## 1. Introduction



There are two competing mathematical strategies that are used in
connection with physical theory; one emphasizes rigor and the other
pragmatics. The pragmatic approach often compromises mathematical
rigor, but offers instead expediency of calculation and elegance of
expression. A case in point is the notion of an infinitesimal, a
non-zero quantity that is smaller than any finite quantity.
Infinitesimals were used by Kepler, Galileo, Newton, Leibniz and many
others in developing and using their respective physical theories,
despite lacking a mathematically rigorous foundation, as Berkeley
clearly showed in his famous 1734 treatise *The Analyst*
criticizing infinitesimals. Such criticisms did not prevent various
18th Century mathematicians, scientists, and engineers such as Euler
and Lagrange from using infinitesimals to get accurate answers from
their calculations. Nevertheless, the pull towards rigor led to the
development in the 19th century of the concept of a limit by Cauchy
and others, which provided a rigorous mathematical framework that
effectively replaced the theory of infinitesimals. A rigorous
foundation was eventually provided for infinitesimals by Robinson
during the second half of the 20th Century, but infinitesimals are
rarely used in contemporary physics. For more on the history of
infinitesimals, see the entry on
 continuity and infinitesimals.



The competing mathematical strategies are manifest in a more recent
discussion concerning the mathematical foundations of quantum
mechanics. In the preface to von Neumann's (1955) treatise on
that topic, he notes that Dirac provides a very elegant and powerful
formal framework for quantum mechanics, but complains about the
central role in that framework of an "improper function with
self-contradictory properties," which he also characterizes as a
"mathematical fiction." He is referring to the Dirac
\(\delta\) function, which has the following incompatible properties:
it is defined over the real line, is zero everywhere except for one
point at which it is infinite, and yields unity when integrated over
the real line. Von Neumann promotes an alternative framework, which he
characterizes as being "just as clear and unified, but without
mathematical objections." He emphasizes that his framework is
not merely a refinement of Dirac's; rather, it is a radically
different framework that is based on Hilbert's theory of
operators.



Dirac is of course fully aware that the \(\delta\) function is not a
well-defined expression. But he is not troubled by this for two
reasons. First, as long as one follows the rules governing the
\(\delta\) function (such as using the \(\delta\) function only under
an integral sign, meaning in part not asking the value of a \(\delta\)
function at a given point), then no inconsistencies will arise.
Second, the \(\delta\) function can be eliminated, meaning that it can
be replaced with a well-defined mathematical expression. However, the
drawback in that case is, according to Dirac, that the substitution
leads to a more cumbersome expression that obscures the argument. In
short, when pragmatics and rigor lead to the same conclusion,
pragmatics trumps rigor due to the resulting simplicity, efficiency,
and increase in understanding.



As in the case of the notion of an infinitesimal, the Dirac \(\delta\)
function was eventually given a mathematically rigorous foundation.
That was done within Schwartz's theory of distributions, which
was later used in developing the notion of a rigged Hilbert space. The
theory of distributions was used to provide a mathematical framework
for quantum field theory (Wightman 1964). The rigged Hilbert space was
used to do so for quantum mechanics (Bohm 1966) and then for
quantum field theory (Bogoluliubov et al. 1975).



The complementary approaches, rigor and pragmatics, which are
exhibited in the development of quantum mechanics, later came about in
a more striking way in connection with the development of quantum
electrodynamics (QED) and, more generally, quantum field theory (QFT).
The emphasis on rigor emerges in connection with two frameworks,
algebraic QFT and Wightman's axiomatic QFT. Algebraic QFT has
its roots in the work of von Neumann on operator algebras, which was
developed by him in an attempt to generalize the Hilbert space
framework. Wightman's axiomatic QFT has its roots in
Schwartz's theory of distributions, and it was later developed
in the rigged Hilbert space framework. Roughly, the basic distinction
between the two approaches is that the algebra of operators is the
basic mathematical concept in algebraic QFT, while operator-valued
distributions (the quantum analogues of field quantities) are
fundamental in Wightman's axiomatic QFT. It is worth noting that
algebraic QFT is generally formulated axiomatically, and that it is
just as deserving of the name "axiomatic" QFT. However,
that term is often taken to refer specifically to the approach based
on operator-valued distributions. To avoid any possible confusion,
that approach is referred to here as "Wightman's
axiomatic" QFT. The emphasis on pragmatics arises most notably
in Lagrangian QFT, which uses perturbation theory, path integrals, and
renormalization techniques. Although some elements of the theory were
eventually placed on a firmer mathematical foundation, there are still
serious questions about its being a fully rigorous approach on a par
with algebraic and Wightman's axiomatic QFT. Nevertheless, it
has been spectacularly successful in providing numerical results that
are exceptionally accurate with respect to experimentally determined
quantities, and in making possible expedient calculations that are
unrivaled by other approaches.



The two approaches to QFT continue to develop in parallel. Fleming
(2002, pp. 135-136) brings this into focus in his discussion of
differences between Haag's *Local Quantum Physics* (1996)
and Weinberg's *Quantum Field Theory* (1995);
Haag's book presents algebraic QFT, and Weinberg's book
presents Lagrangian QFT. While both books are ostensibly about the
same subject, Haag gives a precise formulation of QFT and its
mathematical structure, but does not provide any techniques for
connecting with experimentally determined quantities, such as
scattering cross sections. Weinberg gives a pragmatic formulation that
engages with physical intuition and provides heuristics that are
important for performing calculations; however, it is not as
mathematically rigorous. Moreover, there are a number of important
topics that are examined in one book while not even mentioned in the
other. For example, unitarily inequivalent representations are
discussed by Haag, but not by Weinberg. By contrast, Weinberg
discusses Feynman's rules for path integrals, which are not
mentioned at all by Haag. There is also the issue of demographics.
Most particle and experimental physicists will read and study
Weinberg's book, but very few will read Haag's book.
Because of these differences, Fleming (2002, p. 136) suggests that one
might question whether the two books are really about the same
subject. This gives rise to the question whether any formulation of
QFT is worthy of philosophical attention to its foundations. In
particular, there is a debate between Wallace (2006, 2011) and Fraser
(2009, 2011) over whether an interpretation of QFT should be based on
the standard textbook treatment of QFT or an axiomatic formulation of
QFT.


