In Weyl (1918a, 1919a), and in 
the third edition (1920) of *Raum-Zeit-Materie*, Weyl 
presented his ingenious attempt to unify gravitation and 
electromagnetism by constructing a gauge-invariant geometry (see 
below), or what he called a *purely infinitesimal 
'metric' geometry*. Since the conformal structure 
\(C\) (see below) of spacetime does not determine a unique 
symmetric linear connection \(\Gamma\) but only an equivalence class 
\(K = [\Gamma]\) of *conformally equivalent symmetric linear 
connections*, Weyl was able to show that this degree of freedom 
in a conformal structure of spacetime provides just enough room for 
the geometrization of the electromagnetic potentials. The resulting 
geometry, called a *Weyl geometry*, is an intermediate 
geometric structure that lies between the conformal and Riemannian 
 structures.[46]





The metric tensor field that is locally described by 



\[\tag{8}
ds^{2} = g\_{ij}(x(p))dx^{i}dx^{j},
\]



is characteristic of a Riemannian geometry. That geometry requires of
the symmetric linear connection \(\Gamma\) that the infinitesimal 
parallel transport of a vector always preserves the length of the 
vector. Therefore, the metric field in Riemannian geometry determines
a unique symmetric linear connection, a "metric 
connection" that satisfies the length-preserving condition of 
parallel transport. This means that the metric field, locally 
represented by (8), is invariant under parallel transport. The 
coefficients of this unique symmetric linear *metric*connection are given by 



\[\tag{9}
\Gamma^i\_{jk} = \frac{1}{2} g^{ir}(g\_{rj,k} + g\_{kr,j} - g\_{jk,r}).

\]



If \(v\_{p}\) is a vector at \(p \in M\), its length 



\[\tag{10}
\lvert v\_p \rvert^2 = g\_{ij}(x(p))v^{\,i}\_p v^{\,j}\_p.
\]



Moreover, the angle between two vectors \(v\_{p}\) 
and \(w\_{p}\) at \(p\in M\) is given by 



\[\tag{11}
\cos \theta = \frac{g\_{ij}(x(p))v^{\,i}\_p w^{\,j}\_p}{\lvert v\_p \rvert \lvert w\_p \rvert}.
\]



While in Riemannian geometry the parallel transport of length is 
*path independent*, that is, it is possible to compare the 
lengths of any two vectors, even if they are located at two finitely 
different points, a vector suffers a *path-dependent* change 
in *direction* under parallel transport; that is, it is not 
possible to define the angle between two vectors, located at 
different points, in a path-independent way. Consequently, the angle 
between two vectors at a given point is invariant under parallel 
transport if and only if both vectors are transported *along the 
same path*. In particular, a vector which is carried around a 
closed circuit by a continual parallel displacement back to the 
starting point, will have the same length, but will not in general 
return to its initial direction. 



![figure](fig5.png)
Figure 5: The parallel transport of a
vector by a two-dimensional creature, from \(A \rightarrow B
\rightarrow C \rightarrow A\) around a geodesic triangle on a
two-dimensional surface \(S^{2}\), ends up pointing in a different
direction upon returning to \(A\).





For a closed loop which circumscribes an infinitesimally small 
portion of space, the rotation of the vector per unit area 
constitutes the measure of the local curvature of space. 
Consequently, whether or not finite parallel displacement of 
direction is *integrable*, that is, path-independent, depends 
on whether or not the curvature tensor vanishes. 




According to Weyl, Riemannian geometry, is not a pure or genuine 
*infinitesimal* differential (metric) geometry, since it
permits the comparison of length at a finite distance. In his seminal
1918 paper entitled *Gravitation und Elektrizitat*
(*Gravitation and Electricity*) Weyl (1918a) says: 



>  
> 
> However, in the Riemannian geometry described above, there remains a
> last distant-geometric [ferngeometrisches] element--without any
> sound reason, as far as I can see; the only cause of this appears to
> be the development of Riemannian geometry from the theory of
> surfaces. The metric permits the comparison of length of two vectors
> not only at the same point, but also at any arbitrarily separated
> points. *A true near-geometry (Nahegeometrie), however, may
> recognize only a principle of transferring a length at a point to an
> infinitesimal neighbouring point*, and then it is no more
> reasonable to assume that the transfer of length from a point to a
> finitely distant point is integrable, then it was to assume that the
> transfer of direction is integrable.
> 
> 
> 




Weyl wanted a metric geometry which would not permit *distance 
comparison of length* between two vectors located at finitely 
different points. In a *pure infinitesimal geometry*, Weyl 
argued, if attention is restricted to a single point of the manifold,
then some standard of length or *gauge* must be chosen 
arbitrarily before the lengths of vectors can be determined. 
Therefore, all that is intrinsic to the notion of a pure 
infinitesimal metric differential geometry is the ability to 
determine the *ratios of the lengths* of any two vectors and 
the angle between any two vectors, *at a point*. Such a pure 
*infinitesimal* metric manifold must have at least a 
*conformal* structure \(C\). 




The defining characteristic of a conformal spacetime structure is 
given by the equation 



\[\tag{12}
0 = ds^2 = g\_{ij}(x(p))dx^i dx^{\,j},
\]



which determines the light cone at \(p\). A gauge transformation 
of the metric is a map 



\[
g\_{ij}(x(p)) \rightarrow \lambda(x(p))g\_{ij}(x(p)) 
 = \overline{g}\_{ij}(x(p)),
\]



which preserves the metric up to a positive and smooth but otherwise 
arbitrary scalar factor or gauge function 
\(\lambda(x(p))\). In the case of a pseudo-Riemannian 
structure such a gauge transformation leaves the light cones 
unaltered. The angle between two vectors at \(p\) is given by 
(11). Clearly, the gauge transformation 
\(\overline{g}\_{ij}(x(p)) = \lambda(x(p))g\_{ij}(x(p))\)
is angle preserving, that is, conformal. Two metrics which are 
related by a conformal gauge transformation are called 
*conformally equivalent.* A conformal structure does not 
determine the length of any one vector at a point. Only the relative 
lengths, the ratio of lengths, of any two vectors 
\(\bfrac{\lvert v\_p \rvert}{\lvert w\_p \rvert}\)
is determined. 




Weyl exploited these features of the conformal structure, and 
suggested that given a conformal structure, a *gauge* could be
chosen at each point in a smooth but otherwise arbitrary manner, such
that the metric (8) at any point of the manifold is conventional or 
undetermined to the extent that the metric 



\[\tag{13}
d\overline{s}^2 = \lambda(x(p))g\_{ij}(x(p))dx^{i}dx^{\,j}
\]



is equally valid. 




However, a conformal structure by itself does not determine a unique 
symmetric linear connection; it only determines an equivalence class 
of conformally equivalent connections \(K = [\Gamma]\), namely, 
connections which preserve the conformal structure \(C\) during 
parallel transport. The difference between any two conformally 
equivalent symmetric linear connections \(\overline{\Gamma}^{\,i}\_{jk}\), 
\(\Gamma^i\_{jk} \in [\Gamma]\) is given by 



\[\tag{14}
\overline{\Gamma}^{\,i}\_{jk} - \Gamma^i\_{jk}
 = \frac{1}{2}(\delta^{\,i}\_j \theta\_k + \delta^{\,i}\_k \theta\_j
 - g\_{jk}g^{ir}\theta\_{r}),
\]



where 



\[\tag{15}
\theta\_{j}(x(p))dx^{\,j}
\]


is an arbitrary one-form field. 




Since the conformal structure determines only an equivalence class of
conformally equivalent symmetric linear connections \(K = [\Gamma]\),
the affine connection in this type of geometry is not uniquely
determined, and the parallel transport of vectors is not generally
well defined. Moreover, the ratio of the lengths of two vectors
located at different points is not determined even in a path-dependent
way. According to Weyl, it is a fundamental principle of infinitesimal
geometry that the metric structure on a manifold \(M\) determines a
unique affine structure on \(M\). As was pointed out earlier, this
principle is satisfied in Riemannian geometry where the metric
determines a unique symmetric linear connection, namely, the metric
connection according to (9). Evidently this fundamental principle of
infinitesimal geometry is not satisfied for a structure which is
merely a conformal structure, since the conformal structure only
determines an equivalence class of conformally equivalent symmetric
connections. Weyl showed that besides the conformal structure an
additional structure is required in order to determine a unique
symmetric linear connection from the equivalence class 
\(K = [\Gamma]\) of conformally equivalent symmetric linear
connections. Weyl showed that this additional structure is provided by
the *length connection* or *gauge field* \(A\_{j}\) that
governs the *congruent displacement of lengths*. Weyl called
this additional structure the "metric connection" on a
manifold; however, we shall use the term "length
connection" instead, in order to avoid confusion with the modern
usage of the term "metric connection", which today denotes
the symmetric linear connection that is uniquely determined by a
Riemannian metric tensor according to (9). 



**Weyl's Length Connection:**  

A point \(p\) is *length connected* with its infinitesimal
neighborhood, if and only if for every length at \(p\), there is
determined at every point \(q\) infinitesimally close to \(p\) a
length to which the length at \(p\) gives rise when it
is *congruently* displaced from \(p\) to \(q\).




This definition merely says that a manifold is "length 
connected" if it admits the process of infinitesimal congruent 
displacement of length. The only condition imposed on the concept of 
congruent displacement of length is the following: 



**Congruent Displacement of Length:**  
 
With respect to a choice of gauge for a neighborhood of \(p\), the
transport of a length \(l\_{p}\) at \(p\) to an infinitesimally
neighboring point \(q\) constitutes a
*congruent* displacement if and only if there exists a choice 
of gauge for the neighborhood of \(p\) relative to which the 
transported length 
\(\overline{l}\_{q}\) has the same value as 
\(\overline{l}\_{p}\); that is 



\[\tag{16}
\overline{l}\_q - \overline{l}\_p = d\overline{l}\_p = 0.
\]



Weyl called such a gauge at \(p\) a *geodesic gauge* at 
 \(p\).[47]
 Weyl's proof of the following theorem closely parallels the proof of 
theorem A.3 in 
 the supplement on Weyl's metric independent construction of the affine connection.




**Theorem 4.1:**  
 If for every point \(p\) in a 
neighborhood \(U\) of \(M\), there exists a choice of gauge 
such that the change in an arbitrary length at \(p\) under 
congruent displacement to an infinitesimally near point \(q\) is 
given by 



\[\tag{17}
d\overline{l}\_p = 0,
\]



then locally with respect to any other choice of gauge, 



\[\tag{18}
dl = -lA\_j(x(p))dx^{\,j},
\]


and conversely.




Making use of 



\[\begin{align}
dv^{\,i}\_p &= -\Gamma^i\_{jk}(x(p))v^{\,j}\_pdx^k \\
 l\_p &= g\_{ij}(x(p))v^{\,i}\_p v^{\,j}\_p \\
 dl\_p &= -l\_p A\_j x(p)dx^{\,j},
\end{align}\]



Weyl (1923a, 124-125) shows that the conformal structure 
supplemented with the structure of a *length connection* or 
*gauge field* \(A\_{j}(x)\) singles 
out a unique connection from the equivalence class \(K = [\Gamma]\) of 
conformally equivalent 
 connections.[48]
 This unique connection, which is called the *Weyl connection*,
is given by 



\[\begin{align}
\Gamma^{\,i}\_{jk} &= \frac{1}{2}g^{ir}(g\_{rj,k} + g\_{kr,j} - g\_{jk,r})
 +\frac{1}{2} g^{ir}(g\_{rj}A\_{k} + g\_{kr}A\_{j} - g\_{jk}A\_{r}) \\

\tag{19}
 &= \frac{1}{2}g^{ir}(g\_{rj,k} + g\_{kr,j} - g\_{jk,r})
 +\frac{1}{2}(\delta^{\,i}\_j A\_{k} + \delta^{\,i}\_k A\_{j} - g\_{jk}g^{ir}A\_{r}),
\end{align}\]



which is analogous to (14). The first term of the Weyl connection is 
identical to the metric connection (9) of Riemannian geometry, 
whereas the second term represents what is new in a Weyl geometry. 
The Weyl connection is invariant under the gauge transformation 



\[\begin{align}
\overline{g}\_{ij}(x) &= e^{\theta(x)}g\_{ij}(x) \\

\tag{20}
 \overline{A}\_j(x) &= A\_j(x) - \partial\_j \theta(x),
\end{align}\]



where the gauge function is \(\lambda(x) = e^{\theta(x)}\). Thus, a
conformal structure plus *length connection* or *gauge
field* \(A\_{j}(x)\) determines a *Weyl geometry*equipped with a unique *Weyl connection*. Therefore, the 
fundamental principle of infinitesimal geometry also holds in a Weyl 
geometry; that is, the metric structure of a Weyl geometry determines
a unique affine connection, namely, the Weyl connection. 




In Weyl's physical interpretation of his purely infinitesimal metric 
geometry (Weyl geometry), the gauge field 
\(A\_{j}(x)\) is identified with the 
electromagnetic four potential, and the electromagnetic field tensor 
is given by 



\[\tag{21}
F\_{jk}(x) = \partial\_{j} A\_{k}(x) - \partial\_{k} A\_{j}(x).
\]



A spacetime that is formally characterizable as a Weyl geometry, would
not only have a *curvature of direction*
(*Richtungskrummung*) but would also have a *curvature
of length* (*Streckenkrummung*). Because of the latter
property the formal characterization of the congruent displacement of
length would be non-integrable, that is, path-dependent, in a Weyl
geometry. 



![figure](fig6.png)
Figure 6: 
In a Weyl geometry parallel displacement of a vector along different
paths not only changes its direction but also its
length





Suppose physical spacetime corresponds to a Weyl geometry. Then two 
identical clocks \(A\) and \(B\) at an event \(p\) with a 
common unit of time, that is, a timelike vector of given length 
\(l\_{p}\), which are separated and moved along 
different world lines to an event \(q\), will not only differ with
respect to the elapsed time (first clock effect (i.e., relativistic 
effect)), but in general the clocks will differ with respect to their
common unit of time (rate of ticking) at \(q\) (second clock 
effect). That is, congruent time displacement in a Weyl geometry is 
such that two congruent time intervals at \(p\) will not in 
general be congruent at \(q\), when congruently displaced in 
parallel along different world lines from \(p\) to \(q\), that 
is, \(l^{A}\_{q} \ne l^{B}\_{q}\). 
This means that a twin who travels to a distant star and then returns
to earth would not only discover that the other twin on earth had 
aged much more, but also that all the clocks on earth tick at a 
different rate. Hence, in the presence of a non-vanishing 
electromagnetic field \(F\_{jk}(x)\) the 
clock rates will not in general be the same; that is, there will be a
second clock effect in addition to the relativistic effect (first 
clock effect). Thus, \(l^{A}\_{q} = l^{B}\_{q}\) if 
and only if the curl of \(A\_{j}(x)\) 
vanishes, that is, if and only if the electromagnetic field tensor 
\(F\_{jk}(x)\) vanishes, namely, 



\[
 F\_{jk}(x) = \partial\_{j} A\_{k}(x) - \partial\_{k} A\_{j}(x) = 0.
\]



In that case the second term of the Weyl connection vanishes and (19)
reduces to the metric connection (9) of Riemannian geometry. 




In a Weyl geometry there are no ideal absolute "meter 
sticks" or "clocks". For example, the rate at which
any clock measures time is a function of its history. However, as 
Einstein pointed out in a *Nachtrag* (addendum) to Weyl 
(1918a), it is precisely this situation which suggests that Weyl's 
geometry conflicts with experience. In Weyl's geometry, the frequency
of the spectral lines of atomic clocks would depend on the location 
and past histories of the atoms. But experience indicates otherwise. 
The spectral lines are well-defined and sharp; they appear to be 
independent of an atom's history. Atomic clocks define units of time,
and experience shows they are integrably transported. Thus, if we 
assume that the atomic time and the gravitational standard time are 
identical, and that the gravitational standard time is determined by 
the Weyl geometry, then the electromagnetic field tensor is zero. But
if that is the case, then a Weyl geometry reduces to the standard 
Riemannian geometry that underlies general relativity, since the 
vanishing of Weyl's *Streckenkrummung*(*length curvature*) is necessary and sufficient for the 
existence of a Riemannian metric \(g\_{ij}\). 




When 
quantum theory was developed a few years later it became clear that 
Weyl's theory was in conflict with experience in an even more 
fundamental way since there is a direct relation between clock rates 
and masses of particles in quantum theory. A particle with a certain 
rest mass \(m\) possesses a natural frequency which is a function 
of its rest mass, the speed of light \(c\), and Planck's constant 
\(h\). This means that in a Weyl geometry not only clocks would 
depend on their histories but also the masses of particles. For 
example, if two protons have different histories then they would also
have different masses in a Weyl geometry. But this violates the 
quantum mechanical principle that particles of the same kind--in
this case, protons--have to be exactly identical. 




However, in 1918 it was still possible for Weyl to defend his theory 
in the following way. In response to Einstein's criticism Weyl noted 
that atoms, clocks and meter sticks are complex objects whose real 
behavior in arbitrary gravitational and electromagnetic fields can 
only be inferred from a dynamical theory of matter. Since no detailed
and reliable dynamical models were available at that time, Weyl could
argue that there is no reason to assume that, for example, clock 
rates are correctly modelled by the length of a timelike vector. Weyl
(1919a, 67) said: 



>  
> 
> At first glance it might be surprising that according to the purely
> close-action geometry, length transfer is non-integrable in the
> presence of an electromagnetic field. Does this not clearly contradict
> the behaviour of rigid bodies and clocks? The behaviour of these
> measurement instruments, however, is a physical process whose course
> is determined by natural laws and as such has nothing to do with the
> ideal process of 'congruent displacement of spacetime
> distance' that we employ in the mathematical construction of the
> spacetime geometry. The connection between the metric field and the
> behaviour of rigid rods and clocks is already very unclear in the
> theory of Special Relativity if one does not restrict oneself to
> quasi-stationary motion. Although these instruments play an
> indispensable role in praxis as indicators of the metric field, (for
> this purpose, simpler processes would be preferable, for example, the
> propagation of light waves), it is clearly incorrect
> to *define* the metric field through the data that are directly
> obtained from these instruments.
> 
> 
> 




Weyl elaborated this idea by suggesting that the dynamical nature of
such time keeping systems was such that they
continually *adapt* to the spacetime structure in such a way
that their rates remain constant. He distinguished between quantities
that remain constant as a consequence of such *dynamical
adjustment*, and quantities that remain constant
by *persistence* because they are isolated and undisturbed. He
argued that all quantities that maintain a perfect constancy probably
do so as a result of *dynamical adjustment*. Weyl (1921a, 261)
expressed these ideas in the following way: 



>  
> 
> What is the cause of this discrepancy between the idea of congruent 
> transfer and the behaviour of measuring-rods and clocks? I 
> differentiate between the determination of a magnitude in Nature by 
> "persistence" (*Beharrung*) and by 
> "adjustment" (*Einstellung*). I shall make the 
> difference clear by the following illustration: We can give to the 
> axis of a rotating top any arbitrary direction in space. This 
> arbitrary original direction then determines for all time the 
> direction of the axis of the top when left to itself, by means of a 
> *tendency of persistence* which operates from moment to 
> moment; the axis experiences at every instant a parallel 
> displacement. The exact opposite is the case for a magnetic needle in
> a magnetic field. Its direction is determined at each instant 
> independently of the condition of the system at other instants by the
> fact that, in virtue of its constitution, the system *adjusts*itself in an unequivocally determined manner to the field in 
> which it is situated. *A priori* we have no ground for 
> assuming as integrable a transfer which results purely from the 
> tendency of persistence. ...Thus, although, for example, 
> Maxwell's equations demand the conservational equation 
> \(de\,/\,dt =0\) for the charge \(e\) of an 
> electron, we are unable to understand from this fact why an electron,
> even after an indefinitely long time, always possesses an unaltered 
> charge, and why the same charge \(e\) is associated with all 
> electrons. This circumstance shows that the charge is not determined 
> by persistence, but by adjustment, and that there can exist only 
> *one* state of equilibrium of the negative electricity, to 
> which the corpuscle adjusts itself afresh at every instant. For the 
> same reason we can conclude the same thing for the spectral lines of 
> atoms. The one thing common to atoms emitting the same frequency is 
> their constitution, and not the agreement of their frequencies on the
> occasion of an encounter in the distant past. Similarly, the length 
> of a measuring-rod is obviously determined by adjustment, for I could
> not give *this* measuring-rod in *this* field-position 
> any other length arbitrarily (say double or treble length) in place 
> of the length which it now possesses, in the manner in which I can at
> will pre-determine its direction. The theoretical possibility of a 
> determination of length by adjustment is given as a consequence of 
> the *world-curvature*, which arises from the metrical field 
> according to a complicated mathematical law. As a result of its 
> constitution, the measuring-rod assumes a length which possesses this
> or that value, *in relation to the radius of curvature of the 
> field*.
> 
> 
> 




Weyl's response to Einstein's criticism that a Weyl geometry 
conflicts with experience, took advantage of the fact that the 
underlying dynamical laws of matter which govern clocks and rigid 
rods, were not known at that time. Weyl could thus argue that it is 
at least *theoretically* possible that there exists an 
underlying dynamics of matter, such that a Weyl geometry, according 
to which length transfer is non-integrable, nonetheless coheres with 
*observable* experience, according to which length transfer 
*appears* to be integrable. However, as was clearly pointed 
out by Wolfgang Pauli, Weyl's plausible defence comes at a 
 cost.[49]
 Pauli (1921/1958, 196) argued that Weyl's defence of his theory 
deprives it of its inherent convincing power from a physical point of
view. 



>  
> 
> Weyl's present attitude to this problem is the
> following: *The ideal process of the congruent transference of
> world lengths ... has nothing to do with the real behaviour of
> measuring rods and clocks; the metric field must not be defined by
> means of information taken from these measuring instruments.* In
> this case the quantities \(g\_{ik}\) and \(\varphi\_{i}\) are, be
> definition, no longer observable, in contrast to the line elements of
> Einstein's theory. This relinquishment seems to have very
> serious consequences. While there now no longer exists a direct
> contradiction with experiment, the theory appears nevertheless to have
> been robbed of its inherent convincing power, from a physical point of
> view. For instance, the connexion between electromagnetism and world
> metric is not now essentially physical, but purely formal. For there
> is no longer an immediate connection between the electromagnetic
> phenomena and the behaviour of measuring rods and clocks. There is
> only an interrelation between the former and the ideal process which
> is mathematically defined as congruent transference of
> vectors. Besides, there exists only formal, and not physical, evidence
> for a connection between world metric and
> electricity.[50]
> 
> 
> 
> 




Pauli concluded his critical assessment of Weyl's theory with the 
following statement: 



>  
> *Summarizing, we can say that Weyl's theory has not succeeded in 
> getting any nearer to solving the problem of the structure 
> of matter.* As will be argued in more detail ... there is, 
> on the contrary, something to be said for the view that a solution of
> this problem cannot at all be found in this way.
> 
> 
> 




It should be noted, however, that Weyl's defence of his theory 
implicitly addresses an important methodological consideration 
concerning the relation between theory and evidence. As Pauli puts it
above, according to Weyl "*the metric field must not be 
defined by means of information taken from these measuring*
*instruments* [rigid rods and ideal clocks]". That is, 
Weyl rejects Einstein's *operational* standpoint which gives 
*operational significance* to the metric field in terms of the
*observable* behaviour of ideal rigid rods and ideal 
 clocks.[51]
 Unlike light propagation and freely falling (spherically symmetric, 
neutral) particles, rigid rods and ideal clocks are relativistically 
ill defined probative systems, and are thus unsuitable for the 
determination of the inherent structures of spacetime postulated by 
the theory of relativity. Weyl (1918a) clearly recognized this when 
he said in response to Einstein's critique "because of the 
problematic behaviour of yardsticks and clocks I have in my book 
*Space-Time-Matter* restricted myself for the specific 
measurement of the \(g\_{ik}\), exclusively to the 
observation of the arrival of light signals." It is interesting
to note parenthetically that in the first edition of his book Weyl 
thought that it was possible to have an intrinsic method of comparing
the lengths of arbitrary spacetime intervals with an interval between
two fiducial spacetime events, by using light signals only. It was 
Lorentz who pointed out to Weyl that not only the world lines of 
light rays but also the world lines of material bodies are required 
for an intrinsic method of comparing lengths. Not only did Weyl 
correct this mistake in subsequent editions, but already in 1921, 
Weyl (1921c) discovered the causal-inertial method for determining 
the spacetime metric (see SS4.3) by proving an important theorem 
that shows that the spacetime metric is already fully determined by 
the inertial and causal structure of spacetime. Weyl (1949a, 103) 
remarks "... therefore mensuration need not depend on 
clocks and rigid bodies but ... light signals and mass points 
moving under the influence of inertia alone will suffice." It 
is clear that Weyl regarded the use of clocks and rigid rods as an 
undesirable makeshift within the context of the special and general 
theory. Since neither spatial nor temporal intervals are invariants 
of spacetime, the invariant spacetime interval \(ds\) cannot be 
directly ascertained by means of standard clocks and rigid rods. In 
addition, the latter presuppose quantum theoretical principles for 
their justification and therefore lie outside the relativistic 
framework because the laws which govern their physical processes are 
not 
 known.[52]





Weyl (1929c, 233) abandoned his unified field theory only with the 
advent of the quantum theory of the electron. He did so because in 
that theory a different kind of gauge invariance associated with 
Dirac's theory of the electron was discovered, which more adequately 
accounted for the conservation of electric charge. Weyl's 
contributions to quantum mechanics, and his construction of a new 
principle of gauge invariance, are discussed in 
 SS4.5.3.[53]





Weyl's unified field theory was revived by Dirac (1973) in a slightly
modified form, which incorporated a real scalar field 
\(\beta(x)\). Dirac also argued that the time intervals measured 
by atomic clocks need not be identified with the lengths of timelike 
vectors in the Weyl 
 geometry.[54]



### 4.2 The Riemann-Helmholtz-Lie Problem of Space




Prior to the works of Gauss, Grassmann and Riemann, the study of 
geometry tended to emphasize the employment of empirical intuitions 
and images of the three dimensional physical space. Physical space 
was thought of as having definite metrical attributes. The task of 
the geometer was to take physical mensuration devices in that space 
and work with them. 