## 2. Von Neumann and the Foundations of Quantum Theory



In the late 1920s, von Neumann developed the separable Hilbert space
formulation of quantum mechanics, which later became the definitive
one (from the standpoint of mathematical rigor, at least). In the
mid-1930s, he worked extensively on lattice theory (see the entry on
 quantum logic),
 rings of operators, and continuous geometries. Part of his expressed
motivation for developing these mathematical theories was to develop
an appropriate framework for QFT and a better foundation for quantum
mechanics. During this time, he noted two closely related structures,
modular lattices and finite type-II factors (a special type of ring of
operators), that have what he regarded as desirable features for
quantum theory. These observations led to his developing a more
general framework, continuous geometries, for quantum theory. Matters
did not work out as von Neumann had expected. He soon realized that
such geometries must have a transition probability function, if they
are to be used to describe quantum mechanical phenomena, and that the
resulting structure is not a generalization at all beyond the operator
rings that were already available. Moreover, it was determined much
later that the type-III factors are the most important type of ring of
operators for quantum theory. In addition, a similar verdict was
delivered much later with regards to his expectations concerning
lattice theory. The lattices that are appropriate for quantum theory
are orthomodular - a lattice is orthomodular only if it is
modular, but the converse is false. Of the three mathematical
theories, it is the rings of operators that have proven to be the most
important framework for quantum theory. It is possible to use a ring
of operators to model key features of physical systems in a purely
abstract, algebraic setting (this is discussed in section 4.1). A
related issue concerns whether it is necessary to choose a
representation of the ring in a Hilbert space; see Haag and Kastler
(1964), Ruetsche (2003), and Kronz and Lupher (2005) for further
discussion of this issue. In any case, the separable Hilbert space
remains a crucial framework for quantum theory. The simplest examples
of separable Hilbert spaces are the finite dimensional ones, in which
case the algebra of operators is a type-I\(\_n\) factor (n is a
positive integer). The operators are n-by-n complex matrices, which
are typically used to describe internal degrees of freedom such as
spin. Readers wanting to familiarize themselves with these basic
examples should consult the entry on
 quantum mechanics.


### 2.1 The Separable Hilbert Space Formulation of Quantum Mechanics



Matrix mechanics and wave mechanics were formulated roughly around the
same time between 1925 and 1926. In July 1925, Heisenberg finished his
seminal paper "On a Quantum Theoretical Interpretation of
Kinematical and Mechanical Relations". Two months later, Born
and Jordan finished their paper, "On Quantum Mechanics",
which is the first rigorous formulation of matrix mechanics. Two
months after this, Born, Heisenberg, and Jordan finished "On
Quantum Mechanics II", which is an elaboration of the earlier
Born and Jordan paper; it was published in early 1926. These three
papers are reprinted in van der Waerden (1967). Meanwhile,
Schrodinger was working on what eventually became his four famous
papers on wave mechanics. The first was received by *Annalen der
Physik* in January 1926, the second was received in February, and
then the third in May and the fourth in June. All four are reprinted
in Schrodinger (1928).



Schrodinger was the first to raise the question of the
relationship between matrix mechanics and wave mechanics in
Schrodinger (1926), which was published in *Annalen* in
spring 1926 between the publication of his second and third papers of
the famous four. This paper is also reprinted in Schrodinger
(1928). It contains the germ of a mathematical equivalence proof, but
it does not contain a rigorous proof of equivalency: the mathematical
framework that Schrodinger associated with wave mechanics is a
space of continuous and normalizable functions, which is too small to
establish the appropriate relation with matrix mechanics. Shortly
thereafter, Dirac and Jordan independently provided a unification of
the two frameworks. But their respective approaches required essential
use of \(\delta\) functions, which were suspect from the standpoint of
mathematical rigor. In 1927, von Neumann published three papers in
*Gottinger Nachrichten* that placed quantum mechanics on a
rigorous mathematical foundation and included a rigorous proof (i.e.,
without the use of \(\delta\) functions) of the equivalence of matrix
and wave mechanics. These papers are reprinted in von
Neumann(1961-1963, Volume I, Numbers 8-10). In the preface
to his famous 1932 treatise on quantum mechanics (von Neumann 1955),
which is an elegant summary of the separable Hilbert space formulation
of quantum mechanics that he provided in the earlier papers, he
acknowledges the simplicity and utility of Dirac's formulation
of quantum mechanics, but finds it ultimately unacceptable. He
indicates that he cannot endure the use of what could then only be
regarded as mathematical fictions. Examples of these fictions include
Dirac's assumption that every self-adjoint operator can be put
in diagonal form and his use of \(\delta\) functions, which von
Neumann characterizes as "improper functions with
self-contradictory properties". His stated purpose is to
formulate a framework for quantum mechanics that is mathematically
rigorous.



What follows is a brief sketch of von Neumann's strategy. First,
he recognized the mathematical framework of matrix mechanics as what
would now be characterized as an infinite dimensional, separable
Hilbert space. Here the term "Hilbert space" denotes
a complete vector space with an inner product; von Neumann imposed the
additional requirement of separability (having a countable basis) in
his definition of a Hilbert space. He then attempted to specify a set
of functions that would instantiate an (infinite-dimensional)
separable Hilbert space and could be identified with
Schrodinger's wave mechanics. He began with the space of
square-integrable functions on the real line. To satisfy the
completeness condition, that all Cauchy sequences of functions
converge (in the mean) to some function in that space, he specified
that integration must be defined in the manner of Lebesgue. To define
an inner product operation, he specified that the set of Lebesgue
square-integrable functions must be partitioned into equivalence
classes modulo the relation of differing on a set of measure zero.
That the elements of the space are equivalence classes of functions
rather than functions is sometimes overlooked, and it has interesting
ramifications for interpretive investigations. It has been argued in
Kronz (1999), for example, that separable Hilbert space is not a
suitable framework for quantum mechanics under Bohm's
ontological interpretation (also known as
 Bohmian mechanics).


### 2.2 Rings of Operators, Quantum Logics, and Continuous Geometries



In a letter to Birkhoff from 1935, von Neumann says: "I would
like to make a confession which may seem immoral: I do not believe in
Hilbert space anymore"; the letter is published in von Neumann
(2005). The confession is indeed startling since it comes from the
champion of the separable Hilbert space formulation of quantum
mechanics and it is issued just three years after the publication of
his famous treatise, the definitive work on the subject. The irony is
compounded by the fact that less than two years after his confession
to Birkhoff, his mathematical theorizing about the abstract
mathematical structure that was to supersede the separable Hilbert
space, continuous geometries with a transition probability, turned out
not to provide a generalization of the separable Hilbert space
framework. It is compounded again with interest in that subsequent
developments in mathematical physics initiated and developed by von
Neumann ultimately served to strengthen the entrenchment of the
separable Hilbert space framework in mathematical physics (especially
with regards to quantum theory). These matters are explained in more
detail in Section 4.1.



Three theoretical developments come together for von Neumann in his
theory of continuous geometries during the seven years following 1932:
the algebraic approach to quantum mechanics, quantum logics, and rings
of operators. By 1934, von Neumann had already made substantial moves
towards an algebraic approach to quantum mechanics with the help of
Jordan and Wigner - their article, "On an Algebraic
Generalization of the Quantum Mechanical Formalism",  is
reprinted in von Neumann (1961-1963, Vol. II, No. 21). In 1936,
he published a second paper on this topic, "On an Algebraic
Generalization of the  Quantum Mechanical Formalism (Part
I)", which is reprinted in von Neumann (1961-1963, Vol.
III, No. 9). Neither work was particularly influential, as it turns
out. A related paper by von Neumann and Birkhoff, "The Logic of
Quantum Mechanics", was also published in 1936, and it is
reprinted in von Neumann (1961-1963, Vol. IV, No. 7). It was
seminal to the development of a sizeable body of literature on
 quantum logics.
 It should be noted, however, that this happens only after modularity,
a key postulate for von Neumann, is replaced with orthomodularity (a
weaker condition). The nature of the shift is clearly explained in
Holland (1970): modularity is in effect a weakening of the
distributive laws (limiting their validity to certain selected triples
of lattice elements), and orthomodularity is a weakening of modularity
(limiting the validity of the distributive laws to an even smaller set
of triples of lattice elements). The shift from modularity to
orthomodularity was first made in (Loomis 1955). Rapid growth of
literature on orthomodular lattices and the foundations of quantum
mechanics soon followed. For example, see Pavicic (1992) for
a fairly exhaustive bibliography of quantum logic up to 1990, which
has over 1800 entries.



Of substantially greater note for the foundations of quantum theory
are six papers by von Neumann (three jointly published with Murray) on
rings of operators, which are reprinted in von Neumann
(1961-1963, Vol. III, Nos 2-7). The first two, "On
Rings of Operators" and a sequel "On Rings of Operators
II", were published in 1936 and 1937, and they were seminal to
the development of the other four. The third, "On Rings of
Operators: Reduction Theory", was written during 1937-1938
but not published until 1949. The fourth, "On Infinite Direct
Products", was published in 1938. The remaining two, "On
Rings of Operators III" and "On Rings of Operators
IV" were published in 1941 and 1943, respectively. This massive
work on rings of operators was very influential and continues to have
an impact in pure mathematics, mathematical physics, and the
foundations of physics. Rings of operators are now referred to as
"von Neumann algebras" following Dixmier (1981), who first
referred to them by this name (stating that he did so following a
suggestion made to him by Dieudonne) in the introduction to his
1957 treatise on operator algebras (Dixmier 1981).



A von Neumann algebra is a \(\*\)-subalgebra of the set
of bounded operators B(H) on a Hilbert space H that is closed in the
weak operator topology. It is usually assumed that the von Neumann
algebra contains the identity operator. A \(\*\)-subalgebra
contains the adjoint of every operator in
the algebra, where the "\(\*\)" denotes the
adjoint. There are special types of von Neumann algebras that are
called "factors". A von Neumann algebra is a factor, if
its center (which is the set of elements that commute with all
elements of the algebra) is trivial, meaning that it only contains
scalar multiples of the identity element. Moreover, von Neumann showed
in his reduction-theory paper that all von Neumann algebras that are
not factors can be decomposed as a direct sum (or integral) of
factors. There are three mutually exclusive and exhaustive factor
types: type-I, type-II, and type-III. Each type has been classified
into (mutually exclusive and exhaustive) sub-types: types I\(\_n\) \((n
= 1,2,\ldots ,\infty),\) II\(\_n\) \((n = 1,\infty),\) III\(\_z\)
\((0\le z\le 1).\) As mentioned above, type-I\(\_n\) correspond to
finite dimensional Hilbert spaces, while type-I\(\_{\infty}\)
corresponds to the infinite dimensional separable Hilbert space that
provides the rigorous framework for wave and matrix mechanics. Von
Neumann and Murray distinguished the subtypes for type-I and type-II,
but were not able to do so for the type-III factors. Subtypes were not
distinguished for these factors until the 1960s and 1970s - see
Chapter 3 of Sunder (1987) or Chapter 5 of Connes (1994) for
details.



As a result of his earlier work on the foundations of quantum
mechanics and his work on quantum logic with Birkhoff, von Neumann
came to regard the type-II\(\_1\) factors as likely to be the most
relevant for physics. This is a substantial shift since the most
important class of algebra of observables for quantum mechanics was
thought at the time to be the set of bounded operators on an
infinite-dimensional separable Hilbert space, which is a
type-I\(\_{\infty}\) factor. A brief explanation for this shift is
provided below. See the well-informed and lucid account presented in
(Redei 1998) for a much fuller discussion of von
Neumann's views on fundamental connections between quantum
logic, rings of operators (particularly type-II\(\_1\) factors),
foundations of probability theory, and quantum physics. It is worth
noting that von Neumann regarded the type-III factors as a catch-all
class for the "pathological" operator algebras; indeed, it
took several years after the classificatory scheme was introduced to
demonstrate the existence of such factors. It is ironic that the
predominant view now seems to be that the type-III factors are the
most relevant class for physics (particularly for QFT and quantum
statistical mechanics). This point is elaborated further in Section
4.1 after explaining below why von Neumann's program never came
to fruition.