Under the influence of Gauss and Grassmann, Riemann's great 
philosophical contribution consisted in the demonstration that, 
unlike the case of a discrete manifold, where the determination of a 
set necessarily implies the determination of its quantity or cardinal
number, in the case of a continuous manifold, the concept of such a 
manifold and of its continuity properties, can be separated form its 
metrical structure. Using modern terminology, Riemann separated a 
manifold's local differential topological structure from its metrical
structure. Thus Riemann's separation thesis gave rise to the 
*space problem*, or as Weyl called it, *das 
Raumproblem*: how can metric relations be determined on a 
continuous manifold \(M\)? 




A metric manifold is a manifold on which a distance function 
\(f : M \times M \rightarrow \mathbb{R}\) is defined. Such a 
distance function must satisfy the following minimal conditions: for 
all \(p, q, r \in M\), 


1. \(f(p, q) \ge 0\), and if \(f(p,q) = 0\), then \(p = q\),
2. \(f(p, q) = f(q,p)\), (symmetry)
3. \(f(p, q) + f(q,r) \ge f(p,r)\) (triangle inequality).




In his famous inaugural lecture at Gottingen, entitled 
*Uber die Hypothesen, welche der Geometrie zu*
*Grunde liegen* (*About the hypotheses which lie at the 
basis of geometry*), Riemann (1854) examined how metric relations
can be determined on a continuous manifold; that is, what specific 
form \(f : M \times M \rightarrow \mathbb{R}\) should 
have. Consider the coordinates \(x^{i}(p)\) 
and \(x^{i} (p) + dx^{i} (p)\) of two neighboring 
points \(p, q \in M\). The measure of the 
distance \(ds = f(p,q)\) must 
be some function \(F\_{p}\) at \(p\) of the 
differential increments \(dx^{i}(p)\); that is, 



\[\tag{22}
ds = F\_{p}(dx^{1}(p), \ldots ,dx^{n}(p)).
\]



Riemann states that \(F\_{p}\) should satisfy the 
following requirements: 



**Functional Homogeneity:**  
 If \(\lambda \gt 0\) and 
\(ds = F\_{p}(dx(p))\), then



\[\tag{23}
\lambda ds = \lambda F\_{p}(dx(p)) = F\_{p}(\lambda dx(p)).
\]


**Sign Invariance:**  
 A change in sign of the 
differentials should leave the value of \(ds\) invariant.




Sign invariance is satisfied by every positive homogeneous function 
of degree \(2m\) \((m = 1, 2, 3, \ldots)\). In the simple 
case \(m = 1\), and the length element \(ds\) is the square 
root of a homogeneous function of second degree, which can be 
expressed in the standard form 



\[\tag{24}
ds = \left[
 \sum\_{i=1}^n (dx^{\,i}(p))^{2}
 \right]^{\bfrac{1}{2}}.
\]



That is, at each point of \(M\) there exists a coordinate system 
(defined up to an element of the orthogonal group \(O(n)\) 
in which the square root of the homogeneous function of second degree
can be expressed in the above standard form. Riemann's well-known 
general expression for the measure of length at \(p \in M\) with 
respect to an arbitrary coordinate system is given by 



\[\tag{25}
ds^{2} = g\_{ij}(x(p))dx^{i}(p)dx^{\,j}(p),
\]



where the components of the metric tensor satisfy the symmetry 
condition \(g\_{ij} = g\_{ji}\). 




The assumption that \(ds^{2} = F^{2}\_{p}\) is a 
quadratic differential form is not only the simplest one, but also 
the preferred one for other important reasons. Riemann himself was 
well aware of other possibilities; for example, the possibility that 
\(ds\) could be the 4th root of a homogeneous polynomial of 4th 
order in the differentials. But Riemann restricted himself to the 
special case \(m = 1\) because he was pressed for time and because he 
wanted to give specific geometric interpretations of his results. As 
Weyl points out Riemann's own answer to the space problem is 
inadequate since Riemann's mathematical justification for the 
restriction to the Pythagorean case are not very compelling. The 
first satisfactory justification of the Pythagorean form of the 
Riemannian metric, although limited in scope because it presupposed 
the full homogeneity of Euclidean space, was provided by the 
investigations of Hermann von Helmholtz. Helmholtz diverged from 
Riemann's analytic approach and made use merely of the fundamental 
concept of geometry, namely, the concept of *congruent 
mapping*, and characterized the geometric structure of space by 
requiring of space the full homogeneity of Euclidean space. His 
analysis was thereby restricted to the cases of constant positive, 
zero, or negative curvature. Abstracting from our experience of the 
movement of rigid bodies, Helmholtz was able to mathematically derive
Riemann's distance formula from a number of axioms about rigid body 
motion in space. Helmholtz (1868) argued that Riemann's hypothesis 
that the metric structure of space is determined locally by a 
quadratic differential form, is really a consequence of the facts 
(*Tatsachen*) of rigid-body motion. 




Considering the general case of \(n\) dimensions, and using Lie 
groups and Lie algebras, Sophus Lie, (Lie (1886/1935, 1890a,b)), 
later developed and improved Helmholtz's justification. However, the 
Helmholtz-Lie treatment of, and solution to, the problem of space, 
lost its relevance with the arrival of Einstein's theory of general 
relativity. As Weyl (1922b) points out, instead of a 
three-dimensional continuum we must now consider a four-dimensional 
continuum, the metric of which is not *positive definite* but 
is given instead by an *indefinite* quadratic form. In 
addition, Helmholtz's presupposition of metric homogeneity no longer 
holds, since we are now dealing with an inhomogeneous metric field 
that causally depends on the distribution of matter. Consequently, 
Weyl provided a reformulation of the space problem that is compatible
with the causal and metric structures postulated by the theory of 
general relativity. But Weyl went further. Such a reformulation 
should not only incorporate Riemann's *infinitesimal* 
standpoint, as required by Einstein's general theory, it should also 
cohere with Weyl's requirements of a **pure***infinitesimal geometry* developed earlier in the 
context of Weyl's construction of a unified field theory. 




More precisely, Weyl generalized the so-called 
*Riemann-Helmholtz-Lie* problem of space in two ways: First, 
he allowed for *indefinite metrics* in order to encompass the 
general theory of relativity. Secondly, he considered metrics with 
variable gauge \(\lambda(x(p))\) together with an 
associated *length connection*, in order to obtain a 
*purely* infinitesimal geometry. Thus each member of a general
class of geometries under consideration is locally determined 
relative to a choice of variable gauge by two structural fields 
(Strukturfelder): (1) a possibly indefinite *Finsler* metric 
 field[55]
 \(F\_{p}(dx)\), and (2) a *length connection* that is
determined by a 1-form field \(\theta\_{i}dx^{i}\). Weyl's task
was to prove: 



>  
> 
> If the geometry satisfies the *Postulate of Freedom*, (the 
> nature of space imposes no restrictions on admissible metrical 
> relations), and determines a unique, symmetric, linear connection 
> \(\Gamma\), then the Finsler metric field 
> \(F\_{p}(dx)\) must be a Riemannian metric field of 
> some 
>  signature.[56]
> 
> 
> 
> 




In a Riemannian space the concept of parallel displacement is defined
by two conditions: 


1. The components of a vector remain unchanged during an infinitesimal
parallel displacement in a suitably chosen coordinate system (geodesic
coordinate).[57]
This condition is satisfied if

\[\begin{matrix}
dv^{\,i}\_{p} = - \Gamma^{i}\_{jk} v^{\,j}\_{p} dx^{k}\_{p} 
 & \text{and} &
 \Gamma^{i}\_{jk} = \Gamma^{i}\_{kj}\,.
\end{matrix}\]
2. The length of a vector remains unchanged during an infinitesimal
parallel displacement.




It follows from these conditions that a Riemannian space possesses a 
definite symmetric linear connection--a symmetric linear 
*metric* 
 connection[58]--which
 is uniquely determined by the Pythagorean-Riemannian metric. Weyl 
calls this: 



**The Fundamental Postulate of Riemannian Geometry:**  
 Among the *possible* systems of parallel displacements of a
vector to infinitely near points, that is among the *possible*sets of symmetric linear connection coefficients, there exists 
one and only one set, and hence one and only one system of parallel 
displacement, which is *length preserving*.




In his 
 lectures[59]
 on the mathematical analysis of the problem of space delivered in 
1922 at Barcelona and Madrid, Weyl sketched a proof demonstrating 
that the following is also true: 



**Uniqueness of the Pythagorean-Riemannian Metric:**  
 
Among all the possible infinitesimal metrics that can be put on a
differentiable manifold, the Pythagorean-Riemannian metric is the only
type of metric that uniquely determines a symmetric linear
connection.




Weyl begins his proof with two natural assumptions. First, the 
*nature* of the metric should be coordinate independent. If 
\(ds\) is given by an expression 
\(F\_{p}(dx^{1}, \ldots ,dx^{n})\) with respect to a given system 
of coordinates, then with respect to another system of coordinates, 
\(ds\) is given by a function that is related to 
\(F\_{p}(dx^{1}, \ldots ,dx^{n})\) by a linear, homogeneous 
transformation of its arguments \(dx^{i}\). 
Second, it is reasonable to assume that the *nature* of the 
metric is the same everywhere, in the sense that at every point of 
the manifold, and with respect to every coordinate system for a 
neighborhood of the point in question, \(ds\) is represented by an
element of the equivalence class \([F]\) of functions generated by
any one such function, say 
\(F\_{p}(dx^{1}, \ldots ,dx^{n})\), by all linear, homogeneous 
transformations of its arguments \(dx^{i}\).





For the case in which \(F\_{p}\) is Pythagorean in 
form, namely the square root of a positive-definite quadratic form, 
there exists just one possible equivalence class [\(F\)], because 
every function that is the square root of a positive-definite 
quadratic form can be transformed to the standard expression 



\[\tag{26}
F = \left[(dx^{1})^{2} + \cdots + (dx^{n})^{2}\right]^{\bfrac{1}{2}}
\]



by means of a linear, homogeneous transformation. 




To every possible equivalence class [\(F\)] of homogeneous 
functions, there corresponds a *type* of metrical space. The 
Pythagorean-Riemannian space, for which 
\(F^{2}\_{p} = (dx^{1})^{2} + \cdots + (dx^{n})^{2}\), is one among 
several types of possible metrical spaces. The problem, therefore, is
to single out the equivalence class \([F]\), where \(F\) 
corresponds to 
\(F^{2}\_{p} = (dx^{1})^{2} + \cdots + (dx^{n})^{2}\), from the other 
possibilities, and to provide arguments for this preference. 




By the term 'metric' Weyl means *any*infinitesimal *distance function* 
\(F\_{p} \in [F]\), where the equivalence 
class \([F]\) represents a *type* of metric structure or 
metric field. Any such type of metric field structure has a 
microsymmetry group \(G\_{p}\) at each \(p \in M\). 



**Definition 4.1 (Microsymmetry Group)**  

A microsymmetry of a structural field (Strukturfeld) at a point \(p
\in M\) is a local diffeomorphism that takes \(p \in M\) into \(p\)
and preserves the structural field at \(p \in M\). The microsymmetry
group of a field at \(p \in M\) is the group of its microsymmetries at
\(p \in M\) under the operation of composition.





A microsymmetry group \(G\_{p}\), at \(p \in M\), of a metric
structure, is a set of invertible, linear maps of the tangent space
\(T(M\_{p})\) onto itself, which preserve the infinitesimal distance
function at \(p \in M\). For every \(p \in M, G\_{p}\) is isomorphic to
one and the same abstract group. 




For a Riemannian type of metric structure the congruent linear maps 
of the tangent space T\((M\_{p})\) onto itself form 
a group \(G\_{p}\) which is isomorphic to the 
orthogonal group \(O(n)\). The Pythagorean-Riemannian 
metric at \(p\) is therefore determined through the concrete 
realization of the *orthogonal* group at \(p\) which leaves
the fundamental quadratic differential form at \(p\) invariant. 
Thus the Pythagorean-Riemannian *type* of metric is 
characterized by the abstract microsymmetry group \(O(n)\).
For a metric which is not of the Pythagorean-Riemannian metric type, 
the abstract microsymmetry group \(G\_{p}\) will be 
different from \(O(n)\) and will be some other subgroup of 
\(GL(n)\). At each point of the manifold the microsymmetry 
group will be a concrete realization of this subgroup of 
\(GL(n)\). Weyl now states what he calls 



**The Postulate of Freedom:**  

If only the *nature* (of whatever type) of the metric is
specified, that is, if only the corresponding abstract microsymmetry
group \(G\_{p}\) is specified, and the metric in
question is otherwise left arbitrary, then the *mutual
orientations* of the corresponding microsymmetry groups at
different points are also left arbitrary.




Weyl emphasizes that the *Postulate of Freedom* provides the 
general framework for a concise formulation of 



**The Hypothesis of Dynamical Geometry:**  

Whatever the nature or type of the metric may be--provided it
is the same everywhere--the variations in the *mutual
orientations* of the concrete microsymmetry groups from point to
point are causally determined by the material content that fills
space.




In contrast with Helmholtz's analysis, which presupposes the 
homogeneity of space, the *Postulate of Freedom* allows for 
the possibility of replacing Helmholtz's homogeneity requirement with
the possibility of subjecting the metric field to arbitrary, 
infinitesimal change. *To assert this dynamical possibility does 
not require that the nature of the metric be specified*. 




Next, Weyl points out that what has been provided so far is merely an
explication of the concepts *metric*, *length*
*connection* and *symmetric linear connection*. Some 
claim which goes beyond conceptual analysis has to be made, according
to Weyl, in order to prove that among the various types of possible 
metrical structures that can be put on a differentiable manifold 
representing physical space, the Pythagorean-Riemannian form is 
unique. Weyl suggests the following hypothesis: 



**Weyl's Hypothesis:**  

Whatever determination the essentially free length connection at some
point \(p\) of the manifold may realize with the points in its
infinitesimal neighborhood, there always exists among the possible
systems of parallel displacements of the tangent space
\(T(M\_{p})\), one and only one, which is at the same
time a system of infinitesimal *congruent transport*.





Weyl shows that this hypothesis does in fact single out metrics of 
the Pythagorean-Riemannian type by proving the following theorem: 



**Theorem 4.2**  

If a specific length connection is such that it determines a unique
symmetric linear connection, then the metric must be of the
Pythagorean-Riemannian form (for some signature).




Thus the *Postulate of Freedom* and *Weyl's
Hypothesis* together entail the existence, at each \(p \in M\), of
a non-degenerate quadratic form that is unique up to a choice of gauge
at \(p \in M\), and that is invariant under the action of the
microsymmetry group \(G\_{p}\) that is isomorphic to an orthogonal
group of some signature. 




This formulation suggests, according to Weyl, an intuitive contrast 
between Euclidean 'distance-geometry' and the 
'*near*-geometry' (*Nahegeometrie*) or 
'*field*-geometry' of Riemann. Weyl (1949a, 88) 
compared Euclidean 'distance-geometry' to a crystal 
"built up from uniform unchangeable atoms in the rigid and 
unchangeable arrangement of a lattice", and the latter 
[Riemannian *field-geometry*] to a liquid, "consisting 
of the same indiscernible unchangeable atoms, whose arrangement and 
orientation, however, are mobile and yielding to forces acting upon 
them." 




The *nature* of the metric field, that is the *nature*of the metric everywhere, is the same and is, therefore, 
absolutely determined. It reflects according to Weyl, the *a 
priori* structure of space or spacetime. In contrast, what is 
*posteriori*, that is, accidental and capable of continuous 
change being causally dependent on the material content that fills 
space, are the mutual *orientations* of the metrics at 
different points. Hence, the demarcation between the \(a\) 
*priori* and the *a posteriori* has shifted, according 
to Weyl: Euclidean geometry is still preserved for the infinitesimal 
neighborhood of any given point, but the coordinate system in which 
the metrical law assumes the standard form 
\(ds^{2} =\sum^{n}\_{i=1}(dx^{i})^{2}\)
is in general different from place to place. 




Weyl's *a priori* and *a posteriori* distinction must 
not be confused with Kant's distinction. Weyl (1949a, 134) remarks: 
"In the case of physical space it is possible to 
counterdistinguish aprioristic and aposterioristic features in a 
certain objective sense without, like Kant, referring to their 
cognitive source or their cognitive character." Weyl makes the 
same remark in (Weyl, 1922b, 266). See also the discussion in 
SS4.5.8. 




In the context of his group-theoretical analysis, Weyl (1922b, p. 
266) makes the following interesting and important statement: 



>  
> 
> I remark from an epistemological point of view: it is not correct to 
> say that space or the world [spacetime] is in itself, prior to any 
> material content, merely a formless continuous manifold in the sense 
> of analysis situs; the *nature* of the metric [its 
> infinitesimal Pythagorean-Riemannian character] is characteristic of 
> space in itself, only the mutual orientation of the metrics at the 
> various points is contingent, a posteriori and dependent on the 
> material content.
> 
> 
> 




Within the context of general relativity, empty spacetime is 
impossible, if 'empty' is understood to mean not merely 
empty of all *matter* but also empty of all *fields*. 
At another place, Weyl (1949a, Engl. edn, 172) says: 



>  
> 
> Geometry unites organically with the field theory; space is not 
> opposed to things (as it is in substance theory) like an empty vessel
> into which they are placed and which endows them with far-geometrical
> relationships. No empty space exists here; the assumption that the 
> field omits a portion of the space is absurd.
> 
> 
> 




According to Weyl, the metric field does not cease to exist in a world
devoid of matter but is in a state of rest: As a *rest* field
it would possess the property of *metric homogeneity*; the
mutual *orientations* of the *orthogonal* groups
characterizing the Pythagorean-Riemannian nature of the metric
everywhere would not differ from point to point. *This means that
in a matter-empty universe the metric is fixed. Consequently, the set
of congruence relations on spacetime is uniquely determined.*
Since the metric uniquely determines the symmetric linear connection,
the homogeneous *metric* field (rest field) determines
an *integrable* affine structure. Therefore, a flat Minkowski
spacetime consistent with the complete absence of matter is endowed
with an *integrable connection* and thus determines all
(hypothetical) free motions. According to Weyl, there exists in the
absence of matter a homogeneous metric field, a structural field
(*Strukturfeld*), which has the character of a *rest*
field, and which constitutes an all pervasive background that cannot
be eliminated. The structure of this *rest* field determines
the *extension* of the spacetime congruence relations and
determines Lorentz invariance. The *rest*
field possesses no net energy and makes no contribution to 
curvature. 




The contrast with Helmholtz and Lie is this: both of them require 
homogeneity and isotropy for physical space. From a general 
Riemannian standpoint, the latter characteristics are valid only for 
a matter-empty universe. Such a universe is flat and Euclidean, 
whereas a universe that contains matter is *inhomogeneous, 
anisotropic and of variable curvature*. 




It is important to note here that the validity of Weyl's assertion 
that the metric field does not cease to exist but is in a state of 
rest, has its source in the mathematical fact that the metric field 
is a \(G\)-structure. A \(G\)-structure may be flat or non-flat; but a 
\(G\)-structure can never vanish. Consequently, geometric fields 
characterizable as \(G\)-structures, such as the projective, conformal, 
affine and metric structures, do not 
 vanish.[60]



### 4.3 Weyl's Causal-Inertial Method for determining the Spacetime Metric


#### 4.3.1 Weyl's Field Ontology of Spacetime Geometry




Riemann searched for the most general type of an \(n\)-dimensional
manifold. On this manifold, Euclidean geometry turns out to be a 
special case resulting from a certain form of the metric. Weyl takes 
this general structure, the manifold structure, which has certain 
continuity and order properties, as basic, but leaves the 
determination of the other geometrical structures, such as the 
projective, conformal, affine and metric structures, open. The 
metrical axioms are no longer dictated, as they were for Kant, by 
pure intuition. According to Weyl (1949a, 87), for Riemann the metric
is not, as it was for Kant, "part of the static homogeneous 
form of phenomena, but of their ever changing material 
content". Weyl (1931a, 338) says: 



>  
> 
> We differentiate now between the amorphous continuum and its metrical
> structure. The first has retained its *a priori* 
>  character,[61]
>  ... whereas the structural field [Strukturfeld] is completely 
> subjected to the power-play of the world; being a real entity, 
> Einstein prefers to call it the ether.
> 
> 
> 




There is no indication in Riemann's work on gravitation and
electromagnetism that would indicate that he anticipated the
conceptual revolution underlying Einstein's theory. However,
Weyl's interpretation of Riemann's work suggests that
Riemann foresaw something like its possibility in the following
sense: 



>  
> 
> By formally separating the post-topological structures such as the 
> affine, projective, conformal and metric structures from the 
> manifold, so that these structures are no longer rigidly tied to it, 
> Riemann deprived them of their formal geometric rigidity and, on the 
> basis of his infinitesimal geometric standpoint or 
> "near-geometry", allowed for the possibility of 
> interpreting them as mathematical representations of flexible, 
> dynamical physical structural fields [Strukturfelder] on the manifold
> of spacetime, geometrical fields that reciprocally interact with 
> matter.
> 
> 
> 




Riemann's separation thesis together with his adoption of the 
infinitesimal standpoint, were prerequisite steps for the development
of differential geometry as the mathematics of differentiable 
geometric fields on manifolds. When interpreted physically, these 
mathematical structures or geometrical fields correspond, as Weyl 
says, to physical structural fields (Strukturfelder). Analogous to 
the electromagnetic field, these structural fields act on matter and 
are in turn acted on by matter. Weyl (1931a, 337) remarks: 



>  
> 
> I now come to the crucial idea of the theory of General Relativity. 
> *Whatever exerts as powerful and real effects as does the
> metric structure of the world, cannot be a rigid, once and for 
> all, fixed geometrical structure of the world, but must 
> itself be something real which not only exerts effects on 
> matter but which in turn suffers them through matter.* Riemann 
> already suggested for space the idea that the structural field, like 
> the electromagnetic field, reciprocally interacts with matter.
> 
> 
> 




Weyl (1931a, 338) continues: 



>  
> 
> We already explained with the example of inertia, that the structural
> field [Strukturfeld] must, as a close-action [Nahewirkung], be 
> understood infinitesimally. How this can occur with the metric 
> structure of space, Riemann abstracted from Gauss's theory of curved 
> surfaces.
> 
> 
> 




The various geometrical fields are not "intrinsic" to the
manifold structure of spacetime. The manifold represents an amorphous
four-dimensional differentiable continuum in the sense of 
*analysis situs* and has no properties besides those that fall
under the concept of a manifold. 