In the introduction to the first paper in the series of four entitled
"On Rings of Operators", Murray and von Neumann list two
reasons why they are dissatisfied with the separable Hilbert space
formulation of quantum mechanics. One has to do with a property of the
trace operation, which is the operation appearing in the definition of
the probabilities for measurement results (the Born rule), and the
other with domain problems that arise for unbounded observable
operators. The trace of the identity is infinite when the separable
Hilbert space is infinite-dimensional, which means that it is not
possible to define a correctly normalized *a priori*
probability for the outcome of an experiment (i.e., a measurement of
an observable). By definition, the *a priori* probability for
an experiment is that in which any two distinct outcomes are equally
likely. Thus, the probability must be zero for each distinct outcome
when there is an infinite number of such outcomes, which can occur if
and only if the space is infinite dimensional. It is not clear why von
Neumann believed that it is necessary to have an *a priori*
probability for every experiment, especially since von Mises clearly
believed that a priori probabilities ("uniform
distributions" in his terminology) do not always exist (von
Mises 1981, pp. 68 ff.) and von Neumann was influenced substantially
by von Mises on the foundations of probability (von Neumann 1955, p.
198 fn.). Later, von Neumann changed the basis for his expressed
reason for dissatisfaction with infinite dimensional Hilbert spaces
from probabilistic to algebraic considerations (Birkhoff and von
Neumann 1936, p. 118); namely, that it violates Hankel's
principle of the preservation of formal law, which leads one to try to
preserve modularity - a condition that holds in
finite-dimensional Hilbert spaces but not in infinite-dimensional
Hilbert spaces. The problem with unbounded operators arises from their
only being defined on a merely dense subset of the set elements of the
space. This means that algebraic operations of unbounded operators
(sums and products) cannot be generally defined; for example, it is
possible that two unbounded operators \(A\), \(B\)
are such that the range of \(B\) and
the domain of \(A\) are disjoint, in which case the
product \(AB\) is meaningless.



The problems mentioned above do not arise for type-I\(\_n\) factors, if
\(n\lt \infty\), nor do they arise for type-II\(\_1\). That is to say,
these factor types have a finite trace operation and are not plagued
with the domain problems of unbounded operators. Particularly
noteworthy is that the lattice of projections of each of these factor
types (type-I\(\_n\) for \(n\lt \infty\) and type-II\(\_1)\) is modular.
By contrast, the set of bounded operators on an infinite-dimensional
separable Hilbert space, a type-I\(\_{\infty}\) factor, is not modular;
rather, it is only orthomodular. These considerations serve to explain
why von Neumann regarded the type-II\(\_1\) factor as the proper
generalization of the type-I\(\_n\) \((n\lt \infty)\) for quantum
physics rather than the type-I\(\_{\infty}\) factors. The shift in the
literature from modular to orthomodular lattices that was
characterized above is in effect a shift back to von Neumann's
earlier position (prior to his confession). But, as was already
mentioned, it now seems that this was not the best move either.



It was von Neumann's hope that his program for generalizing
quantum theory would emerge from a new mathematical structure known as
"continuous geometry". He wanted to use this structure to
bring together the three key elements that were mentioned above: the
algebraic approach to quantum mechanics, quantum logics, and rings of
operators. He sought to forge a strong conceptual link between these
elements and thereby provide a proper foundation for generalizing
quantum mechanics that does not make essential use of Hilbert space
(unlike rings of operators). Unfortunately, it turns out that the
class of continuous geometries is too broad for the purposes of
axiomatizing quantum mechanics. The class must be suitably restricted
to those having a transition probability. It turns out that there is
then no substantial generalization beyond the separable Hilbert space
framework. An unpublished manuscript that was finished by von Neumann
in 1937 was prepared and edited by Israel Halperin, and then published
as von Neumann (1981). A review of the manuscript by Halperin was
published in von Neumann (1961-1963, Vol. IV, No. 16) years
before the manuscript itself was published. In that review, Halperin
notes the following:



> 
> The final result, after 200 pages of deep reasoning is (essentially):
> every such geometry with transition probability can be identified with
> the projection geometry of a finite factor in some finite or infinite
> dimensional Hilbert space (I\(\_m\) or II\(\_1)\). This result indicates
> that continuous geometries do not provide new useful mathematical
> descriptions of quantum mechanical phenomena beyond that already
> available from rings of operators.
> 



This unfortunate development does not, however, completely undermine
von Neumann's efforts to generalize quantum mechanics. On the
contrary, his work on rings of operators does provide significant
light to the way forward. The upshot of subsequent developments is
that von Neumann settled on the wrong factor type for the foundations
of physics.


## 3. Dirac and the Foundations of Quantum Theory



Dirac's formal framework for quantum mechanics was very useful
and influential despite its lack of mathematical rigor. It was used
extensively by physicists and it inspired some powerful mathematical
developments in functional analysis. Eventually, mathematicians
developed a suitable framework for placing Dirac's formal
framework on a firm mathematical foundation, which is known as a
*rigged Hilbert space* (and is also referred to as a
*Gelfand Triplet*). This came about as follows. A rigorous
definition of the \(\delta\) function became possible in distribution
theory, which was developed by Schwartz from the mid-1940s to the
early 1950s. Distribution theory inspired Gelfand and collaborators
during the mid-to-late 1950s to formulate the notion of a rigged
Hilbert space, the firm foundation for Dirac's formal framework.
This development was facilitated by Grothendiek's notion of a
nuclear space, which he introduced in the mid-1950s. The rigged
Hilbert space formulation of quantum mechanics was then developed
independently by Bohm and by Roberts in 1966. Since then, it has
been extended to a variety of different contexts in the quantum domain
including decay phenomena and the arrow of time. The mathematical
developments of Schwartz, Gelfand, and others had a substantial effect
on QFT as well. Distribution theory was taken forward by Wightman in
developing the axiomatic approach to QFT from the mid-1950s to the
mid-1960s. In the late 1960s,  the axiomatic approach was
explicitly put into the rigged Hilbert space framework by Bogoliubov
and co-workers.



Although these developments were only indirectly influenced by Dirac,
by way of the mathematical developments that are associated with his
formal approach to quantum mechanics, there are other elements of his
work that had a more direct and very substantial impact on the
development of QFT. In the 1930s, Dirac (1933) developed a Lagrangian
formulation of quantum mechanics and applied it to quantum fields ,
and the latter inspired Feynman (1948) to develop the path-integral
approach to QFT. The mathematical foundation for path-integral
functionals is still lacking (Rivers 1987, pp, 109-134), though
substantial progress has been made (DeWitt-Morette *et al*.
1979). Despite such shortcomings, it remains the most useful and
influential approach to QFT to date. In the 1940s, Dirac (1943)
developed a form of quantum electrodynamics that involved an
indefinite metric - see also Pauli (1943) in that connection.
This had a substantial influence on later developments, first in
quantum electrodynamics in the early 1950s with the Gupta-Bluer
formalism, and in a variety of QFT models such as vector meson fields
and quantum gravity fields by the late 1950s - see Chapter 2 of
Nagy (1966) for examples and references.


### 3.1 Dirac's \(\delta\) Function, *Principles*, and Bra-Ket Notation



Dirac's attempt to prove the equivalence of matrix mechanics and
wave mechanics made essential use of the \(\delta\) function, as
indicated above. The \(\delta\) function was used by physicists before
Dirac, but it became a standard tool in many areas of physics only
after Dirac very effectively put it to use in quantum mechanics. It
then became widely known by way of his textbook (Dirac 1930), which
was based on a series of lectures on quantum mechanics given by Dirac
at Cambridge University. This textbook saw three later editions: the
second in 1935, the third in 1947, and the fourth in 1958. The fourth
edition has been reprinted many times. Its staying power is due, in
part, to another innovation that was introduced by Dirac in the third
edition, his bra-ket formalism. He first published this formalism in
(Dirac 1939), but the formalism did not become widely used until after
the publication of the third edition of his book. There is no question
that these tools, first the \(\delta\) function and then the bra-ket
notation, were extremely effective for physicists practicing and
teaching quantum mechanics both with regards to setting up equations
and to the performance of calculations. Most quantum mechanics
textbooks use \(\delta\) functions and plane waves, which are key
elements of Dirac's formal framework, but they are not included
in von Neumann's rigorous mathematical framework for quantum
mechanics. Working physicists as well as teachers and students of
quantum mechanics often use Dirac's framework because of its
simplicity, elegance, power, and relative ease of use. Thus, from the
standpoint of pragmatics, Dirac's framework is much preferred
over von Neumann's. The notion of a rigged Hilbert space placed
Dirac's framework on a firm mathematical foundation.


### 3.2 The Rigged Hilbert Space Formulation of Quantum Mechanics



Mathematicians worked very hard to provide a rigorous foundation for
Dirac's formal framework. One key element was Schwartz's
(1945; 1950-1951) theory of distributions. Another key element,
the notion of a nuclear space, was developed by Grothendieck (1955).
This notion made possible the generalized-eigenvector decomposition
theorem for self-adjoint operators in rigged Hilbert space - for
the theorem see Gelfand and Vilenken (1964, pp. 119-127), and
for a brief historical account of the convoluted path leading to it
see Berezanskii (1968, pp. 756-760). The decomposition principle
provides a rigorous way to handle observables such as position and
momentum in the manner in which they are presented in Dirac's
formal framework. These mathematical developments culminated in the
early 1960s with Gelfand and Vilenkin's characterization of a
structure that they referred to as a *rigged Hilbert space*
(Gelfand and Vilenkin 1964, pp. 103-127). It is unfortunate that
their chosen name for this mathematical structure is doubly
misleading. First, there is a natural inclination to regard it as
denoting a type of Hilbert space, one that is *rigged* in some
sense, but this inclination must be resisted. Second, the term
*rigged* has an unfortunate connotation of illegitimacy, as in
the terms *rigged election* or *rigged roulette table*,
and this connotation must be dismissed as prejudicial. There is
nothing illegitimate about a rigged Hilbert space from the standpoint
of mathematical rigor (or any other relevant standpoint). A more
appropriate analogy may be drawn using the notion of a rigged ship:
the term *rigged* in this context means fully equipped. But
this analogy has its limitations since a rigged ship is a fully
equipped ship, but (as the first point indicates) a rigged Hilbert
space is not a Hilbert space, though it is generated from a Hilbert
space in the manner now to be described.



A rigged Hilbert space is a dual pair of spaces \((\Phi , \Phi^x)\)
that can generated from a separable Hilbert space \(\Eta\) using a
sequence of norms (or semi-norms); the sequence of norms is generated
using a nuclear operator (a good approximate meaning is an operator of
trace-class, meaning that the trace of the modulus of the operator is
finite). In the mathematical theory of topological vector spaces, the
space \(\Phi\) is characterized in technical terms as a *nuclear
Frechet space*. To say that \(\Phi\) is a
*Frechet space* means that it is a complete metric
space, and to say that it is *nuclear* means that it is the
projective limit of a sequence of Hilbert spaces in which the
associated topologies get rapidly finer with increasing n (i.e., the
convergence conditions are increasingly strict); the term
*nuclear* is used because the Hilbert-space topologies are
generated using a nuclear operator. In distribution theory, the space
\(\Phi\) is characterized as a test-function space, where a
test-function is thought of as a very well-behaved function (being
continuous, n-times differentiable, having a bounded domain or at
least dropping off exponentially beyond some finite range, etc).
\(\Phi^x\) is a space of distributions, and it is the topological dual
of \(\Phi\), meaning that it corresponds to the complete space of
continuous linear functionals on \(\Phi\). It is also the inductive
limit of a sequence of Hilbert spaces in which the topologies get
rapidly coarser with increasing n. Because the elements of \(\Phi\)
are so well-behaved, \(\Phi^x\) may contain elements that are not so
well-behaved, some being singular or improper functions (such as
Dirac's \(\delta\) function). \(\Phi\) is the topological
anti-dual of \(\Phi^x\), meaning that it is the complete set of
continuous anti-linear functionals on \(\Phi^x\); it is anti-linear
rather than linear because multiplication by a scalar is defined in
terms of the scalar's complex conjugate.



It is worth noting that neither \(\Phi\) nor \(\Phi^x\) is a Hilbert
space in that each lacks an inner product that induces a metric with
respect to which the space is complete, though for each space there is
a topology with respect to which the space is complete. Nevertheless,
each of them is closely related to the Hilbert space \(\Eta\) from
which they are generated: \(\Phi\) is densely embedded in \(\Eta\),
which in turn is densely embedded in \(\Phi^x\). Two other points are
worth noting. First, dual pairs of this sort can also be generated
from a pre-Hilbert space, which is a space that has all the features
of a Hilbert space except that it is not complete, and doing so has
the distinct advantage of avoiding the partitioning of functions into
equivalence classes (in the case of functions spaces). The term
*rigged Hilbert space* is typically used broadly to include
dual pairs generated from either a Hilbert space or a pre-Hilbert
space. Second, the term *Gelfand triplet* is sometimes used
instead of the term *rigged Hilbert space*, though it refers to
the ordered set \((\Phi , \Eta , \Phi^x)\), where \(\Eta\) is the
Hilbert space used to generate \(\Phi\) and \(\Phi^x\).