The amorphous four-dimensional differentiable manifold possesses a 
high degree of symmetry. Because of its homogeneity, all points are 
alike; there are no objective geometric properties that enable one to
distinguish one point from another. This full homogeneity or symmetry
of space must be described by its group of *automorphisms*, 
the one-to-one mappings of the point field onto itself which leave 
all relations of objective significance between points undisturbed. 
If a geometric object \(F\), that is a point set with a definite 
relational structure is given, then those automorphisms of space that
leave \(F\) invariant, constitute a group and this group describes
exactly the symmetry which \(F\) possesses. For instance, to use 
an example by Weyl (1938b) (see also Weyl (1949a, 72-73) and 
Weyl (1952)), if 
\(R(p\_{1},p\_{2},p\_{3})\) is a ternary relation that asserts 
\(p\_{1},p\_{2},p\_{3}\)
lie on a straight line, then we require that any three points, 
satisfying this relation \(R\), are mapped by an automorphism into
three other points 
\(p\_{1}',p\_{2}',p\_{3}'\),
fulfilling the same relation. 




The group of automorphisms of the \(n\)-dimensional number space 
contains only the identity map, since all numbers of 
\(\mathbb{R}^{n}\) are distinct individuals. It is 
essentially for this reason that the real numbers are used for 
coordinate descriptions. Whereas the continuum of real numbers 
consists of individuals, the continua of space, time, and spacetime 
are homogeneous. Spacetime points do not admit of an absolute 
characterization; they can be distinguished, according to Weyl, only 
by "a demonstrative act, by pointing and saying 
here-now". 




In a little book entitled *Riemanns geometrische Ideen, ihre 
Auswirkung und ihre Verknupfung mit 
der Gruppentheorie*, published posthumously in 1988, Weyl (1988, 
4-5) makes this interesting comment: 



>  
> 
> Coordinates are introduced on the Mf [manifold] in the most direct 
> way through the mapping onto the number space, in such a way, that 
> all coordinates, which arise through one-to-one continuous 
> transformations, are equally possible. With this the coordinate 
> concept breaks loose from all special constructions to which it was 
> bound earlier in geometry. In the language of relativity this means: 
> The coordinates are not measured, their values are not read off from 
> real measuring rods which react in a definite way to physical fields 
> and the metrical structure, rather they are a priori placed in the 
> world arbitrarily, in order to characterize those physical fields 
> including the metric structure numerically. The metric structure 
> becomes through this, so to speak, freed from space; it becomes an 
> existing field within the remaining structure-less space. Through 
> this, space as form of appearance contrasts more clearly with its 
> real content: The content is measured after the form is arbitrarily 
> related to coordinates.
> 
> 
> 




By mapping a given spacetime homeomorphically onto the real number 
space, providing through the arbitrariness of the mapping, what Weyl 
calls, a qualitatively non-differentiated field of free 
possibilities--the continuum of all possible 
coincidences--we represent spacetime points by their coordinates
corresponding to some coordinate system. The four-dimensional 
arithmetical space can be utilized as a four-dimensional schema for 
the localization of events of all possible "here-nows". 




Physical dynamical quantities in spacetime, such as the geometrical 
structural fields on the four-dimensional spacetime continuum, are 
describable as functions of a variable point which ranges over the 
four-dimensional number space \(\mathbb{R}^{4}\). Instead of 
thinking of the spacetime points as real substantival entities, and 
any talk of fields as just a convenient way of describing geometrical
relations between points, one thinks of the geometrical fields such 
as the projective, conformal causal, affine and metric fields, as 
real physical entities with dynamical properties, such as energy, 
momentum and angular momentum, and the field points as mere 
mathematical abstractions. Spacetime is not a medium in the sense of 
the old ether concept. No ether in that sense exists here. Just as 
the electromagnetic fields are not states of a medium but constitute 
independent realities which are not reducible to anything else, so, 
according to Weyl, the geometrical fields are independent irreducible
physical 
 fields.[62]





A class of geometric structural fields of a given type is 
characterized by a particular Lie group. A geometric structural field
belonging to a given class has a microsymmetry group (see definition 
4.1) at each point \(p \in M\) which is isomorphic to 
the Lie group that is characteristic of the class. In relativity 
theory, this microsymmetry group is isomorphic to the Lorentz group 
and leaves invariant a pseudo-Riemannian metric of Lorentzian 
signature. 




The different types of geometric, structural fields may be 
represented from a modern mathematical point of view as cross 
sections of appropriate fiber bundles over the manifold \(M\); 
that is, the amorphous manifold \(M\) has associated with it 
various geometric fields in terms of a mapping of a certain kind 
(called a cross section) from the manifold \(M\) to the 
corresponding bundle space over 
 \(M\).[63]
 In particular, Einstein's general theory of relativity postulates a 
physical field, the metrical field, which, mathematically speaking, 
may be characterized as a cross section of the bundle of 
non-degenerate, second-order, symmetric, covariant tensors of Lorentz
signature over \(M\). Weyl (1931a, 336) says of this world 
structure: 



>  
> 
> However this structure is to be exactly and completely described and 
> whatever its inner ground might be, all laws of nature show that it 
> constitutes the most decisive influence on the evolution of physical 
> events: the behavior of rigid bodies and clocks is almost exclusively
> determined through the metric structure, as is the pattern of the 
> motion of a force-free mass point and the propagation of a light 
> source. And only through these effects on the concrete natural 
> processes can we recognize this structure.
> 
> 
> 




The views of Weyl are diametrically opposed to geometrical
conventionalism and some forms of relationalism. According to Weyl,
we *discover* through the behavior of physical phenomena an
already determined metrical structure of spacetime. The metrical
relations of physical objects are determined by a physical field, the
metric field, which is represented by the second rank metric tensor
field. Contrary to geometric conventionalism, spacetime geometry is
not about rigid rods, ideal clocks, light rays or freely falling
particles, except in the derivative sense of providing information
about the physically real metric field which, according to Weyl, is as
physically real as is the electromagnetic field, and which determines
and explains the metrical behavior of congruence standards under
transport. The metrical field has physical *and* metrical
significance, and the metrical significance does not consist in the
mere articulation of relations obtaining between, say, rigid rods or
ideal clocks. 




The special and general, as well as the non-relativistic spacetime 
theories postulate various structural constraints which events are 
held to satisfy. When interpreted physically, these mathematical 
structures or constraints correspond to physical structural fields 
(*Strukturfelder*). Analogous to the electromagnetic field, 
these structural fields act on matter and are, within the context of 
the general theory of relativity, in turn acted on by matter. An 
\(n\)-dimensional manifold \(M\) whose sole properties are 
those that fall under the concept of a manifold, Weyl (1918b) 
physically interprets as an \(n\)-dimensional empty world, that 
is, a world empty of both matter and fields. On the other hand, an 
\(n\)-dimensional manifold \(M\) that is an affinely connected 
manifold, Weyl physically interprets as an \(n\)-dimensional world
filled with a gravitational field, and an \(n\)-dimensional 
manifold \(M\) endowed with a projective structure represents an 
\(n\)-dimensional non-empty world filled with an 
inertial-gravitational field, or what Weyl calls the *guiding 
field Fuhrungsfeld*). In a 
similar vein, an \(n\)-dimensional manifold \(M\) that 
possesses a conformal structure of Lorentz type, represents a 
non-empty \(n\)-dimensional world filled with a causal field. 
Finally, an \(n\)-dimensional manifold \(M\) endowed with a 
metrical structure may be interpreted physically as an 
\(n\)-dimensional non-empty world filled with a metric field. 


#### 4.3.2 The Causal and Inertial Field uniquely determine the Metric




The mathematical model of physical spacetime is the four-dimensional 
pseudo-Riemannian manifold. Weyl (1921c) distinguished between two 
primitive substructures of that model: the *conformal* and 
*projective* structures and showed that the conformal 
structure, modelling the *causal field* governing light 
propagation, and the projective structure, modelling the inertial or 
*guiding field* governing all free (fall) motions, uniquely 
determine the metric. That is, Weyl (1921c) proved 



**Theorem 4.3**  

The projective and conformal structure of a metric space determine the
metric uniquely.




A metric \(g\) on a manifold determines a first-order conformal 
structure on the manifold, namely, an equivalence class of 
conformally related metrics 



\[\tag{27}
[g] = \{\overline{g} \mid \overline{g} = e^{\theta} g \}.
\]



A metric \(g\) also uniquely determines a symmetric linear 
connection \(\Gamma\) on the manifold. Under a conformal transformation 



\[\tag{28}
g \rightarrow e^{\theta} g = \overline{g},
\]



the change of the components of the symmetric linear connection is 
given by (14), that is, 



\[\tag{29}
\Gamma^{i}\_{jk} \rightarrow \overline{\Gamma}^{i}\_{jk}
 = \Gamma^{i}\_{jk} + 
 \frac{1}{2}(\delta^{i}\_{j}\theta\_{k} + \delta^{i}\_{k}\theta\_{j} - g\_{jk}g^{ir}\theta\_{r}).
\]



Thus the set of all arbitrary conformal transformations of the metric
induces an equivalence class \(K\) of conformally related 
symmetric linear connections. This equivalence class \(K\) 
constitutes a second-order conformal structure on the manifold and 
the difference between any two connections in the equivalence class 
is given by (29). Weyl shows that a conformal transformation (29) 
preserves the projective structure and hence is a projective 
transformation (that is, a conformal transformation which also 
satisfies (7)), if and only if \(\theta\_{j} = 0\), in 
which case the conformal and projective structures are compatible. 
Weyl remarks after the proof: 



>  
> 
> If it is possible for us, in the real world, to discern causal 
> propagation, and in particular light propagation, and if moreover, we
> are able to recognize and observe as such the motion of free mass 
> points which follow the guiding field, then we are able to read off 
> the metric field from this alone, without reliance on clocks and 
> rigid rods.
> 
> 
> 




Elsewhere, Weyl (1949a, 103) says: 



>  
> 
> As a matter of fact it can be shown that the metrical structure of 
> the world is already fully determined by its inertial and causal 
> structure, that therefore mensuration need not depend on clocks and 
> rigid bodies but that light signals and mass points moving under the 
> influence of inertia alone will suffice.
> 
> 
> 




The use of clocks and rigid rods is, within the context of either 
theory, an undesirable makeshift for two reasons. First, since 
neither spatial nor temporal intervals are invariants of the 
four-dimensional spacetime of the special theory of relativity and 
the general theory of relativity, the invariant spacetime interval 
\(ds\) cannot be directly ascertained by means of standard clocks 
and rigid rods. Second, the concepts of a rigid body and a periodic 
system (such as pendulums or atomic clocks) are not fundamental or 
theoretically self-sufficient, but involve assumptions that 
presuppose quantum theoretical principles for their justification and
thus lie outside the present conceptual relativistic framework. 
Therefore, methodological and ontological considerations decidedly 
favor Weyl's causal-inertial method for determining the spacetime 
metric. 




From the physical point of view, Weyl emphasized the roles of light 
propagation and free (fall) motion in revealing the conformal-causal 
and the projective structures respectively. However, from the 
mathematical point of view, Weyl did not use these two structures 
directly in order to derive from them and their compatibility 
relation, the metric field. Rather, Weyl regarded the metric and 
affine structures as fundamental and showed that the conformal and 
the projective structures respectively arise from those structures by
mathematical abstraction. 



![figure](fig7.png)
Figure 7:
Weyl took the metric and affines structures as fundamental and showed
that the conformal and projective structures respectively arise from
them by mathematical abstraction.



#### 4.3.3 The Ehlers, Pirani, Schild Construction of the Causal-Inertial Method




Ehlers et al. (1972) generalized Weyl's causal-inertial method by 
deriving the metric field *directly* from the conformal and 
projective fields and derived a unique pseudo-Riemannian spacetime 
metric solely as a consequence of a set of natural, physically 
well-motivated, constructive, "geometry-free" axioms 
concerning the incidence and differential properties of light 
propagation and free (fall) motion. Ehlers, Pirani and Schild adopt 
Reichenbach's (1924) term, *constructive axiomatics* to 
describe the nature of their approach. The 
"geometry-free" axioms are propositions about a few 
general qualitative assumptions concerning free (fall) motion and 
light propagation that can be verified directly through experience in
a way that does not presuppose the full blown edifice of the general 
theory of relativity. From these axioms, the theoretical basis of the
theory is reconstructed step by step. 




The constructive axiomatic approach to spacetime structure is roughly
this: 


1. **Primitive Notions.** The constructive axiomatic 
approach is based on a triple of sets



\[
\langle M, \mathcal{P}, \mathcal{L}\rangle
\]



of objects corresponding respectively to the notions of *events, 
particle paths* and *light rays*, which are taken as 
primitive. The set \(M\) of events is assumed to have a Hausdorff 
topology with a countable basis in order to state local axioms 
through the use of such terms as "neighborhood". Members 
of the sets \(\mathcal{P} = \{P, Q, P\_{1}, Q\_{1}, \ldots \}\) and 
\(\mathcal{L} = \{ L, N, L\_{1}, N\_{1}, \ldots \}\) are subsets of 
\(M\) that represent the possible or actual paths of massive 
particles and light rays in spacetime.
2. **Differential Structure.** The differential structure 
is not presupposed; rather through the first few axioms a 
differential-manifold structure is introduced on the set of events 
\(M\) that is sufficient for the localization of events by means 
of local coordinates, such as radar coordinates. Once \(M\) is 
given a differential-manifold structure through the introduction of 
local radar coordinates by means of particles and light rays (such 
that any two radar coordinates are smoothly related to one another), 
one can do calculus on \(M\) and one may speak of tangent and 
direction spaces.




It is important to emphasize that the members of \(\mathcal{P}\)
represent possible or actual paths of *arbitrary* massive
particles that may have some internal structure such as higher order
gravitational and electromagnetic multipole moments and that may
therefore interact in complicated ways with various physical
fields. In order to constructively establish the projective structure
of spacetime, it is necessary to single out a subset of
\(\mathcal{P}\), namely \(\mathcal{P}\_{f}\), the set of possible or
actual paths of spherically symmetric, electrically neutral particles
(that is, the world lines of freely falling particles). However, the
set \(\mathcal{P}\_{f} \subset \mathcal{P}\), can be properly 
characterized only after a coordinate system (differential structure)
is available. *Consequently, one must employ* arbitrary *particles in the statement of those axioms that lead to the local 
differential structure of spacetime.*
3. **Second-Order Conformal Structure.** The *Law of 
Causality* asserts the existence of a unique first-order 
conformal structure on spacetime (27), that is, a field of 
infinitesimal light cones.




Only null one-directions are determined. Therefore no special choice 
of parameters along light rays is determined by this structure. The 
first-order conformal structure can be measured using only the local 
differential-topological structure. Moreover, by a purely 
mathematical process involving only differentiation, the first-order 
conformal structure determines a second-order conformal structure, 
namely, an equivalence class \(K\) of conformally related 
symmetric linear connections.
4. **Projective Structure.** The motions of freely falling 
particles governed by the guiding field reveal the geodesics of 
spacetime, that is, the geodesics corresponding to an equivalence 
class \(\Pi\) of projectively equivalent symmetric linear connections. 
Only geodesic one-directions are determined, that is, no special 
choice of parameters is involved in characterizing free fall motion.
5. **Compatibility between the Conformal and Projective 
Structures.** That the conformal and projective structures are
compatible is suggested by high energy experiments, according to 
Ehlers, Pirani and Schild: "A massive particle \((m \gt 0)\), 
though always slower than a photon, can be made to chase a photon 
arbitrarily closely." Ehlers, Pirani and Schild 
therefore assume an axiom of compatibility between the conformal and 
projective structures, and this leads to a *Weyl space*: If 
the projective and conformal structures are compatible, then the 
intersection



\[
\Pi \cap K = \Gamma\ \text{ (Weyl connection)}
\]



of the equivalence class \(K\) of conformally equivalent symmetric
linear connections, and the equivalence class \(\Pi\) of projectively 
equivalent symmetric linear connections, contains a unique symmetric 
linear connection, a *Weyl connection*. Thus light propagation
and free (fall) motion reveal on spacetime a unique Weyl connection 
which determines the parallel transport of vectors, preserving their 
timelike, null or spacelike character, and for any pair of non-null 
vectors, the Weyl connection leaves invariant the ratio of their 
lengths and the angle between them, provided the vectors are 
transported along the same path.
6. **Pseudo-Riemannian Metric.** Since length transfer is 
non-integrable (i.e., path-dependent) in a Weyl space, a Weyl 
geometry reduces to a pseudo-Riemannian geometry if and only if 
Weyl's *length-curvature* 
(*Streckenkrummung*) tensor equals 
zero, in which case the length of a vector is path-independent under 
parallel transport, and there exists no second clock effect.


#### 4.3.4 The Philosophical Significance of the Causal-Inertial Method




Can it be argued that Ehlers, Pirani and Schild's generalization of 
Weyl's causal-inertial method for determining the spacetime metric 
constitutes a convention-free, and - in relevant respects 
- theory-independent body of evidence that can adjudicate 
between spacetime geometries, and hence between spacetime theories 
that postulate them? As Weyl showed, we can empirically determine the
metric field, provided certain epistemic conditions are satisfied, 
that is, provided we can measure the conformal-causal structure, and 
provided "we are able to recognize and observe as such the 
motion of free mass points which follow the guiding field." 
Criticisms of Ehlers, Pirani and Schild's constructive axiomatics 
suggest that the causal-inertial method is not convention-free and 
that it is ineffective epistemologically in providing a possible 
solution to the controversy between geometrical realism and 
conventionalism in favor of realism. Basically, all of the charges 
laid against Ehlers, Pirani and Schild's constructive axiomatics 
concentrate on the roles which massive particles play in their 
construction. One of the constructive axioms employed by Ehlers, 
Pirani and Schild, the projective axiom, is a statement of the 
infinitesimal version of the Law of Inertia, the law of free (fall) 
motion which contains Newton's first law of motion as a special case 
in the absence of gravitation. Since Ehlers, Pirani and Schild do not
provide an independent, non-circular criterion by which to 
characterize free (fall) motion, their approach has been charged with
circularity by philosophers such as Grunbaum (1973), Salmon 
(1977), Sklar (1977) and Winnie (1977). 




The problem is a familiar one; how to introduce a class of preferred
motions, that is, how to characterize that particular path structure
that would govern the motions of free particles \(\mathcal{P}\_{f}\),
that is, neutral, spherically symmetric, non-rotating test bodies,
while avoiding the circularity problem surrounding the notion of
a *free particle*: The only way of knowing when no forces act
on a body is by observing that it moves as a free particle along the
geodesics of spacetime. But how, without already knowing the geodesics
or the projective structure of spacetime is it possible to determine
which particles are free and which are not? And to determine the
projective structure of spacetime it is necessary to use free
particles. 




Coleman and Korte (1980) have addressed these and related 
difficulties by providing a non-conventional procedure for the 
empirical determination of the projective 
 structure.[64]





It is worth emphasizing that Weyl's approach to differential 
geometry, in which the affine, projective and conformal structures 
are treated in their own right rather than as mere aspects of the 
metric, was instrumental for his discovery of the non-circular and 
non-conventional geodesic method for the empirical determination of 
the spacetimet metric. The old notion of a 'geodesic 
path' had its inception in the context of classical 
*metrical* geometry and 'geodesicity' was 
characterized in terms of *extremal* paths of curves, which 
presupposed a metric. It was Weyl's metric-independent construction 
of the symmetric linear connection that led him to introduce the 
geometry of paths and the metric-independent characterization of a 
*geodesic* path in terms of the process of autoparallelism of 
its tangent direction. 


### 4.4 The Laws of Motion, Mach's Principle, and Weyl's Cosmological Postulate




Weyl provided a general conceptual/mathematical clarification of the 
concept of motion that applies to any spacetime theory that is based 
on a differential manifold. In particular, Weyl's penetrating 
analysis shows that Einstein's understanding of the role and 
significance of Mach's Principle for the general theory of relativity
and cosmology is actually inconsistent with the basic principles of 
general relativity. 




Weyl's major contribution to cosmology is known as "Weyl's 
Hypothesis". The name was coined by Weyl (1926d) himself in an 
article in the *Encyclopedia of*
*Britannica*.[65]
 According to Weyl's Postulate, the worldlines of all galaxies are 
non-intersecting diverging geodesics that have a common origin in the
distant past. From this system of worldlines Weyl derived a common 
cosmic time. On the basis of his postulate, Weyl (1923c, Appendix 
III) was also the first to show that there is an approximately 
*linear* relation between the redshift of galactic spectra and
distance. Weyl had basically discovered *Hubble's Law* six 
years prior to Hubble's formulation of it in 1929. Another 
contribution to cosmology is Weyl's (1919b) spherically symmetric 
static exact solution to Einstein's 
 linearized[66]
 field equations. 


#### 4.4.1 The Laws of Motion and Mach's Principle




There are essentially two ways to understand Mach's Principle: (1) 
Mach's Principle rejects the absolute character of the inertial 
structure of spacetime, and (2) Mach's Principle rejects the inertial
structure of spacetime *per se*. Version (2) might be 
characterized as *Leibnizian relativity* or *body 
relationalism*; that is, one understands by relative motion the 
motion of *bodies* with respect only to other observable 
*bodies* or observable *bodily* reference frames. The 
relative motion of a body with respect to absolute space or to the 
inertial structure of space (Newton) or spacetime is ruled out on 
epistemological and/or metaphysical grounds. 




In the context of his general theory of relativity, what Einstein is
objecting to in Newtonian Mechanics, and by implication, the theory of
special relativity, is the *absolute* character of the inertial
structure; he is not asserting its *fictitious* character. That
is, the general theory of relativity incorporates Mach's
Principle as expressed in version (1) by treating the inertial
structure as dynamical and not as absolute. 




However, Einstein also tried to extend and generalize the special
theory of relativity by incorporating version (2) of Mach's
Principle into the general theory of relativity. Einstein was deeply
influenced by Mach's empiricist programme and accepted
Mach's insistence on the primacy of observable facts of
experience: *only observable facts of experience may be invoked to
account for the phenomena of motion*. As a consequence, Einstein
restricted the concept of relative motions to relative motions between
bodies. Newton thought that the plane of Foucault's pendulum
remains aligned with respect to absolute space. Since the fixed stars
are at rest with respect to absolute space the plane of
Foucault's pendulum remains aligned to them as well, and rotates
relative to the earth. But according to Einstein, Newton's
intermediary notion of absolute space is as questionable as it is
unnecessary in explaining the behaviour of Foucault's
pendulum. Not absolute space, but the actually existing masses of the
fixed stars of the whole cosmos guide the plane of Foucault's
pendulum. 