The dual pair \((\Phi , \Phi^x)\) possesses the means to represent
important operators for quantum mechanics that are problematic in a
separable Hilbert space, particularly the unbounded operators that
correspond to the observables position and momentum, and it does so in
a particularly effective and unproblematic manner. As already noted,
these operators have no eigenvalues or eigenvectors in a separable
Hilbert space; moreover, they are only defined on a dense subset of
the elements of the space and this leads to domain problems. These
undesirable features also motivated von Neumann to seek an alternative
to the separable Hilbert space framework for quantum mechanics, as
noted above. In a rigged Hilbert space, the
operators corresponding to position and momentum can have a
complete set of eigenfunctionals (i.e., generalized eigenfunctions).
The key result is known as the nuclear spectral theorem (and it is
also known as the Gelfand-Maurin theorem). One version of the theorem
says that if A is a symmetric linear operator defined on the space
\(\Phi\) and it admits a self-adjoint extension to the Hilbert space
H, then A possesses a complete system of eigenfunctionals belonging to
the dual space \(\Phi^x\) (Gelfand and Shilov 1977, chapter 4). That
is to say, provided that the stated condition is satisfied, A can be
extended by duality to \(\Phi^x\), its extension \(A^x\) is continuous
on \(\Phi^x\) (in the operator topology in \(\Phi^x)\), and \(A^x\)
satisfies a completeness relation (meaning that it can be decomposed
in terms of its eigenfunctionals and their associated eigenvalues).
The duality formula for extending \(A\) to \(\Phi^x\)
is \(\braket{\phi}{A^x\kappa} = \braket{A\phi}{\kappa}\), for all
\(\phi \in \Phi\) and for all \(\kappa \in \Phi^x\). The completeness
relation says that for all \(\phi ,\theta \in \Phi\):

 

\[ \braket{A\phi}{\theta} = \int\_{v(A)} \lambda \braket{\phi}{\lambda} \braket{\lambda}{\theta}^\* \mathrm{d}\mu(\lambda), \]


where \(v(A)\) is the set of all generalized eigenvalues of \(A^x\)
(i.e., the set of all scalars \(\lambda\) for which there is \(\lambda
\in \Phi^x\) such that \(\braket{\phi}{A^x\lambda} = \lambda
\braket{\phi}{\lambda}\) for all \(\phi \in \Phi)\).



The rigged Hilbert space representation of these observables is about
as close as one can get to Dirac's elegant and extremely useful
formal representation with the added feature of being placed within a
mathematically rigorous framework. It should be noted, however, that
there is a sense in which it is a proper generalization of
Dirac's framework. The rigging (based on the choice of a nuclear
operator that determines the test function space) can result in
different sets of generalized eigenvalues being associated with an
operator. For example, the set of (generalized) eigenvalues for the
momentum operator (in one dimension) corresponds to the real line, if
the space of test functions is the set \(S\) of
infinitely differentiable functions of \(x\) which
together with all derivatives vanish faster than any inverse power of
\(x\) as \(x\) goes to infinity, whereas
its associated set of eigenvalues is the complex plane, if the space
of test functions is the set \(D\) of infinitely
differentiable functions with compact support (i.e., vanishing outside
of a bounded region of the real line). If complex eigenvalues are not
desired, then \(S\) would be a more appropriate choice
than \(D\) - see Nagel (1989) for a brief
discussion. But there are situations in which it is desirable for an
operator to have complex eigenvalues. This is so, for example, when a
system exhibits resonance scattering (a type of decay phenomenon), in
which case one would like the Hamiltonian to have complex eigenvalues
- see Bohm & Gadella (1989). (Of course, it is
impossible for a self-adjoint operator to have complex eigenvalues in
a Hilbert space.)



Soon after the development of the theory of rigged Hilbert spaces by
Gelfand and his associates, the theory was used to develop a new
formulation of quantum mechanics. This was done independently by
Bohm (1966) and Roberts (1966). It was later demonstrated that
the rigged Hilbert space formulation of quantum mechanics can handle a
broader range of phenomena than the separable Hilbert space
formulation. That broader range includes scattering resonances and
decay phenomena (Bohm and Gadella 1989), as already noted.
Bohm (1997) later extended this range to include a quantum
mechanical characterization of the arrow of time. The Prigogine school
developed an alternative characterization of the arrow of time using
the rigged Hilbert space formulation of quantum mechanics (Antoniou
and Prigogine 1993). Kronz (1998, 2000) used this formulation to
characterize quantum chaos in open quantum systems. Castagnino and
Gadella (2003) used it to characterize
 decoherence
 in closed quantum systems.


## 4 Mathematical Rigor: Two Paths


### 4.1 Algebraic Quantum Field Theory



In 1943, Gelfand and Neumark published an important paper on an
important class of normed rings, which are now known as abstract
\(C^\*\)-algebras. Their paper was influenced by Murray and von
Neumann's work on rings of operators, which was discussed in the
previous section. In their paper, Gelfand and Neumark focus attention
on abstract normed \(\*\)-rings. They show that any
\(C^\*\)-algebra can be given a concrete representation in a Hilbert
space (which need not be separable). That is to say, there is an
isomorphic mapping of the elements of a \(C^\*\)-algebra into the set
of bounded operators of the Hilbert space. Four years later, Segal
(1947a) published a paper that served to complete the work of Gelfand
and Neumark by specifying the definitive procedure for constructing
concrete (Hilbert space) representations of an abstract
\(C^\*\)-algebra. It is called the GNS construction (after Gelfand,
Neumark, and Segal). That same year, Segal (1947b) published an
algebraic formulation of quantum mechanics, which was substantially
influenced by (though deviating somewhat from) von Neumann's
(1963, Vol. III, No. 9) algebraic formulation of quantum mechanics,
which is cited in the previous section. It is worth noting that
although \(C^\*\)-algebras satisfy Segal's postulates, the
algebra that is specified by his postulates is a more general
structure known as a Segal algebra. Every \(C^\*\)-algebra is a Segal
algebra, but the converse is false since Segal's postulates do
not require an adjoint operation to be defined. If a Segal algebra is
isomorphic to the set of all self-adjoint elements of a
\(C^\*\)-algebra, then it is a special or exceptional Segal algebra.
Although the mathematical theory of Segal algebras has been fairly
well developed, a \(C^\*\)-algebra is the most important type of
algebra that satisfies Segal's postulates.



The algebraic formulations of quantum mechanics that were developed by
von Neumann and Segal did not change the way that quantum mechanics
was done. Nevertheless, they did have a substantial impact in two
related contexts: QFT and quantum statistical mechanics. The key
difference leading to the impact has to do with the domain of
applicability. The domain of quantum mechanics consists of finite
quantum systems, meaning quantum systems that have a finite number of
degrees of freedom. Whereas in QFT and quantum statistical mechanics,
the systems of special interest - i.e., quantum fields and
particle systems in the thermodynamic limit, respectively - are
infinite quantum systems, meaning quantum systems that have an
infinite number of degrees of freedom. Dirac (1927) was the first to
recognize the importance of infinite quantum systems for QFT, which is
reprinted in Schwinger (1958).



Segal (1959, p. 5) was the first to suggest that the beauty and power
of the algebraic approach becomes evident when working with an
infinite quantum system . The key advantage of the algebraic approach,
according to Segal (1959, pp. 5-6), is that one may work in the
abstract algebraic setting where it is possible to obtain interacting
fields from free fields by an automorphism on the algebra, one that
need not be unitarily implementable. Segal notes (1959, p. 6) that von
Neumann (1937) had a similar idea (that field dynamics are to be
expressed as an automorphism on the algebra) in an unpublished
manuscript. Segal notes this advantage in response to a result
obtained by Haag (1955), that field theory representations of free
fields are unitarily inequivalent to representations of interacting
fields. Haag mentions that von Neumann (1938) first discovered
'different' (unitarily inequivalent) representations much
earlier. A different way of approaching unitarily equivalent
representations, by contrast with Segal's approach, was later
presented by Haag and Kastler (1964), who argued that unitarilty
inequivalent representations are physically equivalent. Their notion
of physical equivalence was based on Fell's mathematical idea of
weak equivalence (Fell 1960).



After indicating important similarities between his and von
Neumann's approaches to infinite quantum systems, Segal draws an
important contrast that serves to give the advantage to his approach
over von Neumann's. The key mathematical difference, according
to Segal, is that von Neumann was working with a weakly closed ring of
operators (meaning that the ring of operators is closed with respect
to the weak operator topology), whereas Segal is working with a
uniformly closed ring of operators (closed with respect to the uniform
topology). It is crucial because it has the following interpretive
significance, which rests on operational considerations:



> 
> The present intuitive idea is roughly that the only measurable
> field-theoretic variables are those that can be expressed in terms of
> a *finite* number of canonical operators, or *uniformly*
> approximated by such; the technical basis is a *uniformly*
> closed ring (more exactly, an abstract \(C^\*\)-algebra). The crucial
> difference between the two varieties of approximation arises from the
> fact that, in general, weak approximation has only analytical
> significance, while uniform approximation may be defined
> operationally, two observables being close if the maximum (spectral)
> value of the difference is small (Segal 1959, p. 7).
> 



Initially, it appeared that Segal's assessment of the relative
merits of von Neumann algebras and \(C^\*\)-algebras with respect to
physics was substantiated by a seminal paper, (Haag and Kastler 1964).
Among other things, Haag and Kastler introduced the key axioms of the
algebraic approach to QFT. They also argued that unitarily
inequivalent representations are "physically equivalent"
to each other. However, the use of physical equivalence to show that
unitarily inequivalent representations are not physically significant
has been challenged; see Kronz and Lupher (2005), Lupher (2018), and
Ruetsche (2011). The prominent role of type-III factor von Neumann
algebras within the algebraic approach to quantum statistical
mechanics and QFT raises further doubts about Segal's
assessment.



The algebraic approach has proven most effective in quantum
statistical mechanics. It is extremely useful for characterizing many
important macroscopic quantum effects including crystallization,
ferromagnetism, superfluidity, structural phase transition,
Bose-Einstein condensation, and superconductivity. A good introductory
presentation is Sewell (1986), and for a more advanced discussion see
Bratteli and Robinson (1979, 1981). In algebraic quantum statistical
mechanics, an infinite quantum system is defined by specifying an
abstract algebra of observables. A particular state may then be used
to specify a concrete representation of the algebra as a set of
bounded operators in a Hilbert space. Among the most important types
of states that are considered in algebraic statistical mechanics are
the equilibrium states, which are often referred to as "KMS
states" (since they were first introduced by the physicists
Kubo, Martin, and Schwinger). There is a continuum of KMS states since
there is at least one KMS state for each possible temperature value
\(\tau\) of the system, for \(0\le \tau \le +\infty\). Given an
automorphism group, each KMS state corresponds to a representation of
the algebra of observables that defines the system, and each of these
representations is unitarily inequivalent to any other. It turns out
that each representation that corresponds to a KMS state is a factor:
if \(\tau = 0\) then it is a type-I factor, if \(\tau = +\infty\) then
it is a type-II factor, and if \(0\lt \tau \lt +\infty\) then it is a
type-III factor. Thus, type-III factors play a predominant role in
algebraic quantum statistical mechanics.



In algebraic QFT, an algebra of observables is associated with bounded
regions of Minkowski spacetime (and unbounded regions including all of
spacetime by way of certain limiting operations) that are required to
satisfy standard axioms of local structure: isotony, locality,
covariance, additivity, positive spectrum, and a unique invariant
vacuum state. The resulting set of algebras on Minkowski spacetime
that satisfy these axioms is referred to as the *net of local
algebras*. It has been shown that special subsets of the net of
local algebras - those corresponding to various types of
unbounded spacetime regions such as tubes, monotones (a tube that
extends infinitely in one direction only), and wedges - are
type-III factors. Of particular interest for the foundations of
physics are the algebras that are associated with bounded spacetime
regions, such as a double cone (the finite region of intersection of a
forward and a backward light cone). As a result of work done over the
last thirty years, local algebras of relativistic QFT appear to be
type III von Neuman algebras see Halvorson (2007, pp. 749-752)
for more details.



One important area for interpretive investigation is the existence of
a continuum of unitarily inequivalent representations of an algebra of
observables. Attitudes towards unitarily inequivalent representations
differ drastically in the philosophical literature. In (Wallace 2006)
unitarily inequivalent representations are not considered a
foundational problem for QFT, while in Ruetsche (2011), Lupher (2018)
and Kronz and Lupher (2005) unitarily inequivalent representations are
considered physically significant.