Einstein (1916) argued that the general theory of relativity removes 
from the special theory of relativity and Newton's theory an inherent
epistemological defect. The latter is brought to light by Mach's 
paradox, namely, Einstein's example of two fluid bodies, \(A\) and
\(B\), which are in constant *relative* rotation about a 
common axis. With regard to the extent to which each of the spheres 
bulges at its equator, infinitely many different states are possible 
although the relative rotation of the two bodies is the same in every
case. Einstein considered the case in which \(A\) is a sphere and 
\(B\) is an oblate spheroid. The paradox consists in the fact that
there is no readily discernible reason that accounts for the fact 
that one of the bodies bulges and the other does not. According to 
Einstein, an epistemological satisfactory solution to this paradox 
must be based on 'an observable fact of experience'. 
Einstein wanted to implement a Leibnizian-Machian *relational*conception of motion according to which all motion is to be 
interpreted as the motion of some bodies in relation to other bodies.
Einstein wished to extend the body-relative concept of uniform 
inertial motion to the concept of a body-relative accelerated motion.



#### 4.4.2 Weyl's Critique of Einstein's Machian Ideas




Weyl was very critical of Einstein's attempt to incorporate version 
(2) of Mach's Principle into the theory of general relativity and 
relativistic cosmology because he considered the Leibnizian-Machian 
*relational* conception of motion--according to which all
motion is to be interpreted as the motion of some bodies in relation 
to other bodies--to be an incoherent notion within the context 
of the general theory of relativity. 




In a paper entitled *Massentragheit und 
Kosmos. Ein Dialog* [*Inertial Mass and Cosmos. A 
Dialogue*] Weyl (1924b) articulates his overall position on the 
concept of motion and the role of Mach's Principle in general 
relativity and 
 cosmology.[67]
 Weyl defines Mach's Principle as follows: 



**M (Mach's Principle):**  

The inertia of a body is determined through the interaction of all the masses in the universe.





Weyl (1924b) then makes the observation that the *kinematic 
principle of relative motion* is by itself without any content, 
unless one also makes the additional physical causal assumption that 



**C (Physical Causality):**  

All events or processes are uniquely causally determined through matter, that is, through charge, mass and the state of motion of the elementary particles of 
 matter.[68]





The underlying motivation for assumption \(\mathbf{C}\) of physical
causality is essentially Mach's empiricist programme, namely,
Mach's insistence on the *primacy of observable facts of
experience*. Addressing Einstein's formulation of
Mach's paradox, Weyl (1924b) says: Only if we conjoin
the *kinematic principle of relative motion* with the physical
assumption \(\mathbf{C}\) does it appear groundless or impossible on
the basis of the kinematic principle that in the absence of any
external forces a stationary body of fluid has the form of a sphere
"at rest", while on the other hand it has the form of a
"rotating" flattened ellipsoid. Weyl rejects principle
\(\mathbf{C}\) of physical causality because he denies the feasibility
of \(\mathbf{M}\) (Mach's Principle), as defined above, on \(a\)
 *priori*[69]
 grounds. According to Weyl (1924b) 



**A:**  

The concept of relative motion of several isolated bodies with respect
to each other is as untenable according to the theory of general
relativity as is the concept of absolute motion of a single body.





Weyl notes that what we seem to observe as the rotation of the stars,
is in reality not the rotation of the stars themselves but the 
rotation of the "star compass" [Sternenkompass] which 
consists of light signals from the stars that meet our eyes at our 
present location from a certain direction. It is crucial, Weyl 
reminds us, to be cognisant of the existence of the metric field 
between the stars and our eyes. This metric field determines the 
propagation of light, and, like the electromagnetic field, it is 
capable of change and variation. Weyl (1924b) says that "the 
metric field is no less important for the direction in which I see 
the star then is the location of the star itself." How is it 
possible, Weyl asks, to compare within the context of the general 
theory of relativity, the state of motions of two separate bodies? Of
course, Weyl notes, prior to the general theory of relativity, during
Mach's time, one could rely on a rigid frame of reference such as the
earth, and indefinitely extend such a frame throughout space. One 
could then postulate the relative motion of the stars with respect to
this frame. However, under the hands of Einstein the coordinate 
system has lost its rigidity to such a degree, that it can always 
"cling to the motion of all bodies simultaneously"; that 
is, whatever the motions of the bodies are, there exists a coordinate
system such that all bodies are at rest with respect to that 
coordinate system. Weyl then clarifies and illustrates the above with
the plasticine example, which Weyl (1949a, 105) elsewhere describes 
as follows: 



>  
> 
> Incidentally, without a world structure the concept of relative 
> motion of several bodies has, as the postulate of general relativity 
> shows, no more foundation than the concept of absolute motion of a 
> single body. Let us imagine the four-dimensional world as a mass of 
> plasticine traversed by individual fibers, the world lines of the 
> material particles. Except for the condition that no two world lines 
> intersect, their pattern may be arbitrarily given. The plasticine can
> then be continuously deformed so that not only one but all fibers 
> become vertical straight lines. Thus no solution of the problem is 
> possible as long as in adherence to the tendencies of Huyghens and 
> Mach one disregards the structure of the world. But once the inertial
> structure of the world is accepted as the cause for the dynamical 
> inequivalence of motions, we recognize clearly why the situation 
> appeared so unsatisfactory. ... Hence the solution is attained 
> as soon as we dare to *acknowledge the inertial structure as a 
> real thing that not only exerts effects upon matter but in*
> *turn suffers such effects.*
> 
> 
> 



![figure](fig8.png)
Figure 8:
Weyl's plasticine example





Applying these considerations to the fixed stars and assuming that it
is possible that the (conformal) metrical field which determines the 
cones of light propagation (light cones) at each point of the 
plasticine, is carried along by the continuous transformation of the 
plasticine, then both the earth and the fixed stars will be at rest 
with respect to the plasticine's coordinate system. Yet despite this 
the "star compass" is rotating with respect to the earth,
exactly as we observe! 




Employing the concept of the microsymmetry group (definition 4.1), 
Coleman and Korte (1982) have analyzed Weyl's plasticine 
example in the following way: Consider a space-time manifold equipped
only with a differentiable structure, the plasticine of Weyl's 
example. Then our spacetime does not have an affine, conformal, 
projective or metric structure defined on it. In such a world it is 
possible do define curves and paths; however, there are no preferred 
curves or paths. Since there is only the differentiable structure, 
one may apply any diffeomorphism; that is, *all*diffeomorphisms preserve this structure; consequently, in the 
absence of a post-differential-topological structure, the 
microsymmetry group at any event \(p\) is an infinite-parameter 
group isomorphic to the group of all invertible formal power series 
in four variables. If there is no post-differentiable topological 
geometric field in the neighbourhood of a point, then all of these 
infinite parameters may be chosen freely within rather broad limits. 
Clearly then, given an infinite number of parameters, one can, as 
Weyl says, straighten out an arbitrary pattern of world lines 
(fibers) in the neighbourhood of any event. Now suppose that there 
exists a post-differentiable topological geometric field, namely, the
projective structure at any event of spacetime. Then the 
microsymmetry group that preserves that structure is a 20-parameter 
Lie group (see Coleman and Korte (1981)). Thus instead of an 
infinity of degrees of freedom, only twenty degrees of freedom may be
used to actively deform the neighbouring region of spacetime. The 
fact that only a finite number of parameter are available prevents an
arbitrary realignment of the worldlines of material bodies in the 
neighbourhood of any given event. 




Other post-differential topological geometrical field structures are 
similarly restrictive. For example, the microsymmetry group of the 
conformal structure, which determines the causal structure of 
spacetime, permits 7 degrees of freedom (6 Lorentz transformations 
and a dilatation), and permits four more degrees of freedom in second 
order. Consequently, the existence of the conformal metrical field 
which determines at each point the cones of light propagation would 
prevent an arbitrary realignment of light-like fibers, that is, it 
would be impossible to realign the earth and the fixed stars such 
that both are at rest with the coordinate system of the plasticine. 




Weyl's plasticine example shows that the Leibnizian-Machian view of 
relative motion, namely the view according to which all motion must 
be defined as motion relative to bodies, is self-defeating in the 
general theory of relativity. The fact that a stationary, homogeneous
elastic sphere will, when set in rotation, bulge at the equator and 
flatten at the poles is, according to Weyl (1924b), to be accounted 
for in the following way. The complete physical system consisting of 
both the body and the local inertial-gravitational field is not the 
same in the two situations. The cause of the effect is the state of 
motion of the body *with respect to* the local 
inertial-gravitational field, the guiding field, and is not, indeed 
as Weyl's plasticine example shows, cannot be the state of motion of 
the body relative to other bodies. To attribute the effect as 
Einstein and Mach did to the rotation of the body with respect to the
other *bodies* in the universe is, according to Weyl, to 
endorse a remnant of the unjustified monopoly of the older body 
ontology, namely, the sovereign right of material bodies to play the 
role of physically real and acceptable causal 
 agents.[70]



#### 4.4.3 Coordinate Transformation Laws of Acceleration




Weyl's view that there must be an *inertial structure field*on spacetime, which governs material bodies in *free*
*motion*, follows from the mathematical nature of the 
coordinate-transformation laws for acceleration. In a world equipped 
with only a differential structure, it is possible to do calculus; 
one can define curves and paths and differentiate, etc. However, as 
was already pointed out, in such a world, the world of Weyl's 
plasticine example, there would be no preferred curves or paths. 
Consequently, *the motion of material bodies would not be
predictable*. However, experience overwhelmingly indicates that 
the acceleration of a massive body cannot be freely chosen. In 
particular, consider a simple type of particle, a monopole 
(unstructured) particle. Experience overwhelmingly tells us that such
a particle is characterized by the fact that at any event on its 
world line, its velocity at that event is sufficient to determine its
acceleration at that event. Predictability of motion, therefore, 
entails that corresponding to every type of massive monopole, there 
exists a geometric structure field, or what Weyl calls a 
*Strukturfeld* that governs the motion of that type of 
particle. The basic reason which explains this brute fact of 
experience is a simple mathematical fact about how the acceleration 
of bodies transforms under a coordinate transformation. Moreover, 
this simple mathematical fact, involving no more than the basic 
techniques of partial differentiation, holds in all relativistic, 
non-relativistic, curved or flat, dynamic or non-dynamic spacetime 
theories that are based on a local differential topological 
structure, the minimal structure required for the possibility of 
assigning arbitrary local coordinates on a differential manifold. 



**Transformation law for acceleration:**  

The transformation law for acceleration is linear, but is *not*
homogeneous in the acceleration variable.




As an example consider the transformation laws for the 4-velocity and
the 4-acceleration. Recall that a *curve* in the
four-dimensional spacetime manifold \(M\) is a map \(\gamma :
\mathbb{R} \rightarrow M\). For convenience we restrict our attention
to those curves which satisfy \(\gamma(0) = p\). If we set
\(\gamma^{i} = x^{i} \circ \gamma(0)\), then the components of the
4-velocity and 4-acceleration at \(p \in M\) are respectively given
by 



\[\tag{30}
\gamma^i\_1 =\_{def} \frac{d}{dt}\gamma^i(0),
\]

\[\tag{31}
\gamma^i\_2 =\_{def} \frac{d^2}{dt^2}\gamma^i(0), 
\]



The transformation laws of the 4-velocity components
\(\gamma^{i}\_{1}\) and of the 4-acceleration components
\(\gamma^{i}\_{2}\) under a change of coordinate chart from
\((U,x)\_{p}\) to \((\overline{U}, \overline{x})\_p\), follow from their
pointwise definition. From 



\[
\overline{\gamma}^i(t) = \overline{X}^i(\gamma^i(t)),
\]



where \(\overline{X}^{i} = \overline{x}^i \circ x^{- 1}\), 
one obtains the transformation law for the 4-velocity and the
4-acceleration respectively: 



\[\begin{align}
\tag{32}
\overline{\gamma}^i\_1 &= \overline{X}^i\_j \gamma^{\,j}\_1 \\

\tag{33}
\overline{\gamma}^i\_2 &= \overline{X}^i\_j \gamma^{\,j}\_2
 + \overline{X}^i\_{jk} \gamma^{\,j}\_1 \gamma^k\_1.
\end{align}\]



The \(\overline{X}^{i}\_{j}\) and \(\overline{X}^{i}\_{jk}\) denote the
first and second partial derivatives of \(\overline{X}^{i}(x^{i})\) at
\(x^{i} (p)\), namely,



\[
\frac{\partial \overline{x}^i}{\partial x^{j}}
 \text{ and }
\frac{\partial \overline{x}^i}{\partial x^{j} \partial x^{k} }
\]



The expression \(\overline{X}^{i}\_{jk}\gamma^{\,j}\_{1}\gamma^{k}\_{1}\) in 
equation (33) represents the inhomogeneous term of the transformation
of the 4-acceleration. The inhomogeneity of the transformation law 
entails that a 4-acceleration that is zero with respect to one 
coordinate system is not zero with respect to another coordinate 
system. This means that there does not exist a unique standard of 
zero 4-acceleration that is intrinsic to the differential topological
structure of spacetime. Moreover, even the difference of the 
4-accelerations of two bodies at the same spacetime point has no 
absolute meaning, unless their 4-velocities happen to be the same. 
This shows that while the differential topological structure of 
spacetime gives us sufficient structure to do calculus and to derive 
the transformation laws for 4-velocities and 4-accelerations by way 
of simple differentiation, it does not provide sufficient structure 
with which to determine a standard of zero 4-acceleration. Therefore,
as Weyl repeatedly emphasized, no solution to the problem of motion 
is possible, unless "we dare to *acknowledge the 
inertial structure as a real thing that not only exerts 
effects upon matter but in turn suffers such effects*". In 
other words there must exist a structure in addition to the 
differential topological structure in the form of a geometric 
structure field, or in Weyl's words, *geometrisches 
Strukturfeld*, which constitutes the inertial structure of 
spacetime, and which provides the standard of zero 4-acceleration. 
Since this field provides the standard of zero 4-acceleration we can 
call it a geodesic 4-acceleration field, or simply, geodesic 
acceleration field. A particle in *free motion* is one that is
exclusively governed by this geodesic acceleration field. 




An acceleration field, geodesic or non-geodesic, can be constructed 
in the following way. Since the terms that are independent of the 
4-acceleration depend on both the spacetime location and on the 
corresponding 4-velocity of the particle, it is necessary to specify 
a *geometric field standard* for zero 4-acceleration that also
depends on those independent variables. 




The transformation law for a 4-acceleration field can be obtained 
from (33) by replacing 
\(\overline{\gamma}^{i}\_{2}\) by 
\(\overline{A}^{i}\_{2}(\overline{x}^{i},\overline{\gamma}^{i}\_{1})\)
and \(\gamma^{j}\_{2}\) by \(A^{j}\_{2}(x^{i}, \gamma^{i}\_{1}\)) to yield



\[
\overline{A}^{i}\_{2}(\overline{x}^{i}, \overline{\gamma}^{i}\_{1}) =
 \overline{X}^{i}\_{j} A^{j}\_{2}(x^{i},\gamma^{i}\_{1}) +
 \overline{X}^{i}\_{jk}\gamma^{j}\_{1}\gamma^{k}\_{1}.
\]



The important special case for which the function \(A^{i}\_{2}(x^{i}, 
\gamma^{i}\_{1})\) is a geodesic 4-acceleration field 
corresponds to the affine structure of spacetime. For this special 
case the function \(A^{i}\_{2}(x^{i}, \gamma^{i}\_{1}\)) 
is denoted by \(\Gamma^{i}\_{2}(x^{i}, \gamma^{i}\_{1}\)) and is given
by 



\[
\Gamma^{i}\_{2}(x^{i},\gamma^{i}\_{1}) = -\Gamma^{i}\_{jk}(x^{i},\gamma^{i}\_{1})
\gamma^{j}\_{1}\gamma^{k}\_{1}.
\]



The familiar transformation law for the affine structure (geodesic 
4-acceleration field) is then given by 



\[\tag{34}
\overline{\Gamma}^{i}\_{2}(\overline{x}^{1},\overline{\gamma}^{i}\_{1}) 
 = \overline{X}^{i}\_{j}\Gamma^{j}\_{2}(x^{i},\gamma^{i}\_{1}) 
 +\overline{X}^{i}\_{jk}\gamma^{j}\_{1}\gamma^{k}\_{1}.
\]



Note that the inhomogeneous term 
\(\overline{X}^{i}\_{jk}\gamma^{j}\_{1}\gamma^{k}\_{1}\) of the 
geodesic 4-acceleration field is identical to the inhomogeneous term 
of the transformation law (33) for the 4-acceleration of body motion.
The differences 



\[\tag{35}
\overline{\gamma}^{i}\_{2} - \overline{\Gamma}^{i}\_{2}(\overline{x}^{i}, 
\overline{\gamma}^{i}\_{1}) = \overline{X}^{i}\_{j}(\gamma^{j}\_{2} 
 - \Gamma^{j}\_{2}(x^{i},\gamma^{i}\_{1}))
\]



then transform linearly and homogeneously; consequently, the 
vanishing or non-vanishing of body accelerations relative to the 
standard of zero acceleration provided by the geodesic 4-acceleration
field (the affine structure), is coordinate independent. That is, the
4-accelerations of bodies and the corresponding 4-forces, are 
tensorial quantities in concordance with experience. 




The above argument for the necessity of geometric fields also holds
for 3-velocity and 3-acceleration, denoted respectively by
\(\xi^{\alpha}\_{1}\) and \(\xi^{\alpha}\_{2}\). The transformation law
for the 3-acceleration is much more complicated than that of the
4-acceleration. However, analogous to the case of 4-acceleration, the
transformation law of 3-acceleration is linear and is inhomogeneous in
the 3-acceleration variable \(\xi^{\alpha}\_{2}\). Consequently, there
does not exist a unique standard of zero 3-acceleration that is
intrinsic to the differential topological structure of spacetime. The
standard of zero 3-acceleration must be provided by a geodesic
3-acceleration field or geodesic directing field, or what Weyl calls
the guiding field. The guiding field is also referred to as the
projective structure of spacetime and is denoted by
\(\Pi^{\alpha}\_{2}(x^{i}, \xi^{\alpha}\_{1}\)). It is a function of
spacetime location and the 3-velocity, both variables of which are
independent of the 3-acceleration, as is required. Since the
transformation law of the projective structure
\(\Pi^{\alpha}\_{2}(x^{i}, \xi^{\alpha}\_{1}\)) has the same
inhomogeneous form as the 3-acceleration \(\xi^{\alpha}\_{2}\), the
difference 



\[\tag{36}
\xi^{\alpha}\_{2} - \Pi^{\alpha}\_{2}(x^{i}, \xi^{\alpha}\_{1})
\]



also transforms linearly and homogeneously. 




The components \(\gamma^{i}\_{2}\) and \(\xi^{\alpha}\_{2}\) of the 
4-acceleration and 3-acceleration can be thought of as the dynamic 
descriptors of a material body. On the other hand, the components 
\(\Gamma^{i}\_{2}(x^{i},\gamma^{i}\_{1}\)) and 
\(\Pi^{\alpha}\_{2}(x^{i}, \xi^{\alpha}\_{1}\)) of the 
geodesic acceleration field, and the geodesic directing field, 
respectively, are *field* quantities. The differences 



\[\tag{37}
\gamma^{i}\_{2} - \Gamma^{i}\_{2}(x^{i},\gamma^{i}\_{1})
\]



and 



\[\tag{38}
\xi^{\alpha}\_{2} -\Pi^{\alpha}\_{2}(x^{i}, \xi^{\alpha}\_{1})
\]



denote the components of a coordinate independent *field-body*
*relation*.[71]





Weyl (1924b) remarks: 



>  
> 
> We have known since Galileo and Newton, that the motion of a body 
> involves an inherent struggle between inertia and force. According to
> the old view, the inertial tendency of persistence, the 
> "guidance", which gives a body its natural inertial 
> motion, is based on a formal geometric structure of the spacetime 
> (uniform motion in a straight line) which resides once and for all in
> spacetime independently of any natural processes. This assumption 
> Einstein rejects; because whatever exerts as powerful effects as 
> inertia--for example, in opposition to the molecular forces of 
> two colliding trains it rips apart their freight cars--must be 
> something real which itself suffers effect from matter. Moreover, 
> Einstein recognized that the guiding field's variability and 
> dependence on matter is revealed in gravitational effects. Therefore,
> the dualism between guidance and force is maintained; but 
> 
> 
> 
> **(G)** *Guidance is a physical field, like the 
> electromagnetic field, which stands in mutual interaction*
> *with matter. Gravitation belongs to the guiding field and not to 
> force.* Only thus is it possible to explain the equivalence 
> between inertial and gravitational mass.
> 
> 
> 




To move from the old conception to the new conception (G) means, 
according to Weyl (1924b) 



>  
> *to replace the geometric difference between uniform and 
> accelerated motion with the dynamic difference between 
> guidance and force.* Opponents of Einstein asked the question: 
> Since the church tower receives a jolt in its motion relative to the 
> train just as the train receives a jolt in its motion relative to the
> church tower, why does the train become a wreckage and not the church
> tower which it passes? Common sense would answer: because the train 
> is ripped out of the pathway of the guiding field, but the church 
> tower is not. ... As long as one ignores the guiding field one 
> can neither speak of absolute nor of relative motion; only if one 
> gives due consideration to the guiding field does the concept of 
> motion acquire content. The theory of relativity, correctly 
> understood, does not eliminate absolute motion in favour of relative 
> motion, rather it eliminates the kinematic concept of motion and 
> replaces it with a dynamic one. The worldview for which Galileo 
> fought is not undermined by it [relativity]; to the contrary, it is 
> more concretely interpreted.
> 
> 
> 


#### 4.4.4 Weyl's Field-Body Relationist Ontology and Newton's Laws of Motion




It is now possible to provide a reformulation of Newton's laws of 
motion which explicitly takes account of Weyl's 
field-body-relationalist spacetime ontology, and his analysis of the 
concept of motion. The law of inertia is an empirically verifiable 
 statement[72]
 which says 



>  
> **The Law of Inertia:** There exists on spacetime a 
> unique projective structure \(\Pi\_{2}\) or equivalently, a 
> unique geodesic directing field \(\Pi\_{2}\).
> 
> 
> 



*Free motion* is defined with reference to the projective 
structure \(\Pi\_{2}\) as follows: 



>  
> **Definition of Free Motion:** A possible or actual 
> material body is in a state of free motion during any part of its 
> history just in case its motion is exclusively governed by the 
> geodesic directing field (projective structure), that is, just in 
> case the corresponding segment of its world path is a solution path 
> of the differential equation determined by the unique projective 
> structure of spacetime.
> 
> 
> 




Newton's second law of motion may be reformulated as follows: 



>  
> **The Law of Motion:** With respect to any coordinate 
> system, the world line path of a possible or actual material body 
> satisfies an equation of the form 
> 
> 
> 
> \[
> m(\xi^{\alpha}\_{2} - \Pi^{\alpha}\_{2}(x^{i}, \xi^{\alpha}\_{1})) 
>  = F^{\alpha}(x^{i},\xi^{\alpha}\_{1}),
> \]
> 
> 
> 
> where \(m\) is a scalar constant characteristic of the material 
> body called its inertial mass, and 
> \(F^{\alpha}(x^{i},\xi^{\alpha}\_{1}\)) is the 
> 3-force acting on the body.
> 
> 
> 




To emphasize, the Law of Inertia and the Law of Motion, as formulated
above, apply to *all*, relativistic or non-relativistic, 
curved or flat, dynamic or non-dynamic, spacetime theories. The 
reason for the general character of these laws consists in the fact 
that they require for their formulation only the local differential 
topological structure of spacetime, a structure which is common to 
all spacetime theories. In addition, as was noted earlier in 
SS4.2, the affine and projective spacetime structures are 
G-structures. Consequently, they may be flat or non-flat; *but 
they can never vanish*. In theories prior to the advent of 
general relativity, the affine and projective structures were flat. 
It was common practice, however, to use coordinate systems that were
*adapted* to these flat G-structures. And since in such 
adapted coordinate systems the components of the affine and 
projective structures vanish, it was difficult to recognize and to 
appreciate the existence of these structures, and their important role
in providing a coherent account of motion. 


#### 4.4.5 Mie's Pure Field Theory, Weyl's 'Agens Theory' and Wormhole Theory of Matter




We saw that Weyl forcefully advocated a *field-body ontological 
dualism*, according to which matter and the guiding field are 
independent physical realities that causally interact with each 
other: matter uniquely generates the various states of the guiding 
field, and the guiding field in turn acts on matter. 




Weyl did not always subscribe to this ontological dualist position. 
For a short period, from 1918 to 1920, he advocated a *pure field 
theory of matter* , developed in 1912 by Gustav Mie, in the context
of Einstein's special theory of relativity: 



**Pure Field Theory of Matter:**  

The physical field has an independent reality that is not reducible to
matter; rather, the physical field is constitutive of all matter in
the sense that the mass (quantity of matter) of a material particle,
such as an electron, consists of a large field energy that is
concentrated in a very small region of spacetime.




Mie's theory of matter is akin to the traditional *geometric
view* of matter: matter is *passive* and *pure*
*extension*. Weyl (1921b) remarks that he adopted the 
standpoint of the classical pure field theory of matter in the first 
three editions of Weyl (1923b) because of its beauty and unity, but 
then gave it up. Weyl (1931a) points out in the Rouse Ball Lecture 
that since the theory of general relativity geometrized a physical 
entity, the gravitational field, it was natural to try to 
*geometrize the whole of physics*. Prior to the advent of 
quantum physics one was justified in regarding gravitation and 
electromagnetism as the only basic entities of nature and to seek 
their unification by geometrizing both. One could hope, following the
example of Gustav Mie, to construct elementary material particles as 
*knots of energy* in the gravitational-electromagnetic field, 
that is, tiny demarcated regions in which the field magnitudes attain
very high values. 




Already in a letter to Felix 
 Klein,[73]
 toward the end of 1920, Weyl indicated that he had finally freed 
himself completely from Mie's theory of matter. It now appeared to 
him that the classical field theory of matter is not the key to 
reality. In the Rouse Ball Lecture Weyl adduces two reasons for this.





First, due to quantum mechanics, there are, in addition to 
electromagnetic waves, *matter waves* (*Materiewellen*)
represented by Schrodinger's wave function \(\psi\). And Pauli and 
Dirac recognized that \(\psi\) is not a scalar but a magnitude with 
several components. Thus, from the point of view of the classical 
field theory of matter not two but three entities would have to be 
unified. Moreover, given the transformation properties of the wave 
function, Weyl says it is certain that the magnitude \(\psi\) cannot be 
reduced to gravitation or electromagnetism. Weyl saw clearly, that 
this *geometric view* of matter or physics--which to a 
certain extent had also motivated his earlier construction and manner
of presentation of a *pure infinitesimal geometry*--was
untenable in light of the new developments in atomic physics. 




The second reason, Weyl says, consists in the radical new 
interpretation of the wave function, which replaces the concept of 
intensity with that of probability. It is only through such a 
statistical interpretation that the corpuscular and atomistic aspect 
of nature is properly recognized. Instead of a *geometric*treatment of the classical field theory of matter, the new 
quantum theory called for a *statistical* treatment of 
 matter.[74]
 Already in 1920, Weyl (1920) addressed the relationship between 
*causal* and *statistical* approaches to 
 physics.[75]





The theory of general relativity, as well as early developments in 
atomic physics, clearly tell us, Weyl (1921b) suggests, that matter 
uniquely determines the field, and that there exist deeper underlying
physical laws with which modern physics, such as quantum theory, is 
concerned, which specify "how the field is affected by 
matter". That is, experience tells us that matter plays the 
role of a *causal agent* which uniquely determines the field, 
and which therefore has an independent physical reality that cannot 
be reduced to the field on which it acts. Weyl (1921b, 1924e) refers 
to his theory of matter as the *Agenstheorie der Materie*(literally, *agent-theory of matter*): 



**Matter-Field Dualism (Weyl's Agens Theory of Matter):**  

Matter and field are independent physical realities that causally
interact with each other: matter uniquely generates the various states
of the field, and the field in turn acts on matter. To excite the field
is the essential primary function of matter. The field's function is
to respond to the action of matter and is thus secondary. The
secondary role of the field is to transmit effects (from body to body)
caused by matter, thereby in return affecting matter.




The view, that *matter uniquely determines the field*, 
was a necessary postulate of an opposing 
ontological standpoint, according to Weyl. The postulate essentially says that *Matter is the only thing which is 
genuinely real*. According to this ontological view, held to a certain degree
by the younger Einstein and others who advocated a form of Machian 
empiricism, the field is relegated to play the role of a feeble 
extensive medium which transmits effects from body to 
 body.[76]
According to this opposing ontological view, the field laws, that is, certain implicit differential 
connections between the various possible states of the field, on the 
basis of which the field alone is capable of transmitting effects 
caused by matter, can essentially have no more significance for 
reality than the laws of geometry could, according to earlier views. 
But as we saw earlier, Weyl held that no satisfactory solution can be given to 
the problem of motion as long as we adhere to the Einstein-Machian 
empiricist position that relegates the field to the role of a feeble
extensive medium, and which does not acknowledge that the guiding 
field is physically real. However, from the standpoint of Weyl's *agens theory of 
matter*, a satisfactory answer to Mach's paradox can be given: 
the reason why a stationary, homogeneous elastic sphere will bulge at the equator and flatten at the poles, when 
set in rotation, is 
due to the fact that the complete physical system consisting of both 
the body *and* the guiding field, differs in the rotating case
from the stationary one. The local guiding field is the real cause of
the inertial forces. 




Weyl lists two reasons in support for his agens theory of matter. 
First, the agens theory of matter is the only theory which coheres 
with the basic experiences of life and physics: matter generates the 
field and all our actions ultimately involve matter. For example, 
only through matter can we change the field. Secondly, in order to 
understand the fact of the existence of charged material particles, 
we have two possibilities: either we follow Mie and adopt a pure 
field theory of matter, or we elevate the ontological status of 
matter and regard it as a *real* singularity of the field and 
not merely as a high concentration of field energy in a tiny region 
of spacetime. Since Mie's approach is necessarily limited to the 
framework of the theory of special relativity, and since there is no 
room in the general theory of relativity for a generalization and 
modification of the classical field laws, as envisaged by Mie in the 
context of the special theory of relativity, Weyl adopted the second 
possibility. He was motivated to do so by his recognition that the 
field equation of an electron at rest contains a finite mass term 
\(m\) that appears to have nothing to do with the energy of the 
associated field. Weyl's subsequent analysis of mass in terms of 
electromagnetic field energy provided a definition of mass and a 
derivation of the basic equations of mechanics, and led Weyl to the 
invention of the topological idea of *wormholes* in spacetime.
Weyl did not use the term 'wormholes'; it was John 
Wheeler who later coined the term 'wormhole' in 1957. 
Weyl spoke of *one-dimensional tubes* instead. 
"Inside" these tubes no space exists, and their 
boundaries are, analogous to infinite distance, inaccessible; they do
not belong to the field. In a chapter entitled "Hermann Weyl 
and the Unity of Knowledge" Wheeler (1994) says, 



>  
> 
> Another insight Weyl gave us on the nature of electricity is 
> topological in character and dates from 1924. We still do not know 
> how to assess it properly or how to fit it into the scheme of 
> physics, although with each passing decade it receives more 
> attention. The idea is simple. Wormholes thread through space as air 
> channels through Swiss cheese. Electricity is not electricity. 
> Electricity is electric lines of force trapped in the topology of 
> space.
> 
> 
> 


#### 4.4.6 Relativistic Cosmology and Weyl's Postulate




A year after Einstein (1916) had established the field equations of 
his new general theory of relativity, Einstein (1917) applied his 
theory for the first time to cosmology. In doing so, Einstein made 
several assumptions: 



**Cosmological Principle:**  
 Like Newton, Einstein 
assumed that the universe is *homogeneous* and 
*isotropic* in its distribution of matter.



**Static Universe:**  
 Einstein assumed, as did 
Newton and most cosmologists at that time, that the universe is 
static on the large scale.



**Mach's Principle:**  
 Einstein believed that the 
metric field is completely determined through the masses of bodies. 
The metric field is determined through the energy-momentum tensor of 
the field equations.




The *cosmological principle* continuous to play an important 
role in cosmological modelling to this day. However, Einstein's 
second assumption that the universe is static was in conflict with 
his field equations, which permitted models of the universe that were
homogeneous and isotropic, but *not static*. In this regard, 
Einstein's difficulties were essentially the same that Newton had 
faced: A static Newtonian model involving an infinite container with 
an infinite number of stars was unstable; that is, local regions 
would collapse under gravity. Because Einstein was committed to 
Mach's Principle he faced a problem concerning the *boundary 
conditions* for *infinite* space containing *finite*amount of 
 matter.[77]
 Einstein recognized that it was impossible to choose boundary 
conditions such that the ten potentials of the metric 
\(g\_{ij}\) are completely determined by the 
energy-momentum tensor \(T\_{ij}\), as required by 
Mach's Principle. That is, the boundary conditions "flat at 
infinity" entail a *global inertial frame* that
is tied to *empty flat space at infinity*, and hence is 
unrelated to the mass-energy content of space, contrary to Mach's 
Principle, according to which *only* mass-energy can influence
inertia. 




Einstein thought that he could solve the difficulties of an unstable 
non-static universe with boundary conditions at infinity that do not 
satisfy Mach's Principle, by introducing the cosmological term 
\(\Lambda\) into his field equations. He showed that for positive values
of the cosmological constant, his modified field equation admitted a 
solution for a 
 static[78]
 universe in which space is curved, unbounded and finite; that is, 
space is a hyper surface of a *sphere in four*
*dimensions*. Einstein's spatially closed universe is often 
referred to as Einstein's "cylinder" world: with two of 
the spatial dimensions suppressed, the model universe can be pictured
as a cylinder where the radius \(A\) represents the space and the 
axis the time coordinate. 



![figure](fig9.png)
Figure 9:
Einstein Universe





According to Einstein's Machian convictions, since inertia is 
determined only by matter, there can be no inertial structure or 
field in the absence of matter. Consequently, it is impossible, 
Einstein conjectured, to find a solution to the field 
equations--that is, to determine the metric 
\(g\_{ij}\)--if the energy-momentum tensor 
\(T\_{ij}\) representing the mass-energy content of 
the universe is zero. The non-existence of 'vacuum 
solutions' for a static universe demonstrated, Einstein 
thought, that Mach's Principle had been successfully incorporated 
into his theory of general relativity. Einstein also believed that 
his solution was unique because of the assumptions of isotropy and 
 homogeneity.[79]





However, Einstein was mistaken. In 1917, the Dutch astronomer Willem 
de Sitter published another solution to Einstein's field equations 
containing the cosmological constant. De Sitter's solution showed 
that Einstein's solution is not a unique solution of his field 
equations. In addition, since de Sitter's universe is empty it 
provided a direct counter-example to Einstein's hope that Mach's 
Principle had been successfully incorporated into his 
 theory.[80]





There are cosmologists who, like Einstein, are favourably disposed 
towards some version of Mach's Principle, and who believe that the 
local laws, which are satisfied by various physical fields, are 
determined by the large scale structure of the universe. On the other
hand, there are those cosmologists who, like Weyl, take a 
conservative approach; they take empirically confirmed *local*laws and investigate what these laws might imply about the 
universe as a whole. Our understanding of the large scale structure 
of the universe, Weyl emphasized, must be based on theories and 
principles which are verified locally. Einstein's general theory is a
*local* field theory; like electromagnetism, it is a *close
action*
*theory*.[81]
 Weyl (1924b) says: 



>  
> 
> It appears to me that one can grasp the concrete physical content of 
> the theory of relativity without taking a position regarding the 
> causal relationship between the masses of the universe and inertia.
> 
> 
> 




And, referring to (G), (see citation at the end of SS4.4.3), 
which says that "*Guidance is a physical field, like 
the electromagnetic field, which stands in mutual 
interaction with matter. Gravitation belongs to the guiding field 
and not to force*", Weyl (1924b) says: 



>  
> 
> What I have so far presented and briefly formulated in the two 
> sentences of G, that alone impacts on physics and underlies the 
> actual individual investigations of problems of the theory of 
> relativity. Mach's Principle, according to which the fixed stars 
> intervene with mysterious power in earthly events, goes far beyond 
> this [G] and is until now pure speculation; it merely has 
> cosmological significance and does not become important for natural 
> science until astronomical observations reach the totality of the 
> cosmos [Weltganze], and not merely one island of stars 
> [Sterneninsel]. We could leave the question unanswered if I did not 
> have to admit that it is tempting to construct, on the basis of the 
> theory of relativity, a picture of the totality of the cosmos.
> 
> 
> 




Weyl's claim is that because general relativity is an inherently 
*local* field theory, its validity and soundness is 
essentially independent of global cosmological considerations. 
However, if we wish to introduce such global considerations into our 
local physics, then we can do so only on the basis of additional 
assumptions, such as, for example, the Cosmological Principle, 
already mentioned. In 1923 Weyl (1923b, SS39) introduced another 
cosmological assumption, namely, the so-called *Weyl 
Postulate*. De Sitter's solution and the new astronomical 
discoveries in the early 1920's, which suggested that the universe is
not static but expanding, led to a drastic change in thinking about 
the nature of the universe and an increased scepticism towards 
Einstein's model of a static universe. In 1923, Weyl (1923b, 
SS39) notes in the fifth edition of *Raum Zeit Materie*, 
that despite its attractiveness, Einstein's cosmology suffers from 
serious defects. Weyl begins by pointing out that spectroscopic 
results indicate that the stars have an age. Weyl continued, 



>  
> 
> all our experiences about the distribution of stars show that the 
> present state of the starry sky has nothing to do with a 
> "*statistical final state*." The small velocities 
> of the stars is due to a common origin rather than some equilibrium; 
> incidentally, it appears, based on observation, that the more distant
> configurations are from each other, the greater the velocities on 
> average. Instead of uniform distribution of matter, astronomical 
> facts lead rather to the view that individual clouds of stars glide 
> by in vast empty space.
> 
> 
> 




Weyl further points out that de Sitter showed that Einstein's 
cosmological equations of gravity have "a very simple regular 
solution" and that an empty spacetime, namely, "a 
metrically homogeneous spacetime of non-vanishing curvature," 
is compatible with these equations after all. Weyl says that de 
Sitter's solution, which on the whole is not static, forces us to 
abandon our predilection for a static universe. 




The Einstein and the de Sitter universe are both spacetimes with two 
separate fringes, the infinitely remote past and the infinitely 
remote future. Dropping two of its spatial dimensions we imagine 
Einstein's universe as the surface of a straight cylinder of a 
certain radius and de Sitter's universe as a one sheeted hyperboloid.
Both surfaces are surfaces of infinite extent in both directions. 
Both the Einstein universe and the de Sitter universe spread from the
eternal past to the eternal future. However, unlike de Sitter's 
universe, in Einstein's universe "the metrical relations are 
such that the light cone issuing from a world point is folded back 
upon itself an infinite number of times. An observer should therefore
see infinitely many images of a star, showing him the star in states 
between which an eon has elapsed, the time needed by the light to 
travel around the sphere of the world." Weyl (1930) says: 



>  
> 
> ... I start from de Sitter's solution: the world, according to 
> its metric constitution, has the character of a four-dimensional 
> "sphere" (hyperboloid) 
> 
> 
> 
> \[\tag{39}
> x^{2}\_{1} + x^{2}\_{2} + x^{2}\_{3} + x^{2}\_{4} - x^{2}\_{5} = a^{2}
> \]
> 
> 
> 
> in a five-dimensional quasi-euclidean space, with the line element 
> 
> 
> 
> \[\tag{40}
> ds^{2} = dx^{2}\_{1} + dx^{2}\_{2} + dx^{2}\_{3} + dx^{2}\_{4} - dx^{2}\_{5}.
> \]
> 
> 
> 
> The sphere has the same degree of metric homogeneity as the world of 
> the special theory of relativity, which can be conceived as a 
> four-dimensional "plane" in the same space. The plane, 
> however, has only one connected infinitely distant 
> "seam," while it is the most prominent topological 
> property of the sphere to be endowed with two--the infinitely 
> distant past and the infinitely distant future. In this sense one may
> say that space is closed in de Sitter's solution. On the other hand, 
> however, it is distinguished from the well-known Einstein solution, 
> which is based on a homogeneous distribution of mass, by the fact 
> that the null cone of future belonging to a world-point does not 
> overlap with itself; in this causal sense, the de Sitter space is 
> open.
> 
> 
> 




On this hyperboloid, a single star (nebula or galaxy, in later 
contexts) \(A\), also called "observer" by Weyl, 
traces a geodesic world line, and from each point of the star's world
line a light cone opens into the future and fills a region \(D\), 
which Weyl calls the *domain of influence of the star.* In de 
Sitter's cosmology this domain of influence covers only half of the 
hyperboloid and Weyl suggests that it is reasonable to assume that 
this half of the hyperboloid corresponds to the real world. 



![figure](fig10.png)
Figure 10:
De Sitter's hyperboloid with domain of influence \(D\) 
covering half of the hyperboloid and world lines of stars.





There are innumerable stars or geodesics, according to Weyl, that 
have the same domain of influence as the arbitrarily chosen star 
\(A\); they form, he says, a *system that has been causally 
interconnected since eternity.* Such a system of causally 
interconnected stars Weyl describes as stars of a common origin that 
lies in an infinitely remote past. The sheaf of world-lines of such a
system of stars converges, in the direction of the infinitely remote 
past, on an infinitely small part of the total extent of the 
hyperboloid, and diverges in the direction of the future on an ever 
increasing extent of the hyperboloid. Weyl's choice of singling out a
particular sheaf of non-intersecting timelike geodesics as 
constituting the cosmological substratum is the content of Weyl's 
Postulate. Weyl (1923b, 295) says: 



>  
> 
> The hypothesis is suggestive, that all the celestial bodies which we 
> know belong to such a single system; this would explain the small 
> velocities of the stars as a consequence of their common origin.
> 
> 
> 




The transition from a static to a dynamic universe opens up the 
possibility of a disorderly universe where galaxies could collide, 
that is, their world lines might intersect. Roughly speaking, Weyl's 
Postulate states that the actual universe is an orderly universe. It 
says that the world lines of the galaxies form a 3-sheaf of 
 non-intersecting[82]
 geodesics orthogonal to layers of spacelike hypersurfaces.



![figure](fig11.png)
Figure 11:
Weyl's Postulate





Since the relative velocities of matter is small in each collection 
of galaxies extending over an astronomical neighbourhood, one can 
approximate a "smeared-out" motion of the galaxies and 
introduce a *substratum* or *fluid* which fills space 
and in which the galaxies move like "fundamental 
 particles".[83]
 Weyl's postulate says that observers associated with this smeared-out
motion constitute a privileged class of observers of the universe. 
Since geodesics do not intersect, according to Weyl's Postulate, 
there exist one and only one geodesic which passes through each 
spacetime point. Consequently, matter possesses a unique velocity at 
any spacetime point. Therefore, the fluid may be regarded as a 
*perfect fluid*; and this is the essential content of Weyl's 
Postulate. 




Since the geodesics of the galaxies are orthogonal to a layer of
spacelike hypersrfaces according to Weyl's Postulate, one can
introduce coordinates \((x^{0}, x^{1}, x^{2}, x^{3})\) such that the
spacelike hypersurfaces are given by \(x^{0} =\) constant, and the
spacelike coordinates \(x^{\alpha}\) \((\alpha = 1, 2, 3)\) are constant
along the geodesics of each galaxy. Therefore, the spacelike
coordinates \(x^{\alpha}\) are *co-moving* coordinates along 
the geodesics of each galaxy. The 
orthogonality condition permits a choice of the time coordinate 
\(x^{0}\) such that the metric or line element has the 
form 



\[\begin{align}
ds^{2} &= (dx^{0})^{2} - g\_{\alpha \beta}dx^{\alpha}dx^{\beta} \\

\tag{41}
 &= c^{2}dt^{2} - g\_{\alpha \beta}dx^{\alpha}dx^{\beta},
\end{align}\]



where \(ct = x^{0}, x^{0}\) is called the 
*cosmic time*, and \(t\) is the *proper time* of any
galaxy. The spacelike hypsersurfaces are therefore the surfaces of 
simultaneity with respect to the cosmic time \(x^{0}\). 
The Cosmological Principle in turn tells us that these hypersurfaces 
of simultaneity are homogeneous and isotropic. 




Independently, Robertson and Walker, were subsequently able to give a
precise mathematical derivation of the most general metric by 
assuming Weyl's Postulate and the Cosmological Principle. 


#### 4.4.7 Discovering Hubble's Law




Weyl's introduction of his Postulate made it possible for him to
provide the first satisfactory treatment of the cosmological
redshift. Consider a light source, say a star \(A\), which emits
monochromatic light that travels along null geodesics \(L, L',\ldots\)
to an observer \(O\). Let \(s\) be the proper time of the light
source, and let \(\sigma\) be the proper time of the observer
\(O\). Then to every point \(s\) on the world line of the light source
\(A\) there corresponds a point on the world line of the observer
\(O\), namely, \(\sigma = \sigma(s)\). 



![figure](fig12.png)
Figure 12:
A body or star \(A\) emits monochromatic light which travels along
null geodesics \(L, L',\ldots\) to an observer \(O\).





Consequently, if one of the generators of the light cone issuing from
\(A\)'s world line at \(A\)'s proper time 
\(s\_{0}\)--the null geodesic \(L\)--reaches 
observer \(O\) at the observer's proper time 
\(\sigma(s\_{0})\), then 



\[\tag{42}
d\sigma = \left.\frac{d\sigma(s)}{ds}\right|\_{s\_0} ds.
\]



Therefore, the frequency \(\nu\_{A}\) of the light that 
would be measured by some hypothetical observer on \(A\) is 
related to the frequency \(\nu\_{O}\) measured on \(O\)
by 



\[\tag{43}
\frac{\nu\_{O}}{\nu\_{A}} = \frac{d\sigma(s)}{ds}.
\]



According to Weyl (1923c) this relationship holds in arbitrary 
spacetimes and for arbitrary motions of source and observer. Weyl 
(1923b, Anhang III) then applied this relationship to de Sitter's 
world and showed, to lowest order, that the redshift is *linear*in distance; that is, Weyl theoretically derived, what was later
called *Hubble's redshift law*. Using Slipher's 
redshift data Weyl estimated a *Hubble constant* six years 
prior to Hubble. Weyl (1923b, Anhang III) remarks: 



>  
> 
> It is noteworthy that neither the elementary nor Einstein's cosmology
> lead to such a redshift. Of course, one cannot claim today, that our 
> explanation hits the right mark, especially since the views about the
> nature and distance of the spiral nebulae are still very much in need
> of further clarification.
> 
> 
> 




In 1933 Weyl gave a lecture in Gottingen in which Weyl (1934b) 
recalls 



>  
> 
> According to the Doppler effect the receding motion of the stars is 
> revealed in a redshift of their spectral lines which is proportional 
> to distance. In this form, where De Sitter's solution of the 
> gravitational equation is augmented by an assumption concerning the 
> undisturbed motion of the stars, I had predicted the redshift in the 
> year 1923.
> 
> 
> 


### 4.5 Quantum Mechanics and Quantum Field Theory


#### 4.5.1 Group Theory




During the period 1925-1926 Weyl published a sequence of 
groundbreaking papers (Weyl (1925, 1926a,b,c)) in which he presented 
a general theory of the representations and invariants of the 
classical Lie groups. In these celebrated papers Weyl drew together 
I. Schur's work on invariants and representations of the 
\(n\)-dimensional rotation group, and E. Cartan's work on 
semisimple Lie algebras. In doing so, Weyl utilized different fields 
of mathematics such as, tensor algebra, invariant theory, Riemann 
surfaces and Hilbert's theory of integral equations. Weyl himself 
considered these papers his greatest work in mathematics. 




The central role that group theoretic techniques played in Weyl's 
analysis of spacetime was one of several factors which led Weyl to 
his general theory of the representations and invariants of the 
classical Lie groups. It was in the context of Weyl's investigation 
of the space-problem (see SS4.2) that Weyl came to appreciate the
value of group theory for investigating the mathematical and 
philosophical foundations of physical theories in general, and for 
dealing with fundamental questions motivated by the general theory of
relativity, in particular. 




A motivation of quite another sort, which led Weyl to his general 
representation theory, was provided by Study when he attacked Weyl 
specifically, as well as other unnamed individuals, by accusing them 
"of having neglected a rich cultural domain (namely, the theory
of invariants), indeed of having completely ignored 
 it".[84]
 Weyl (1924c) replied immediately providing a new foundation for the 
theory of invariants of the special linear groups 

\(SL(n, \mathbb{C})\) and its most important subgroups, the special
orthogonal group \(SO(n, \mathbb{C})\) and the special symplectic group
\(SSp(\bfrac{n}{2}, \mathbb{C})\) (for \(n\) even) based on algebraic
identities due to Capelli. In a footnote, Weyl (1924c) sarcastically
informed Study that "even if he [Weyl] had been as well versed
as Study in the theory of invariants, he would not have used the
symbolic method in his book
*Raum, Zeit, Materie* and even with the last breath of his 
life would not have mentioned the algebraic completeness theorem for 
invariant theory". Weyl's point was that in the context of his 
book *Raum-Zeit-Materie*, the kernel-index method of tensor 
analysis is more appropriate than the methods of the theory of 
algebraic 
 invariants.[85]





While this account of events leading up to Weyl's groundbreaking 
papers on group theory seems reasonable enough, Hawkins (2000) has 
suggested a fuller account, which brings into focus Weyl's deep 
philosophical interest in the mathematical foundations of the theory 
of general relativity by drawing attention to Weyl (1924d) on tensor 
symmetries, which, according to Hawkins, played an important role in 
redirecting Weyl's research interests toward pure 
 mathematics.[86]
 Weyl (1949b, 400) himself noted that his interest in the 
philosophical foundations of the general theory of relativity 
motivated his analysis of the representations and invariants of the 
continuous groups: "I can say that the wish to understand what 
really is the mathematical substance behind the formal apparatus of 
relativity theory led me to the study of representations and 
invariants of groups; and my experience in this regard is probably 
not unique". Weyl's paper (Weyl (1924a)), and the first chapter
Weyl (1925) of his celebrated papers on representation theory, have 
the same title: "The group theoretic foundation of the tensor 
calculus". Hawkins (1998) says, Weyl 



>  
> 
> had obtained through the theory of groups, and in particular through 
> the theory of group representations--as augmented by his own 
> contributions--what he felt was a proper mathematical 
> understanding of tensors, tensor symmetries, and the reason they 
> represent the source of all linear quantities that might arise in 
> mathematics or physics. Once again, he had come to appreciate the 
> importance of the theory of groups--and now especially the 
> theory of group representation--for gaining insight into 
> mathematical questions suggested by relativity theory. Unlike his 
> work on the space problem ...Weyl now found himself drawing upon
> far more than the rudiments of group theory. ... And of course 
>  Cartan[87]
>  had showed that the space problem could also be resolved with the aid
> of results about representations. In short, the representation theory
> of groups had proved itself to be a powerful tool for answering the 
> sort of mathematical questions that grew out of Weyl's involvement 
> with relativity theory.
> 
> 
> 




Somewhat later, Weyl (1939) wrote a book, entitled *The Classical 
Groups, Their Invariants and Representations*, in which 
he returned to the theory of invariants and representations of the 
semisimple Lie groups. In this work, he satisfied his ambition 
"to derive the decisive results for the most important of these
groups by direct algebraic construction, in particular for the full 
group of all non-singular linear transformations and for the 
orthogonal group". He intentionally restricted the discussion 
of the general theory and devoted most of the book to the derivation 
of specific results for the general linear, the special linear, the 
orthogonal and the symplectic groups. 


#### 4.5.2 Weyl's philosophical critique of Cartan's approach to geometry




As far back as the 1920s, the great French mathematician and geometer
Elie Cartan had recognized that the notions
of *parallelism* and *affine connection* admit of an
important generalization in the sense that (1) the spaces for which
the notion of infinitesimal parallel transport is defined need not be
the tangent spaces that intrinsically arise from the differential
structure of a Riemannian manifold \(M\) at each of its points;
rather, the spaces are general spaces that are not intrinsically tied
to the differential manifold structure of \(M\), and (2) relevant
groups operate on these general spaces directly and not on the
manifold \(M\), and therefore groups play a dominant and independent
role. 




Weyl (1938a) published a critical review of Cartan's (1937) book in 
which Cartan further developed his notion of *moving frames*("reperes mobiles") and *generalized 
spaces* ("espaces generalises").
However, Weyl (1988) expressed some of his reservations to Cartan's 
approach as early as 1925; and four years later Weyl (1929e) 
presented a more detailed critique. 




Cartan's approach to differential geometry is in response to the fact
that Euclidean geometry was generalized in two ways resulting in 
essentially two incompatible approaches to 
 geometry.[88]
 The first generalization occurred with the discovery of non-Euclidean
geometries and with Klein's (1921) subsequent Erlanger program in 
1872, which provided a coherent group theoretical framework for the 
various non-Euclidean geometries. The second generalization of 
Euclidean geometry occurred when Riemann (1854) discovered Riemannian
geometry. 




The two generalizations of Euclidean geometry essentially constitute 
incompatible approaches to applied geometry. In particular, while 
Klein's Erlanger program provides an appropriate group theoretical 
framework for Einstein's theory of special relativity, it is 
Riemannian geometry, and not Klein's group theoretic approach, which 
provides the appropriate underlying geometric framework for 
Einstein's theory of general relativity. As Cartan observes: 



>  
> 
> General relativity threw into physics and philosophy the antagonism 
> that existed between the two principle directors of geometry, Riemann
> and Klein. The space-times of classical mechanics and of special 
> relativity are of the type of Klein, those of general relativity are 
> of the type of 
>  Riemann.[89]
> 
> 
> 
> 




Cartan eliminated the incompatibility between the two approaches by 
synthesizing Riemannian geometry and Klein's Erlanger program through
a further generalization of both, resulting in what Cartan called, 
*generalized spaces* (or *generalized geometries*). 




In his Erlanger program, Klein provided a unified approach to the 
various "global" geometries by showing that each of the 
geometries is characterized by a particular group of transformations:
Euclidean geometry is characterized by the group of translations and 
rotations in the plane; the geometry of the sphere 
\(S^{2}\) is characterized by the orthogonal group 
\(O(3)\); and the geometry of the hyperbolic plane is 
characterized by the pseudo-orthogonal group \(O(1, 2)\). In 
Klein's approach each geometry is a (connected) manifold endowed with
a group of automorphisms, that is, a Lie group \(G\) of 
"motions" that acts *transitively* on the 
manifold, such that two figures are regarded as congruent if and only
if there exists an element of the appropriate Lie group \(G\) that
transforms one of the figures into the other. A generalized geometry 
in Klein's sense shifts the emphasis from the underlying manifold or 
space to the group. Thus a Klein geometry (space) consists of, (1) a 
smooth manifold, (2) a Lie group \(G\) (the principal group of the
geometry), and (3) a transitive action of \(G\) on the manifold. 
Besides being "global", a Klein geometry (space) is 
completely homogeneous in the sense that its points cannot be 
distinguished on the basis of geometric relations because the 
transitive group action preserves such relations. 




As Weyl (1949b) describes it, Klein's approach to the various 
"global" geometries is very suited to Einstein's theory 
of special relativity: 



>  
> 
> According to Einstein's special relativity theory the 
> four-dimensional world of the spacetime points is a Klein space 
> characterized by a definite group \(\Gamma\); and that group is the 
> ... group of Euclidean similarities--with one very 
> important difference however. The orthogonal transformations, i.e., 
> the homogeneous linear transformations which leave 
> 
> 
> 
> \[
> x^{2}\_{1} + x^{2}\_{2} + x^{2}\_{3} + x^{2}\_{4}
> \]
> 
> 
> 
> unchanged have to be replaced by the Lorentz transformations leaving 
> 
> 
> 
> \[
> x^{2}\_{1} + x^{2}\_{2} + x^{2}\_{3} - x^{2}\_{4}
> \]
> 
> 
> 
> invariant.
> 
> 
> 




However, with the advent of Einstein's general theory of relativity 
the emphasis shifted from *global* homogeneous geometric 
structures to *local* inhomogeneous structures. Whereas Klein 
spaces are global and fully homogeneous, the Riemannian metric 
structure underlying Einstein's general theory is local and 
inhomogeneous. A general Riemannian space admits of no isometry other
than the identity. 




Referring to Cartan (1923a), Weyl (1929e) says that Cartan's 
generalization of Klein geometries consists in adapting Klein's 
Erlanger program to infinitesimal geometry by applying Klein's 
Erlanger program to the tangent plane rather than to the manifold 
 itself.[90]




>  
> 
> Cartan developed a general scheme of infinitesimal geometry in which 
> Klein's notions were applied to the tangent plane and not to the 
> \(n\)-dimensional manifold \(M\) itself.
> 
> 
> 



![figure](fig13.png)
Figure 13:
Cartan's generalization





Figure 13 above, adapted from Sharpe (1997), may help in clarifying 
the discussion. The generalization of Euclidean geometry to a 
Riemannian space (the left vertical blue arrow) says: 


1. A general Riemannian space approximates Euclidean space only
locally; that is, at each point \(p \in M\) there exists
a **tangent space** \(T(M\_{p})\) *that arises
intrinsically from the underlying differential structure of*
\(M\).
2. In addition, a Riemannian space is inhomogeneous through the
introduction of curvature.




Analogously, Cartan's generalization of a Klein space to a Cartan 
space (the right vertical blue arrow) says: 


1. Cartan's generalized space \(\Sigma(M)\) approximates a
Klein space only locally; that is, at each point \(p \in M\) there
exists a **"Tangent Space"**, that is, a
Klein space \(\Sigma(M\_{p})\). Note that a Klein space
\(\Sigma(M\_{p})\) is itself a generalized space (in the sense of
Cartan) with zero curvature; it possesses perfect homogeneity.
2. In addition, Cartan's generalized space \(\Sigma(M)\) is
inhomogeneous by the introduction of curvature.



![figure](fig14.png)
Figure 14:
Cartan's generalized space





Cartan's generalized space \(\Sigma\)(M) is the space of all 
**"Tangent Spaces"** (i.e., all Klein spaces
\(\Sigma(M\_{p}))\) and contains a mixture of 
homogeneous and inhomogeneous spaces (see figure 14). 




Finally, Cartan's generalization of Riemannian space (lower 
horizontal red arrow) (figure 13) turns on the recognition that the 
**"Tangent Space"** in Cartan's sense is not
the same, or need not be the same, as the ordinary **tangent 
space** that arises naturally from the underlying differential
structure of a Riemannian manifold. Cartan's 
**"Tangent Space"**
\(\Sigma(M\_{p})\) at \(p \in M\) denotes what is known as
a *fiber* in modern fiber bundle language, where the manifold
\(M\) is called the *base space* of the fiber bundle. 




In Weyl (1929e, 1988) and to a lesser extent in Weyl (1938a), Weyl
objected to Cartan's approach by noting that Cartan's
**"Tangent Space"**, namely the Klein space
\(\Sigma(M\_{p})\) associated with each point of the manifold \(M\),
does not arise intrinsically from the differential structure of the
manifold the way the ordinary tangent vector space does. Weyl
therefore noted that it is necessary to impose certain non-intrinsic
embedding conditions on \(\Sigma(M\_{p})\) that specify how the
**"Tangent Space"**
\(\Sigma(M\_{p})\) is associated with each point of the manifold
\(M\). Paraphrasing Weyl, the situation is as follows: We assume that
we can associate a copy \(\Sigma(M\_{p})\) of a given Klein space with
each point \(p\) of the manifold \(M\) and that the displacement of
the Klein space \(\Sigma(M\_{p})\) at \(p \in M\) to the Klein space
\(\Sigma(M\_{p'})\) associated with an infinitely nearby point
\(p'\in M\), constitutes an isomorphic representation of
\(\Sigma(M\_{p})\) on \(\Sigma(M\_{p'})\) by means of an infinitesimal
action of the group \(G\). In choosing an admissible frame of
reference \(f\) for each Klein space \(\Sigma(M\_{p})\), their points
are represented by
*normal* coordinates \(\xi\). Any two frames 
\(f,f'\) are related by a group 
element \(s \in G\), and a succession of transformations
\(f \rightarrow f'\) and \(f' \rightarrow f''\) by \(s \in G\) and 
\(t \in G\) respectively, relates \(f\) and 
\(f''\) by the group composition \(t \circ s \in G\). 




Nothing so far has been said about how specifically the 
**"Tangent Space"** 
\(\Sigma(M\_{p})\) is connected to the manifold. 
Since \(\Sigma(M\_{p})\) is supposed to be a 
generalization of the ordinary tangent space which arises 
intrinsically from the local differential structure of \(M\), Weyl
suggests that certain embedding conditions have to be imposed on the 
*normal* coordinates \(\xi\) of the Klein space 
\(\Sigma(M\_{p})\). 



**Embedding Condition 1:**  
 We must first designate
a point as the center of \(\Sigma(M\_{p})\) and then require that they
coincide or cover the point \(p \in M\). This leads, Weyl says, to a
restriction in the choice of a *normal coordinate system*
\(\xi\) on \(\Sigma(M\_{p})\). And because \(G\) acts transitively, a
normal coordinate system \(\xi\) on \(\Sigma(M\_{p})\) can be chosen
such that the normal coordinates \(\xi\) vanish at the center, that
is, \(\xi^{1} = \xi^{2} = \cdots = 0\). The group \(G\) is therefore
restricted to the subgroup \(G\_{0}\) of all representations of \(G\)
which leave the center invariant.




**Embedding Condition 2:**  
 The notion of a tangent
plane also requires that there is a one-to-one linear mapping between
the line elements of \(\Sigma(M\_{p})\) starting 
from 0, with the line elements of \(M\) starting from \(p\). 
This means that the number of dimension of the Klein space 
\(\Sigma(M\_{p})\) has the same number of dimension
as the manifold \(M\).



**Embedding Condition 3:**  
 The infinitesimal 
displacement \(\Sigma(M\_{p}) \rightarrow \Sigma(M\_{p'})\) will carry an 
infinitesimal vector at the center of 
\(\Sigma(M\_{p})\), which is in one-to-one 
correspondence with a vector at \(p \in M\), to the 
center of \(\Sigma(M\_{p'})\). 




No further conditions need be imposed according to Weyl. If we
displace \(\Sigma(M\_{p})\) by successive steps around a curve
\(\gamma\) back to the point \(p \in M\) then the final position of
\(\Sigma(M\_{p})\) is obtained from its original poition or orientation
by a certain automorphism \(\Sigma(M\_{p}) \rightarrow
\Sigma(M\_{p})\). This automorphism is Cartan's generalization of
Riemann's concept of curvature along the curve \(\gamma\) on
\(M\). 




According to Weyl, the **"Tangent Space"**
\(\Sigma(M\_{p})\) is not uniquely determined by the differential
structure of \(M\). If \(G\) were the affine group, Weyl says, then
the conditions above would fully specify the *normal*
coordinate system \(\xi^{\alpha}\) on \(\Sigma(M\_{p})\) as a function
of the chosen local coordinates \(x^{i}\) on \(M\). Since this is not
the case if \(G\) is a more extensive group than the affine group,
Weyl concludes that the
**"Tangent Space"** 
\(\Sigma(M\_{p})\) "is not as yet uniquely 
determined by the nature of \(M\), and so long as this is not 
accomplished we can not say that Cartan's theory deals only with the 
manifold \(M\)." Weyl adds: 



>  
> 
> Conversely, the tangent plane in \(p\) in the ordinary sense, that
> is, the linear manifold of line elements in \(p\), is a centered 
> affine space; its group \(G\) is not a matter of convention. This 
> has always appeared to me to be a deficiency of the theory ....
> 
> 
> 




The reader may wish to consult Ryckman (2005, 171-173), who 
argues "that a philosophical contention, indeed, 
phenomenological one, underlies the stated mathematical reasons that 
kept him [Weyl] for a number of years from concurring with Cartan's 
"moving frame" approach to differential geometry". 




In 1949 Weyl explicitly acknowledged and praised Cartan's approach. 
Unlike his earlier critical remarks, he now considered it to be a 
virtue that the frame of reference in 
\(\Sigma(M\_{p})\) is independent of the choice of 
coordinates on \(M\). Weyl (1949b) says of the traditional 
approach and Cartan's new approach to geometry: 



>  
> 
> Hence we have here before us the natural general basis on which that 
> notion rests. The infinitesimal trend in geometry initiated by Gauss'
> theory of curved surfaces now merges with that other line of thought 
> that culminated in Klein's Erlanger program. 
> 
> 
> 
> 
> It is not advisable to bind the frame of reference in \(\Sigma\_{p}\)
> to the coordinates \(x^{i}\) covering the neighborhood of \(p\) in
> \(M\). In this respect the old treatment of affinely connected
> manifolds is misleading. ... [I]n the modern development of
> infinitesimal geometry in the large, where it combines with topology
> and the associated Klein spaces appear under the name of fibres, it
> has been found best to keep the *reperes*, the
> frames of the fibre spaces, independent of the coordinates of the
> underlying manifold.
> 
> 
> 




Moreover, in 1949, Weyl also emphasizes that it is necessary to 
employ Cartan's method if one wishes to fit Dirac's theory of the 
electron into general relativity. Weyl (1949b) says: 



>  
> 
> When one tries to fit Dirac's theory of the electron into general 
> relativity, it becomes imperative to adopt the Cartan method. For 
> Dirac's four \(\psi\)-components are relative to a Cartesian (or rather 
> a Lorentz) frame. One knows how they transform under transition from 
> one Lorentz frame to another (spin representation of the Lorentz 
> group); but this law of transformation is of such a nature that it 
> cannot be extended to arbitrary linear transformations mediating 
> between affine frames.
> 
> 
> 




Weyl is here referring to his three important papers, which appeared 
in 1929--the same year in which he had published his detailed 
critique of Cartan's method--in which he investigates the 
adaptation of Dirac's theory of the special relativistic electron to 
the theory of general relativity, and where he develops the 
*tetrad* or *Vierbein* formalism for the representation
of *local* two-component spinor structures on Lorentz 
manifolds. 


#### 4.5.3 Weyl's New Gauge Principle and Dirac's Special Relativistic Electron




Only a year after Pauli's review article in 1921, in which Pauli had 
argued that Weyl's defence of his unified field theory deprives it of
its inherent convincing power from a physical point of view, 
Schrodinger (1922) suggested the possibility that Weyl's 1918 
gauge theory could suitably be employed in the quantum mechanical 
description of the 
 electron.[91]
 Similar proposals were subsequently made by Fock (1926) and London 
(1927). 




With the advent of the quantum theory of the electron around 1927/28 
Weyl abandoned his gauge theory of 1918. He did so because in the new
quantum theory a different kind of gauge invariance associated with 
Dirac's theory of the electron was discovered which, as had been 
suggested by Fock (1926) and London (1927), more adequately accounted
for the conservation of electric 
 charge.[92]
 Why did Weyl hold on to his gauge theory for almost a decade despite 
a preponderance of compelling empirical arguments that were mounted 
against it by Einstein, Pauli and 
 others?[93]
 In one of Weyl's (1918/1998) last letters to Einstein concerning his 
unified field theory, Weyl made it clear that *it was mathematics 
and not physics* that was the driving force behind his unified 
field 
 theory.[94]




>  
> 
> Incidentally, you must not believe that it was because of physics 
> that I introduced the linear differential form d\(\varphi\) in addition 
> to the quadratic form. I wanted rather to eliminate this 
> "inconsistency" which always has been a bone of 
> contention to 
>  me.[95]
>  And then, to my surprise, I realized that it looked as if it might 
> explain electricity. You clap your hands above your head and shout: 
> But physics is not made this way!
> 
> 
> 




As London (1927, 376-377) remarks, one must admire Weyl's 
immense courage in developing his gauge invariant interpretation of 
electromagnetism and holding on to it on the mere basis of purely 
formal considerations. London observes that the principle of 
equivalence of inertial and gravitational mass, which prompted 
Einstein to provide a geometrical interpretation of gravity, was at 
least a physical fact underlying gravitational theory. In contrast, 
an analogous fact was not known in the theory of electricity; 
consequently, it would seem that there was no compelling physical 
reason to think that rigid rods and ideal clocks would be under the 
universal influence of the electromagnetic field. To the contrary, 
London says, experience strongly suggests that atomic clocks exhibit 
sharp spectral lines that are unaffected by their history in the 
presence of a magnetic field, contrary to Weyl's non-integrability 
assumption. London concludes, that in the face of such elementary 
empirical facts it must have been an unusually clear metaphysical 
conviction which prevented Weyl from abandoning his idea that nature 
ought to make use of the beautiful geometrical possibilities that a 
pure infinitesimal geometry offers. 




In 1955, shortly before his death, Weyl wrote an 
 addendum[96]
 to his 1918 paper *Gravitation und 
Elektrizitat*, in which he looks back at 
his early attempt to find a unified field theory and explains why he 
reinterpreted his gauge theory of 1918, a decade later. 



>  
> 
> This work stands at the beginning of attempts to construct a 
> "unified field theory" which subsequently were continued 
> by many, it seems to me, without decisive results. As is known, the 
> problem relentlessly occupied Einstein in particular, until his end. 
> ... The strongest argument for my theory appeared to be that 
> gauge invariance corresponds to the principle of the conservation of 
> electric charge just as coordinate invariance corresponds to the 
> conservation theorem of energy-impulse. Later, quantum theory 
> introduced the Schrodinger-Dirac potential \(\psi\) of the 
> electron-positron field; the latter revealed an experimentally based 
> principle of gauge invariance which guaranteed the conservation of 
> charge and which connected the \(\psi\) with the electromagnetic 
> potentials \(\varphi\_{i}\) in the same way that my 
> speculative theory had connected the gravitational potentials 
> \(g\_{ik}\) with \(\varphi\_{i}\), where,
> in addition, the \(\varphi\_{i}\) are measured in known 
> atomic rather than unknown cosmological units. I have no doubts that 
> the principle of gauge invariance finds its correct place here and 
> not, as I believed in 1918, in the interaction of electromagnetism 
> and gravity.
> 
> 
> 




By the late 1920s Weyl's methodological approach to gauge theory 
underwent an "empirical turn". In contrast to \(a\) 
*priori* geometrical reasoning, which guided his early 
unification attempts--Weyl calls it a "speculative 
theory" in the above citation--by 1928/1929 Weyl 
emphasized *experimentally-based principles* which underlie 
gauge 
 invariance.[97]





In early 1928 P. A. M. Dirac provided the first physically compelling
theoretical account of the dynamics of an electron in the presence of
an electric field. The components \(\psi^{i} (x)\)
of Dirac's four-component wave function or *spinor field* in 
Minkowski space, \(\psi(x) = (\psi^{1}(x), \psi^{2}(x), \psi^{3}(x), 
\psi^{4}(x))\), are complex-valued functions that 
satisfy Dirac's first-order partial differential equation and provide
probabilistic information about the electron's dynamical behaviour, 
such as angular momentum and location. Prior to the appearance of 
*spinor fields* \(\psi\) in Dirac's equation, it was generally 
thought that scalars, vectors and tensors provided an adequate system
of mathematical objects that would allow one to provide a 
mathematical description of reality independently of the choice of 
coordinates or reference 
 frames.[98]
 For example, spin zero particles \((\pi\) mesons, \(\alpha\) particles) 
could be described by means of scalars; spin 1 particles (deuterons) 
by vectors, and spin 2 particles (hypothetical gravitons) by tensors.
However, the most frequently occurring particles in Nature are 
electrons, protons, and neutrons. They are spin 
\(\bfrac{1}{2}\) particles, called *fermions* that 
are properly described by mathematical objects called 
*spinors*, which are neither scalars, vectors or 
 tensors.[99]
 Weyl referred to the \(\psi(x)\) in Dirac's equation as the 
"Dirac quantity" and von Neumann called it the 
"\(\psi(x)\)-vector". Both von Neumann and Weyl, and 
others, immediately recognized that Dirac had introduced something 
that was new in theoretical physics. v. Neumann (1928, 876) remarks: 



>  
> 
> ... \(\psi\) does by no means have the relativistic transformation 
> properties of a common four-vector. ... The case of a quantity 
> with four components that is not a four-vector is a case which has 
> never occurred in relativity theory; the Dirac \(\psi\)-vector is the 
> first example of this type.
> 
> 
> 




(Weyl (1929c)) notes that the spinor representation of the orthogonal
group \(O(1, 3)\) cannot be extended to a representation of the 
general linear group \(GL(n)\), \(n = 4\), with the 
consequence that it is necessary to employ the *Vierbein*, 
*tetrad* or Lorentz-structure formulation of the theory of 
general relativity in order to incorporate Dirac's spinor fields 
\(\psi(x)\): 



>  
> 
> The tensor calculus is not the proper mathematical instrument to use 
> in translating the quantum-theoretic equations of the electron over 
> into the *general theory of relativity.* Vectors and terms 
> [tensors] are so constituted that the law which defines the 
> transformation of their components from one Cartesian set of axes to 
> another can be extended to the most general linear transformation, to
> an affine set of axes. That is not the case for quantity \(\psi\), 
> however; this kind of quantity belongs to a representation of the 
> rotation group which cannot be extended to the affine group. 
> Consequently we cannot introduce components of \(\psi\) relative to an 
> arbitrary coordinate system in general relativity as we can for the 
> electromagnetic potential and field strengths. We must rather 
> describe the metric at a point \(p\) by local Cartesian axes 
> \(e(a)\) instead of by the \(g\_{pq}\). 
> The wave field has definite components 
> \(\psi^{+}\_{1}, \psi^{+}\_{2}, \psi^{-}\_{1}, \psi^{-}\_{2}\) 
> relative to such axes, and we know how they transform on transition to
> any other Cartesian axes in \(p\).
> 
> 
> 




Impressed by the initial success of Dirac's equation of the spinning 
electron within the special relativistic context, Weyl adapted 
Dirac's special relativistic theory of the electron to the general 
theory of relativity in three groundbreaking papers (Weyl 
(1929b,c,d)). A complete exposition of this formalism is presented in
(Weyl (1929b)). O'Raifeartaigh (1997) says of this paper: 



>  
> 
> Although not fully appreciated at the time, Weyl's 1929 paper has 
> turned out to be one of the seminal papers of the century, both from 
> the philosophical and from the technical point of view.
> 
> 
> 




In this ground braking paper, as well as in (Weyl (1929c,d)), Weyl 
explicitly abandons his earlier attempt to unify electromagnetism 
with the theory of general relativity. In his early attempt he 
associated the electromagnetic vector potential 
\(A\_{j}(x)\) with the additional connection 
coefficients that arise when a conformal structure is reduced to a 
Weyl structure (see SS4.1). The important concept of gauge 
invariance, however, is preserved in his 1929 paper. Rather than 
associating gauge transformations with the scale or gauge of the 
spacetime metric tensor, Weyl now associates gauge transformations 
with the phase of the Dirac spinor field \(\psi\) that represents 
matter. In the introduction of (Weyl (1929b)), which presents in 
detail the new formalism, Weyl describes his reinterpretation of the 
gauge principle as follows: 



>  
> 
> The Dirac field-equations for \(\psi\) together with the Maxwell 
> equations for the four potentials \(f\_{p}\) of the 
> electromagnetic field have an invariance property which, from a 
> formal point of view, is similar to the one that I called gauge 
> invariance in my theory of gravitation and electromagnetism of 1918; 
> the equations remain invariant when one makes the simultaneous 
> replacements 
> 
> 
> 
> \[\begin{array}{ccc}
> \psi \text{ by } e^{i\lambda}\psi 
>  & \text{and} &
>  f\_p \text{ by } f\_p - \dfrac{\partial\lambda}{\partial x^p},
> \end{array}\]
> 
> 
> 
> where \(\lambda\) is understood to be an arbitrary function of
> position in the four-dimensional world. Here the factor
> \(\bfrac{e}{ch}\), where \(- e\) is the charge of the electron, \(c\)
> is the speed of light, and \(\bfrac{h}{\pi}\) is the quantum of
> action, has been absorbed in \(f\_{p}\). The connection of this
> "gauge invariance" to the conservation of electric charge
> remains untouched. But an essential difference, which is significant
> for the correspondence to experience, is that the exponent of the
> factor multiplying \(\psi\) is not real but purely imaginary. \(\psi\)
> now assumes the role that \(ds\) played in Einstein's old
> theory. It seems to me that this new principle of gauge invariance,
> which follows not from speculation but from experiment, compellingly
> indicates that *the electromagnetic field is a necessary
> accompanying phenomenon, not of gravitation, but of the material wave
> field represented by* \(\psi\). Since gauge invariance includes an
> arbitrary function \(\lambda\) it has the character of
> "general" relativity and can naturally only be understood
> in that context.
> 
> 
> 




Weyl then introduces his two-component spinor theory in Minkowski
space. Since one of his aims is to adapt Dirac's theory to the
curved spacetime of general relativity, Weyl develops a theory
of *local* spinor structures for curved
 spacetime.[100]
 He achieves this by providing a systematic formulation of 
*local* tetrads or Vierbeins (orthonormal basis
vectors). Orthonormal frames had already been introduced as early as
1900 by Levi-Civita and Ricci. Somewhat later, Cartan had shown the
usefulness of employing local orthonormal-basis vector fields, the
so-called "moving frames" in his investigation of
Riemannian geometry in the 1920s. In addition, Einstein (1928) had
used tetrads or Vierbeins in his attempt to unify gravitation and
electricity by resorting to *distant parallelism*
with *torsion*. In Einstein's theory, the effects of
gravity and electromagnetism are associated with a specialized torsion
of spacetime rather than with the curvature of spacetime. Since the
curvature vanishes everywhere, distant parallelism is a feature of
Einstein's theory. However, distant parallelism appeared to Weyl
to be quite unnatural from the viewpoint of Riemannian geometry. Weyl
expressed his criticism in all three papers (Weyl (1929b,c,d)) and he
contrasted the way in which Vierbeins are employed in his own work
with the way they were used by Einstein. In the introduction Weyl
(1929b) says: 



>  
> 
> I prefer not to believe in distant parallelism for a number of 
> reasons. First, my mathematical attitude resists accepting such an 
> artificial geometry; it is difficult for me to understand the force 
> that would keep the local tetrads at different points and in rotated 
> positions in a rigid relationship. There are, I believe, two 
> important physical reasons as well. In particular, by loosening the 
> rigid relationship between the tetrads at different points, the gauge
> factor \(e^{i\lambda}\), which remains arbitrary 
> with respect to the quantity \(\psi\), changes from a constant to an 
> arbitrary function of spacetime location; that is, only through the 
> loosening of the rigidity does the actual gauge invariance become 
> understandable. Secondly, the possibility to rotate the tetrads at 
> different points independently from each other, is as we shall see, 
> equivalent to the symmetry of the energy-momentum tensor or with the 
> validity of its conservation law.
> 
> 
> 




Every tetrad uniquely determines the pseudo-Riemannian spacetime
metric \(g\_{ij}\). However, the converse does not hold since the
tetrad has 16 independent components whereas the spacetime metric,
\(g\_{ij} = g\_{ji}\), has only 10 independent components. The extra 6
degrees of freedom of the tetrads that are not determined by the
metric may be represented by the elements of a 6-parameter internal
Lorentz group. That is, the local tetrads are determined by the
spacetime metric up to local Lorentz transformations. The tetrad
formalism made it possible, therefore, for Weyl to derive, as a
special case of Noether's second
 theorem[101],
 the energy-momentum conservation laws for general coordinate
transformations and the internal Lorentz transformations of the
tetrads. Moreover, Weyl had always emphasized the strong analogy
between gravitation and electricity. The tetrad formalism and the
conservation laws both made explicit and supported this analogy. 




Weyl introduced the final section of his seminal 1929 paper saying 
"We now come to the critical part of the theory", and 
presented a derivation of electromagnetism from the new gauge 
principle. The initial step in Weyl's derivation exploits the 
intrinsic gauge freedom of his two-component theory of spinors for 
Minkowski space, namely 



\[
\psi(x) \rightarrow e^{i\lambda}\psi(x),
\]



where the gauge factor is a constant. Since Weyl wished to adapt his 
theory to the curved spacetime of general relativity, the above phase
transformation must be generalized to accommodate *local*tetrads. That is, each spacetime point has its own tetrad and 
therefore its own point-dependent gauge factor. The phase 
transformation is thus given by 



\[
\psi(x) \rightarrow e^{i\lambda(x)}\psi(x),
\]



where the \(\lambda(x)\) is a function of spacetime. Weyl says: 



>  
> 
> We come now to the critical part of the theory. In my view the origin
> and the necessity for the electromagnetic field lie in the following 
> justification. The components \(\psi\_{1},\psi\_{2}\) 
> are, in fact, not uniquely determined by the tetrad but only to the 
> extent that they can still be multiplied by an arbitrary 
> "gauge-factor" \(e^{i\lambda}\) of 
> absolute value 1. The transformation of the \(\psi\) induced by a 
> rotation of the tetrad is determined only up to such a factor. In the
> special theory of relativity one must regard this gauge factor as a 
> constant, since we have here only a single point-independent tetrad. 
> This is different in the general theory of relativity. Every point 
> has its own tetrad, and hence its own arbitrary gauge factor, because
> the gauge factor necessarily becomes an arbitrary function of 
> position through the removal of the rigid connection between tetrads 
> at different points.
> 
> 
> 




Today, the concept of gauge invariance plays a central role in 
theoretical physics. Not until 1954 did Yang and Mills (1954) 
generalize Weyl's electromagnetic gauge concept to the case of the 
non-Abelian group 
 \(O(3)\).[102]
 Although Weyl's reinterpretation of gauge invariance had been 
preceded by suggestions from London and Fock, it was Weyl, according 
to O'Raifeartaigh and Straumann (2000), 



>  
> 
> who emphasized the role of gauge invariance as a *symmetry 
> principle* from which electromagnetism can be derived. It took 
> several decades until the importance of this symmetry 
> principle--in its generalized form to non-Abelian gauge groups 
> developed by Yang, Mills, and others--also became fruitful for a
> description of the weak and strong interactions. The mathematics of 
> the non-Abelian generalization of Weyl's 1929 paper would have been 
> an easy task for a mathematician of his rank, but at the time there 
> was no motivation for this from the physics side.
> 
> 
> 




It is interesting in this context to consider the following remarks 
by Yang. Referring to Einstein's objection to Weyl's 1918 gauge 
theory, Yang (1986, 18) asked, "what has happened to Einstein's
original objection after quantum mechanics inserted an \(-i\) into
the scale factor and made it into a phase factor?" Yang 
continuous: 



>  
> 
> Apparently no one had, after 1929, relooked at Einstein's objection 
> until I did in 1983. The result is interesting and deserves perhaps 
> to be a footnote in the history of science: Let us take Einstein's 
> Gedankenexperiment .... When the two clocks come back, because 
> of the insertion of the factor \(-i\), they would not have 
> different scales but different phases. That would not influence their
> rates of time-keeping. Therefore, Einstein's original objection 
> disappears. But you can ask a further question: Can one measure their
> phase difference? Well, to measure a phase difference one must do an 
> interference experiment. Nobody knows how to do an interference 
> experiment with big objects like clocks. However, one can do 
> interference experiments with electrons. So let us change Einstein's 
> Gedankenexperiment to one of bringing electrons back along two 
> different paths and ask: Can one measure the phase difference? The 
> answer is yes. That was in fact a most important development in 1959 
> and 1960 when Aharonov and Bohm realized--completely 
> independently of Weyl--that electromagnetism has some meaning 
> which was not understood 
>  before.[103]
> 
> 
> 
> 




We end the discussion on Weyl's gauge theory by quoting the following
remarks by Dyson (1983). 



>  
> 
> A more recent example of a great discovery in mathematical physics 
> was the idea of a gauge field, invented by Hermann Weyl in 1918. This
> idea has taken only 50 years to find its place as one of the basic 
> concepts of modern particle physics. Quantum chromodynamics, the most
> fashionable theory of the particle physicists in 1981, is 
> conceptually little more than a synthesis of Lie's group-algebras 
> with Weyl's gauge fields. 
> 
> 
> 
> 
> The history of Weyl's discovery is quite unlike the history of Lie 
> groups and Grassmann algebras. Weyl was neither obscure nor 
> unrecognized, and he was working in 1918 in the most fashionable area
> of physics, the newborn theory of general relativity. He invented 
> gauge fields as a solution of the fashionable problem of unifying 
> gravitation with electromagnetism. For a few months gauge fields were
> at the height of fashion. Then it was discovered by Weyl and others 
> that they did not do what was expected of them. Gauge fields were in 
> fact no good for the purpose for which Weyl invented them. They 
> quickly became unfashionable and were almost forgotten. But then, 
> very gradually over the next fifty years, it became clear that gauge 
> fields were important in a quite different context, in the theory of 
> quantum electrodynamics and its extensions leading up to the recent 
> development of quantum chromodynamics. The decisive step in the 
> rehabilitation of gauge fields was taken by our Princeton colleague 
> Frank Yang and his student Bob Mills in 1954, one year before Hermann
> Weyl's death [Yang and Mills, 1954]. There is no evidence that Weyl 
> ever knew or cared what Yang and Mills had done with his brain-child.
> 
> 
> 
> 
> 
> So the story of gauge fields is full of ironies. A fashionable idea, 
> invented for a purpose which turns out to be ephemeral, survives a 
> long period of obscurity and emerges finally as a corner-stone of 
> physics.
> 
> 
> 


#### 4.5.4 Weyl's two-component Neutrino theory




It is remarkable that Weyl's (1929b) two-component spinor formalism 
led him to anticipate the existence of particles that violate 
conservation of parity, that is, left-right symmetry. In 1929 
left-right symmetry was taken for granted and considered a basic fact
of all the laws of Nature. Weyl formulated the four-component Dirac 
spinor \(\psi\) in terms of a two-component left-handed Weyl spinor 
\(\psi\_{L}\) and a two-component right-handed Weyl spinor
\(\psi\_{R}\): 



\[\begin{align}
\psi &= (\psi^{1}, \psi^{2}, \psi^{3}, \psi^{4})^{T} \\
 &= (\psi^{1}\_{L}, \psi^{2}\_{L}, \psi^{1}\_{R}, \psi^{2}\_{R})^{T} \\
 &= (\psi\_L, \psi\_R)^T
\end{align}\]



The four-component Dirac spinor, formulated in terms of the two Weyl 
spinors 



\[
\psi = \left[\matrix{\psi\_L \\ \psi\_R}\right]
\]



preserves parity; it applies to all massive spin \(\bfrac{1}{2}\)
particles (fermions) and all massive fermions are known to obey parity
conservation. However, a single Weyl spinor, either \(\psi\_{L}\) or
\(\psi\_{R}\), does not preserve parity. Weyl noted that instead of the
four-component Dirac spinor "two components suffice if the
requirement of left-right symmetry (parity) is dropped". A
little later he added, "the restriction 2 removes the
equivalence of left and right. It is only the fact that left-right
symmetry actually appears in Nature that forces us to introduce a
second pair of \(\psi\)-components". Weyl's two-spinor
version of the Dirac equation is a *coupled* system of
equations requiring both Weyl spinors \(\psi\_{L}\) and \(\psi\_{R}\) in
order to preserve parity. Weyl considerd massless particles in his
two-spinor version of the Dirac equation. In this case, the equations
of the two-spinor version of Dirac's equation *decouple*,
yielding an equation for \(\psi\_{L}\) and for \(\psi\_{R}\). These
equations are independent of each other, and the equation for the
2-component left-handed Weyl spinor \(\psi\_{L}\) is
called *Weyl's equation*; it is applicable to the
massless particle called the
 *neutrino*[104],
a spin \(\bfrac{1}{2}\) particle, that was discovered in 1956. Yang
(1986, 12) remarks 



>  
> 
> Now I come to another piece of work of Weyl's which dates back to 
> 1929, and is called Weyl's two-component neutrino theory. He invented
> this theory in 1929 in one of his very important articles ... as
> a mathematical possibility satisfying most of the requirements of 
> physics. But it was rejected by him and by subsequent physicists 
> because it did not satisfy left-right symmetry. With the realisation 
> that left-right symmetry was not exactly right in 1957 it became 
> clear that this theory of Weyl's should immediately be re-looked at. 
> So it was and later it was verified theoretically and experimentally 
> that this theory gave, in fact, the correct description of the 
> neutrino.
> 
> 
> 


#### 4.5.5 The Theory of Groups and Quantum Mechanics




During the interval from 1924-26, in which Weyl was intensely 
occupied with the pure mathematics of Lie groups, the essentials of 
the formal apparatus of the new revolutionary theory of quantum 
mechanics had been completed by Heisenberg, Schrodinger and 
others. As if to make up for lost time, Weyl immediately returned 
from pure mathematics to theoretical physics, and applied his new 
group theoretical results to quantum mechanics. As Yang (1986, 9, 10)
describes it, 



>  
> 
> In the midst of Weyl's profound research on Lie groups there
> occurred a great revolution in physics, namely the development of
> quantum mechanics. We shall perhaps never know Weyl's initial
> reaction to this development, but he soon got into the act and studied
> the mathematical structure of the new mechanics. There resulted a
> paper of 1927 and later a book, this book together with Wigner's
> articles and *Gruppen Theorie und Ihre Anwendung auf die Quanten
> Mechanik der Atome* were instrumental in introducing group theory
> into the very language of quantum mechanics.
> 
> 
> 




Mehra and Rechenberg (2000, 482) note in this context: 
"Actually, we have mentioned in previous volumes Weyl's early 
reactions to both matrix mechanics (in 1925) and wave mechanics (in 
early 1926), and they were very enthusiastic. Therefore, we have to 
assume quite firmly that it was only his deep involvement with the 
last stages of his work on the theory of semisimple continuous groups
that prevented Weyl 'to get in the act' 
immediately." 




Weyl was particularly well positioned to handle some of the 
mathematical and foundational problems of the new theory of quantum 
mechanics. Almost every aspect of his mathematical expertise, in 
particular, his recent work on group theory and his very early work 
on the theory of singular differential-integral equations 
(1908-1911), provided him with the precise tools for solving 
many of the concrete problems posed by the new theory: the theory of 
Hilbert space, singular differential equations, eigenfunction 
expansions, the symmetric group, and unitary representations of Lie 
groups. 




Weyl's (1927) paper, referred to by Yang above, is entitled 
*Quantenmechanik und Gruppentheorie* (*Quantum Mechanics 
and Group Theory*). In it, Weyl provides an analysis of the 
foundations of quantum mechanics and he emphasizes the fundamental 
role Lie groups play in that 
 theory.[105]
 Weyl begins the paper by raising two questions: (1) how do I arrive 
at the self-adjoint operators, which represent a given quantity of a 
physical system whose constitution is known, and (2), what is the 
physical interpretation of these operators and which physical 
consequences can be derived from them? Weyl suggests that while the 
second question has been answered by von Neumann, the first question 
has not yet received a satisfactory answer, and Weyl proposes to 
provide one with the help of group theory. 




In a way, Weyl's 1927 paper was programmatic in character; nearly all
the topics of that paper were taken up again a year later in his 
famous book (Weyl (1928)) entitled *Gruppentheorie und 
Quantenmechanik* (*The Theory of Groups and Quantum 
Mechanics*). The book emerged from the lecture notes taken by a 
student named F. Bohnenblust of Weyl's lectures given in Zurich 
during the winter semester 1927-28. A revised edition of that 
book appeared in 1931. In the preface to the first edition Weyl says:




>  
> 
> Another time I venture on stage with a book that belongs only half to
> my professional field of mathematics, the other half to physics. The 
> external reason is not very different from that which led some time 
> ago to the origin of the book *Raum Zeit Materie*. In the 
> winter term 1927/28 Zurich was suddenly deprived of all 
> theoretical physics by the simultaneous departures of Debye and 
> Schrodinger. I tried to fill the gap by changing an already 
> announced lecture course on group theory into one on group theory and
> quantum mechanics. 
>   
>  ...  
>  
> Since I have for some years been deeply occupied with the theory of
> the representation of continuous groups, it appeared to me at this
> point to be a fitting and useful project, to provide an organically
> coherent account of the knowledge in this field won by mathematicians,
> on such a scale and in such a form, that is suitable for the
> requirements of quantum physics.
> 
> 
> 




Weyl's book is one of the first textbooks on the new theory of 
quantum mechanics. As Weyl indicates in the preface it was necessary 
for him to include a short account of the foundation of quantum 
theory in order to be able to show how the theory of groups finds its
application in that theory. If the book fulfils its purpose, Weyl 
suggests, then the reader should be able to learn from it the 
essentials of both the theory of groups and quantum theory. Weyl's 
aim was to explain the mathematics to the physicists and the physics 
to the mathematicians. However, as Yang (1986, 10) points out, 
referring to Weyl's book: 



>  
> 
> Weyl was a mathematician and a philosopher. He liked to deal with 
> concepts and the connection between them. His book was very famous, 
> and was recognized as profound. Almost every theoretical physicist 
> born before 1935 has a copy of it on his bookshelves. But very few 
> read it: Most are not accustomed to Weyl's concentration on the 
> structural aspects of physics and feel uncomfortable with his 
> emphasis on concepts. The book was just too abstract for most 
> physicists.
> 
> 
> 




Weyl's book (Weyl (1931b, 2 edn)) is remarkably complete for such an 
early work and covers many topics. Chapters I and III are mainly 
concerned with preliminary mathematical concepts. The first chapter 
provides an account of the theory of finite dimensional Hilbert 
spaces and the third chapter is an exposition of the unitary 
representation theory of finite groups and compact Lie groups. 
Chapter II is entitled *Quantum Theory*; it is the earliest 
systematic and comprehensive account of the new quantum theory. 
Chapter IV, entitled *Application of the Theory of Groups
to Quantum Mechanics*, is divided into four parts. In part A, 
entitled *The Rotation Group*, Weyl provides a systematic 
explanatory account of the theory of atomic spectra in terms of the 
unitary representation theory of the rotation group, followed by a 
discussion of the selection and intensity rules. Part B is entitled 
*The Lorentz Group*. After discussing the spin of the electron
and its role in accounting for the anomalous Zeeman effect, Weyl 
presents Dirac's theory of the relativistic quantum mechanics of the 
electron and develops in detail the theory of an electron in a 
spherically symmetric field, including an analysis of the fine 
structure of the spectrum. In part C, entitled *The Permutation 
Group*, Weyl applies the Pauli exclusion principle to explicate 
the periodic table of the elements. Next, Weyl develops the second 
quantization of the Maxwell and Dirac fields required for the 
analysis of many body relativistic systems. Weyl noted in the preface
to the second edition that his treatment is in accordance with the 
recent work of Heisenberg and Pauli. It is now customary to include 
such a topic under the heading of relativistic quantum field theory. 
The final part of Chapter IV, part D, is entitled *Quantum*
*Kinematics*; it provides an exposition of part II of Weyl's 
(1927) paper, mentioned earlier. Chapter V, entitled *The 
Symmetric Permutation Group and the Algebra of Symmetric 
Transformations*, is for the most part pure mathematics. It is 
widely regraded to be the most difficult part of the Weyl's book. 




Overall, Weyl's treatment is quite modern except for the confusion 
regarding the positive electron (anti-electron) that at that time was
identified with the proton rather than with the positron, which was 
discovered a few years later. Weyl was quite concerned about the 
identification of the proton with the positive electron because his 
analysis of the discrete symmetries \(\mathbf{C}, \mathbf{P}, \mathbf{T}\) and 
\(\mathbf{CPT}\) led him to conclude that the mass of the positive electron
should equal the mass of the 
 electron.[106]



#### 4.5.6 Weyl's Early Discussion of the Discrete Symmetries \(\mathbf{C}, \mathbf{P}, \mathbf{T}\) and \(\mathbf{CPT}\)




Weyl (1931b, 2 edn) analyzed Dirac's relativistic theory of the 
electron (Dirac (1928a,b)). Although this theory correctly accounted 
for the spin of the electron, there was however a problem because in 
addition to the positive-energy levels, Dirac's theory predicted the 
existence of an equal number of negative-energy levels. Dirac (1930) 
reinterpreted the theory by assuming that all of the negative-energy 
levels were normally occupied. The Pauli Exclusion Principle, which 
asserts that it is impossible for two electrons to occupy the same 
quantum state, would prevent an electron with positive energy from 
falling into a negative- energy state. Dirac's theory also predicted 
that one of the negative-energy electrons could be raised to a state 
of positive energy, thereby creating a 'hole' or 
unoccupied negative-energy state. Such a hole would behave like a 
particle with a positive energy and a positive charge, that is, like 
a positive electron. 




Because the only fundamental particles that were known to exist at 
that time were the electron and the proton, one was justifiably 
reluctant to postulate the existence of new particles that had not 
yet been observed experimentally; consequently, it was suggested that
the positive electron should be identified with the proton. However, 
Weyl was quite concerned about the identification of the proton with 
the anti-electron. In the preface to the second German edition of his
book *Gruppentheorie und Quantenmechanik*, Weyl (1928, 2 edn, 
1931, VII) wrote 



>  
> 
> The problem of the proton and the electron is discussed in connection
> with the symmetry properties of the quantum laws with respect to the 
> interchange of right and left, past and future, and positive and 
> negative electricity. At present no acceptable solution is in sight; 
> I fear, that in the context of this problem, the clouds are rolling 
> together to form a new, serious crisis in quantum physics.
> 
> 
> 




Weyl had good reasons for his concern. He analyzed the invariance of 
the Maxwell-Dirac equations under the discrete symmetries that 
correspond to the transformations now called \(\mathbf{C}, \mathbf{P}, 
\mathbf{T}\) and \(\mathbf{CPT}\) both for the case of relativistic quantum 
mechanics and for the case of relativistic quantum field theory, and 
concluded in both cases that the mass of the anti-electron should be 
the same as the mass of the electron. That the mass of the proton was
so different from the mass of the electron, therefore, appeared to 
Weyl to constitute a new serious crisis in physics. 




In a lecture presented at the Centenary for Hermann Weyl held at the 
ETH in Zurich, Yang (1986, 10) says of the above quote from 
Weyl's preface to the second edition of *Gruppentheorie und 
Quantenmechanik*: 



>  
> 
> This was a most remarkable passage in retrospect. The symmetry that 
> he mentioned here, of physical laws with respect to the interchange 
> of right and left, had been introduced by Weyl and Wigner 
> independently into quantum physics. It was called parity 
> conservation, denoted by the symbol \(P\). The symmetry between 
> the past and future was something that was not well understood in 
> 1930. It was understood later by Wigner, was called time reversal 
> invariance, and was denoted by the symbol \(T\). The symmetry with
> respect to positive and negative electricity was later called charge 
> conjugation invariance \(C\). It is a symmetry of physical laws 
> when you change positive and negative signs of electricity. Nobody, 
> to my knowledge, absolutely nobody in the year 1930, was in any way 
> suspecting that these symmetries were related in any manner. I will 
> come back to this matter later. What had prompted Weyl in 1930 to 
> write the above passage is a great mystery to me.
> 
> 
> 




It would seem that Yang's comment is misleading since it
suggests that Weyl did not have a good reason for his remark. In fact,
however, Weyl's statement was firmly based on a detailed
analysis of the discrete symmetries \(\mathbf{C}, \mathbf{P},
\mathbf{T}\) and \(\mathbf{CPT}\). Coleman and Korte (2001)
have shown in detail that Weyl's treatment of these symmetries
is the *same* as that used today except for the fact that the
symmetry \(\mathbf{T}\) is treated by Weyl as linear and unitary,
rather than as antilinear and antiunitary. Weyl had presented in 1931
a complete analysis, in the context of the quantized Maxwell-Dirac
field equations, of the discrete symmetries that are now called
\(\mathbf{C}, \mathbf{P}, \mathbf{T}\) and \(\mathbf{CPT}\). His
transformations \(\mathbf{C}\) and \(\mathbf{P}\) are the same as
those used today. His transformations \(\mathbf{T}\) and
\(\mathbf{CPT}\) are also very close to those used today except that
Weyl's transformations were linear and unitary rather than
antilinear and antiunitary. Moreover, Weyl drew two very
important conclusions from his analysis of these discrete
symmetries. First, Weyl announced that the important question of the
arrow of time had been solved because the field equations were not
invariant under his time-reversal transformation
\(\mathbf{T}\). Second, Weyl pointed out that the invariance of the
field equations under his charge-conjugation transformation
\(\mathbf{C}\) implied that the mass of the
'anti-electron' is necessarily the same as that of the
electron; moreover, Weyl's result is the primary reason that
Dirac (1931, 61) abandoned the assignment of the proton to the role of
the anti-electron. Many years later Dirac (1977, 145) recalled: 



>  
> 
> Well, what was I to do with these holes? The best I could think of 
> was that maybe the mass was not the same as the mass of the electron.
> After all, my primitive theory did ignore the Coulomb forces between 
> the electrons. I did not know how to bring those into the picture, 
> and it could be that in some obscure way these Coulomb forces would 
> give rise to a difference in the masses. 
> 
> 
> 
> 
> Of course, it is very hard to understand how this difference could be
> so big. We wanted the mass of the proton to be nearly 2000 times the 
> mass of the electron, an enormous difference, and it was very hard to
> understand how it could be connected with just a sort of perturbation
> effect coming from Coulomb forces between the electrons. 
> 
> 
> 
> 
> However, I did not want to abandon my theory altogether, and so I put
> it forward as a theory of electrons and protons. Of course I was very
> soon attacked on this question of the holes having different masses 
> from the original electrons. I think the most definite attack came 
> from Weyl, who pointed out that mathematically the holes would have 
> to have the same mass as the electrons, and that came to be the 
> accepted view.
> 
> 
> 




At another place Dirac (1971, 52-55) remarks: 



>  
> 
> But still, I thought there might be something in the basic idea and 
> so I published it as a theory of electrons and protons, and left it 
> quite unexplained how the protons could have such a different mass 
> from the electrons. 
> 
> 
> 
> 
> This idea was seized upon by Herman [sic] Weyl. He said boldly that 
> the holes had to have the same mass as the electrons. Now Weyl was a 
> mathematician. He was not a physicist at all. He was just concerned 
> with the mathematical consequences of an idea, working out what can 
> be deduced from the various symmetries. And this mathematical 
> approach led directly to the conclusion that the holes would have to 
> have the same mass as the electrons. Weyl just published a blunt 
> statement that the holes must have the same mass as the electrons and
> did not make any comments on the physical implications of this 
> assertion. Perhaps he did not really care what the physical 
> implications were. He was just concerned with achieving consistent 
> mathematics.
> 
> 
> 




Dirac's characterization of Weyl's unconcern for physics seems unfair
in light of Weyl's own statement in the preface of the second edition
of his book, cited earlier, where he expresses the fear "that 
in the context of this problem, the clouds are rolling together to 
form a new, serious crisis in quantum physics"; Weyl did care 
about the physics. 




Weyl's analysis did have a significant impact on the development of 
the Maxwell-Dirac theory; however, as Coleman and Korte (2001)
have argued, Weyl's early analysis of the transformations \(\mathbf{C}, 
\mathbf{P}, \mathbf{T}\) and \(\mathbf{CPT}\) was, for the most part, lost to 
subsequent researchers and had to be essentially re-invented. 
However, it should be noted in this context that Schwinger (1988, 
107-129) was greatly influenced by Weyl's book. Schwinger makes
particular reference to Weyl's work on the discrete symmetries and 
says that this work "... was the starting point of my own 
considerations concerning the connection between spin and statistics,
which culminated in what is now referred to as the TCP--or some
permutation thereof--theorem". 


#### 4.5.7 Weyl's Philosophical Views about Quantum Mechanics




Weyl analyzed the foundations of both the general theory of 
relativity and the theory of quantum mechanics. For both theories, he
provided a coherent exposition of the mathematical structure of the 
theory, elegant characterizations of the entities and laws postulated
by the theory and a lucid account of how these postulates explain the
most significant, more directly observable, lower-level phenomena. In
both cases, he was also concerned with the constructive aspects of 
the theory, that is, with the extent to which the higher-level 
postulates of the theory are necessary. 




There is no doubt that with regard to the general theory of 
relativity, Weyl held strong philosophical views. Some of these views
are couched in a phenomenological language and reveal Husserl's 
influence on Weyl. Ryckman's (2005) study *The Reign of Relativity*provides an extensive account of Weyl's orientation to Husserl's
phenomenology. On the other hand, many of Weyl's philosophical views 
are couched in an unequivocal empiricist-realist language. For 
example, Weyl rejected Poincare's geometrical conventionalism 
and forcefully argued that the spacetime metric field is physically 
real, that it is a physically real structural field (Strukturfeld), 
which is determined by the physically real causal (conformal) 
structure and the physically real inertial (projective) structure or 
guiding field (Fuhrungsfeld) of spacetime. He was not deterred 
in putting forward such ontological claims about the metric structure
of spacetime despite the fact that a complete epistemologically 
satisfactory solution to the measurement problem for the spacetime 
metric field was not then available. In the same manner, Weyl 
forcefully advanced a field-body-relationalist ontology of spacetime 
structure. He argued that a Leibnizian or Einstein-Machian form of 
relationalism that is based on a pure body ontology, is not tenable, 
indeed is incoherent within the context of general relativity, and he
presented a *reductio* argument, the plasticine example, to 
underscore the necessity of the existence of a physically real 
guiding field in addition to the existence of bodies. 




However, in contrast to Weyl's many philosophical views with regard 
to spacetime theories, Weyl's philosophical positions regarding the 
status of quantum mechanics, while not absent, are not as 
transparent. There are passages, such as the following ((Weyl, 1931b,
2 edn, 44), which argue for the reality of photons. 



>  
> 
> The intensity of the monochromatic radiation that is used to generate
> the photoelectric effect has no influence on the speed with which the
> electrons are ejected from the metal but affects only the frequency 
> of this process. Even with intensities so weak that on the classical 
> theory hours would be required before the electromagnetic energy 
> passing through a given atom would attain to an amount equal to that 
> of a photon, the effect begins immediately, the points at which it 
> occurs being distributed irregularly over the entire metal plate. 
> This fact is a proof of the existence of light quanta that is no less
> meaningful than the flashes of light on the scintillation screen are 
> for the corpuscular-discontinuous nature of \(\alpha\)-rays.
> 
> 
> 




On the other hand, Weyl's (1931b) discussion of the problem of 
'directional quantization' in the old quantum theory and 
of the way that this problem is 'resolved' in the new 
quantum theory appears to have a distinctly instrumentalist flavour. 
In a number of places, he describes the essence of the dilemma posed 
by quantum mechanics with a dispassionate precision. Consider, for 
example, the following (Weyl, 1931b, 2 edn, 67): 



>  
> Natural science has a constructive character. The phenomena with which
> it deals are not independent manifestations or qualities which can be
> read off from nature, but can only be determined by means of an
> indirect method, through interaction with other bodies. Their implicit
> definition is bound up with definite natural laws which underlie the
> interactions. Consider, for example, the introduction of the Galilean
> concept of mass which essentially comes down to the following indirect
> definition: "Each body possesses a momentum, that is, a vector
> \(m\overline{v}\) which has the same direction as its velocity
> \(\overline{v}\)--the scalar factor \(m\) is called its mass. The
> law of momentum holds, according to which the sum of the momenta
> before a reaction between several bodies is the same as the sum of
> their momenta after the reaction." By applying this law to the
> observed collision phenomena, one obtains data for the determination
> of the relative masses. The scientific consensus was, however, that
> such *constructive phenomena can nevertheless be attributed to the
> things themselves* even if the manipulations, which alone can lead
> to their recognition, are not being carried out. *In Quantum Theory
> we encounter a fundamental limitation to this epistemological position
> of the constructive natural science.*
> 
> 
> 




It is difficult for many people to accept quantum mechanics as an 
ultimate theory without at the same time giving up some form of 
realism and adopting something like an instrumentalist view of the 
theory. It is clear that Weyl was fully aware of this state of 
affairs, and yet in all of his published work, he refrained from 
making any bold statements of his views on the fundamental questions 
about quantum reality. He did not vigorously participate in the 
debate between Einstein and Schrodinger and the Copenhagen 
School nor did he offer decisive views concerning, for example, the 
*Einstein, Podolsky, Rosen* thought experiment or 
*Schrodinger's Cat*. Since Weyl held 
strong philosophical views within the context of the general theory 
of relativity, it is therefore only natural that one might have 
expected him to take a stand with respect to Schrodinger's cat 
and whether or not one should be fully satisfied with a theory 
according to which the cat is neither alive or dead but is in a 
superposition of these two states. 




The reason for Weyl's seeming reticence concerning the 
ontological/epistemological questions about quantum reality was 
already hinted at in note 5 of SS2, where it was suggested that 
Weyl was not especially bothered by the counterintuitive nature of 
quantum mechanics because he held the view that "objective 
reality cannot be grasped directly, but only through the use of 
symbols". Although Weyl (1948, 1949a, 1953) did express his 
philosophical views about quantum theory, he did so cautiously. Weyl 
(1949a, 263) summarizes some of the features of quantum mechanics 
that he considered of "paramount philosophical 
significance": the measurement problem, the incompatibility of 
quantum physics with classical logic, quantum causality, the 
non-local nature of quantum mechanics, the *Leibniz-Pauli 
Exclusion*
*Principle*[107],
 and the irreducible probabilistic nature of quantum mechanics. At the
end of the summary Weyl remarks: 



>  
> 
> It must be admitted that the meaning of quantum physics, in spite of 
> all its achievements, is not yet clarified as thoroughly as, for 
> instance, the ideas underlying relativity theory. The relation of 
> reality and observation is the central problem. We seem to need a 
> deeper epistemological analysis of what constitutes an experiment, a 
> measurement, and what sort of language is used to communicate its 
> result.
> 
> 
> 


#### 4.5.8 Science as Symbolic Construction




According to Weyl (1948, 295), both the theory of general relativity 
and quantum mechanics force upon us the realization that 
"instead of a real spatio-temporal material being what remains 
for us is only a construction in pure symbols". If it is 
necessary, Weyl (1948, 302) says, that our scientific grasp of an 
objective world must not depend on sense qualities, because of their 
inherent subjective nature, then it is for the same reason necessary 
to eliminate space and time. And Descartes gave us the means to do 
this with his discovery of analytic geometry. 




As Weyl (1953, 529) observes, when Newton explained the experienced 
world through the movements of solid particles in space, he rejected 
sense qualities for the construction of the objective world, but he 
held on to, and used an intuitively given objective space for the 
construction of a real world that lies behind the appearances. It was
Leibniz who recognized the phenomenal character (Phenomenalitat)
of space and time as consisting in the mere ordering of phenomena; 
however, space and time themselves do not have an independent 
reality. 




It is the freely created pure numbers, that is, pure symbols, 
according to Weyl, which serve as coordinates, and which provide the 
material with which to symbolically construct the objective world. In
symbolically constructing the objective world we are forced to 
replace space and time through a pure arithmetical construct. Instead
of spacetime points, \(n\)-tuples of pure numbers corresponding to
a given coordinate system are used. Weyl (1948, 303) says: 



>  
> 
> ... the laws of physics are viewed as arithmetic laws between 
> numerical values of variable magnitudes, in which spatial points and 
> moments of time are represented through their numerical coordinates. 
> Magnitudes such as the temperature of a body or the field strength of
> an electric field, which have at each spacetime point a definite 
> value, appear as functions of four variables, the spacetime 
> coordinates \(x, y, z, t\).
> 
> 
> 




In systematic theorizing we construct a *formal scaffold* that
consists of mere symbols, according to Weyl (1948, 311), without 
explaining initially what the symbols for mass, charge, field 
strength, etc., mean; and only toward the end do we describe how the 
symbolic structure connects directly with experience. 



>  
> 
> It is certain, that on the symbolic side, not space and time but four
> independent variables \(x, y, z, t\) appear; 
> one speaks of space, as one does of sounds and colours, only on the 
> side of conscious experience. A monochromatic light signal ... has 
> now become a mathematical formula in which a certain symbol \(F\),
> called electromagnetic field strength, is expressed as a pure 
> arithmetically constructed function of four other symbols 
> \(x, y, z, t\), called spacetime coordinates.
> 
> 
> 




At another place Weyl (1949a, 113) says: 



>  
> 
> Intuitive space and intuitive time are thus hardly the adequate 
> medium in which physics is to construct the external world. No less 
> than the sense qualities must the intuitions of space and time be 
> relinquished as its building material; they must be replaced by a 
> four-dimensional continuum in the abstract arithmetical sense.
> 
> 
> 




Weyl's point is that while space and time exist within the realm of 
conscious experience, or, according to Kant, as \(a\) *priori*forms underlying all of our conscious experiences, they are 
unsuited as elements with which to construct the objective world and 
must be replaced by means of a purely arithmetical symbolic 
representation. All that we are left with, according to Weyl, is 
symbolic construction. If this still needed any confirmation, Weyl 
(1948, 313) says, it was provided by the theory of relativity and 
quantum 
 theory.[108]
 For ease of reference we repeat a citation of Weyl (1988, 4-5) 
in SS4.3.1: 



>  
> 
> Coordinates are introduced on the Mf [manifold] in the most direct 
> way through the mapping onto the number space, in such a way, that 
> all coordinates, which arise through one-to-one continuous 
> transformations, are equally possible. *With this the coordinate 
> concept breaks loose from all special constructions to which
> it was bound earlier in geometry. In the language of relativity this 
> means: The coordinates are not* measured, *their 
> values are not read off from real measuring rods which react
> in a definite way to physical fields and the metrical structure, 
> rather they are a priori placed in the world arbitrarily, in
> order to characterize those physical fields including the metric
> structure numerically.* The metric structure becomes through 
> this, so to speak, freed from space; it becomes an existing field 
> within the remaining structure-less space. Through this, space as 
> form of appearance contrasts more clearly with its real content: The 
> content is measured after the form is arbitrarily related to 
>  coordinates.[109]
> 
> 
> 
> 




The last two sentences in the above quote suggest that, (a) Weyl 
embraces something close to Kant's position, according to which space
and time are "*a priori* forms of appearances", or
that (b) Weyl adheres to a position called *spacetime 
substantivalism*, according to which, in addition to body and 
fields and their relations, there also exists a 
'container', the spacetime manifold, and this manifold, 
its points and the manifold differential-topological relations are 
physically real. However, this interpretation would contradict Weyl's
basic thesis that in the symbolic construction of the objective word 
we are left with nothing but symbolic arithmetic functional 
relations. Weyl's phrases, do not denote either a physically real 
container or something like Kant's *a priori* form of 
intuition. They merely denote a *conceptual* or *formal 
scaffolding*, a *logical space*, as it were, whose points 
are represented by purely *formal* coordinates \((n\)-tuple
of pure numbers). It is such a formal space which is employed by the 
theorist in the initial stages of constructing an objective world. To
emphasize, in modelling the objective world the theorist begins by 
constructing a *formal scaffold* which consists of mere 
symbols and formal coordinates, without explaining initially what the
symbols for mass, charge, field strength, etc., mean; only toward the
end does the theorist describe how the symbolic structure connects 
directly with experience ((Weyl, 1948, 311)). 




The four-dimensional space-time continuum must be replaced by a 
four-dimensional coordinate space \(\mathbb{R}^{4}\). However, the 
sheer arbitrariness with which we assign coordinates does not affect 
the objective relations and features of the world itself. To the 
contrary, it is only *relative* to a symbolic construction or 
modelling by means of an assignment of coordinates that the state of 
the world, its relations and properties, can be *objectively* 
determined by means of distinct, reproducible symbols. While our 
immediate experiences are *subjective* and *absolute*, 
our symbolic construction of the *objective* world is of 
necessity *relative*. Weyl (1949a, 116) says: 



>  
> 
> Whoever desires the absolute must take the subjectivity and 
> egocentricity into the bargain; whoever feels drawn toward the 
> objective faces the problem of relativity.
> 
> 
> 




Weyl (1949a, 75) notes, "The objectification, by elimination of
the ego and its immediate life of intuition, does not fully succeed, 
and the coordinate system remains as the necessary residue of the 
ego-extinction." However, this residue of ego involvement is 
subsequently rendered harmless through the principle of 
*invariance*. The transition from one admissible coordinate 
system to another can be mathematically described, and the natural 
laws and *measurable quantities* must be 
*invariant* under such transformations. This, Weyl (1948, 336)
says, constitutes the *general principle of 
relativity*. Weyl (1949a, 104) says: 



>  
> 
> ... Only such relations will have objective meaning as are 
> independent of the mapping chosen and therefore remain invariant 
> under deformations of the map. Such a relation is, for instance, the 
> intersection of two world lines. If we wish to characterize a special
> mapping or a special class of mappings, we must do so in terms of the
> real physical events and of the structure revealed in them. That is 
> the content of the *postulate of general relativity*. 
> According to the *special theory of relativity*, it 
> is possible in particular to construct a map of the world such that 
> (1) the world line of each mass point which is subject to no external
> forces appears as a straight line, and (2) the light cone issuing 
> from an arbitrary world point is represented by a circular cone with 
> vertical axis and a vertex angle of 90deg. In this 
> theory the inertial and causal structure and hence also the metrical 
> structure of the world have the character of rigidity, they are 
> absolutely fixed once and for all. It is impossible objectively, 
> without resorting to individual exhibition, to make a narrower 
> selection from among the 'normal mappings' satisfying the
> above conditions (1) and (2).
> 
> 
> 




Weyl (1949a, 115) provides an illustration, which shows how a 
measurement by observer \(B\) of the angular distance \(\delta\) 
between two stars \(\Sigma\) and \(\Sigma^{\*}\) can be 
constructed in the four-dimensional number space, and can be 
expressed as an 
 *invariant*.[110]




![figure](fig15.png)
Figure 15:
Measurement of the angular distance \(\delta\) by an observer \(B\)
between two stars





In figure 15 the stars and observer are represented by their world 
lines, and the past light cone \(K\) issuing from the observation 
event \(O\) intersects the world lines of the stars \(\Sigma\) and 
\(\Sigma^{\*}\) in \(E\) and 
\(E^{\*}\) respectively. The light rays emitted at 
\(E\) and \(E^{\*}\), which arrive at the 
observation event \(O\), are null geodesics laying on the past 
light cone and are respectively denoted by \(\Lambda\) and 
\(\Lambda^{\*}\). This construction of the numerical 
quantity of the angle \(\delta\) observed by \(B\) at \(O\), which
is describable in the form of purely arithmetical relations, is 
invariant under arbitrary coordinate transformations and constitutes 
an objective fact of the 
 world.[111]





On the other hand, the angles between two stars determine the 
*objectively* indescribable *subjective* experience of 
the observer. Moreover, Weyl says, "there is no difference in 
our experiences to which there does not correspond a difference in 
the underlying objective situation." And that difference is 
itself invariant under arbitrary coordinate transformations. In other
words, an observer's subjective experiences *supervene* on the
invariant relationships and structures of a symbolically constructed 
objective world. 




Perhaps no statement captures the contrast between the 
objective-symbolic and the subjective-intuitive more vividly then 
Weyl's famous statement 



>  
> 
> The objective world simply \(is\), it does not *happen*. 
> Only to the gaze of my consciousness, crawling upward along the life 
> line of my body, does a section of this world come to life as a 
> fleeting image in space which continuously changes in 
>  time.[112]
> 
> 
> 
>