### 4.2 Wightman's Axiomatic Quantum Field Theory



In the early 1950s, theoretical physicists were inspired to axiomatize
QFT. One motivation for axiomatizing a theory, not the one for the
case now under discussion, is to express the theory in a completely
rigorous form in order to standardize the expression of the theory as
a mature conceptual edifice. Another motivation, more akin to the case
in point, is to embrace a strategic withdrawal to the foundations to
determine how renovation should proceed on a structure that is
threatening to collapse due to internal inconsistencies. One then
looks for existing piles (fundamental postulates) that penetrate
through the quagmire to solid rock, and attempts to drive home others
at advantageous locations. Properly supported elements of the
superstructure (such as the characterization of free fields,
dispersion relations, etc.) may then be distinguished from those that
are untrustworthy. The latter need not be razed immediately, and may
ultimately glean supportive rigging from components not yet
constructed. In short, the theoretician hopes that the axiomatization
will effectively separate sense from nonsense, and that this will
serve to make possible substantial progress towards the development of
a mature theory. Grounding in a rigorous mathematical framework can be
an important part of the exercise, and that was a key aspect of the
axiomatization of QFT by Wightman.



In the mid-1950s, Schwartz's theory of distributions was used by
Wightman (1956) to develop an abstract formulation of QFT, which later
came to be known known as *axiomatic quantum field theory*.
Mature statements of this formulation are presented in Wightman and
Garding (1964) and in Streater and Wightman (1964). It was
further refined in the late 1960s by Bogoliubov, who explicitly placed
axiomatic QFT in the rigged Hilbert space framework (Bogoliubov *et
al*. 1975, p. 256). It is by now standard within the axiomatic
approach to put forth the following six postulates: spectral condition
(there are no negative energies or imaginary masses), vacuum state (it
exists and is unique), domain axiom for fields (quantum fields
correspond to operator-valued distributions), transformation law
(unitary representation in the field-operator (and state) space of the
restricted inhomogeneous Lorentz group -
"restricted" means inversions are excluded, and
"inhomogeneous" means that translations are included),
local commutativity (field measurements at spacelike separated regions
do not disturb one another), asymptotic completeness (the scattering
matrix is unitary - this assumption is sometimes weakened to
cyclicity of the vacuum state with respect to the polynomial algebra
of free fields). Rigged Hilbert space entered the axiomatic framework
by way of the domain axiom, so this axiom will be discussed in more
detail below.



In classical physics, a field is is characterized as a scalar- (or
vector- or tensor-) valued function \(\phi(x)\) on a domain that
corresponds to some subset of spacetime points. In QFT, a field is
characterized by means of an operator rather than a function. A
*field operator* may be obtained from a classical field
function by quantizing the function in the canonical manner -
see Mandl (1959, pp. 1-17). For convenience, the field operator
associated with \(\phi(x)\) is denoted below by the same expression
(since the discussion below only concerns field operators). Field
operators that are relevant for QFT are too singular to be regarded as
realistic, so they are smoothed out over their respective domains
using elements of a space of well-behaved functions known as *test
functions*. There are many different test-functions spaces
(Gelfand and Shilov 1977, Chapter 4). At first, the test-function
space of choice for axiomatic QFT was the *Schwartz space*
\(\Sigma\), the space of functions whose elements have partial
derivatives of all orders at each point and such that each function
and its derivatives decreases faster than \(x^{-n}\) for any \(n\in
N\) as \(x\rightarrow \infty\). It was later determined that some
realistic models require the use of other test-function spaces. The
smoothed field operators \(\phi[f\)] for \(f \in \Sigma\) are known as
*quantum field operators*, and they are defined as follows



\[ \phi[f] = \int d^4 x f(x)\phi(x). \]


The integral (over the domain of the field operator) of the product of
the test function \(f(x)\) and the field operator \(\phi(x)\) serves
to "smooth out" the field operator over its domain; a more
colloquial description is that the field is "smeared out"
over space or spacetime. It is postulated within the axiomatic
approach that a quantum field operator \(\phi[f\)] may be represented
as an unbounded operator on a separable Hilbert space \(\Eta\), and
that \(\{\phi[f]: f\in \Sigma \}\) (the set of smoothed field
operators associated with \(\phi(x))\) has a dense domain \(\Omega\)
in \(\Eta\). The smoothed field operators are often referred to as
*operator-valued distributions*, and this means that for every
\(\Phi,\Psi \in \Omega\) there is an element of the space of
distributions \(\Sigma^x\), the topological dual of \(\Sigma\), that
may be equated to the expression \(\langle \Phi {\mid} \phi[\
]{\mid}\Psi\rangle\). If \(\Omega'\) denotes the set of functions
obtained by applying all polynomials of elements of \(\{\phi[f]: f\in
\Sigma \}\) onto the unique vacuum state, then the axioms mentioned
above entail that \(\Omega'\) is dense in \(\Eta\) (asymptotic
completeness) and that \(\Omega'\subset \Omega\) (domain axiom). The
elements of \(\Omega\) correspond to possible states of the elements
of \(\{\phi[f]: f\in \Sigma \}\). Though only one field has been
considered thus far, the formalism is easily generalizable to a
countable number of fields with an associated set of countably indexed
field operators \(\phi\_k (x)\) - cf. (Streater and Wightman
1964).



As noted earlier, the appropriateness of the rigged Hilbert space
framework enters by way of the domain axiom. Concerning that axiom,
Wightman says the following (in the notation introduced above, which
differs slightly from that used by Wightman).



> 
> At a more advanced stage in the theory it is likely that one would
> want to introduce a topology into \(\Omega\) such that \(\phi[f\)]
> becomes a continuous mapping of \(\Omega\) into \(\Omega\). It is
> likely that this topology has to be rather strong. We want to
> emphasize that so far we have only required that \(\langle
> \Phi{\mid}\phi[f]{\mid}\Psi\rangle\) be continuous in \(f\)
> for \(\Phi ,\Psi\) *fixed*; continuity in the
> pair \(\Phi ,\Psi\) cannot be expected before we put a suitable strong
> topology on \(\Omega\) (Wightman and Garding 1964, p. 137).
> 



In Bogoliubov *et al.* (1975, p. 256), a topology is introduced
to serve this role, though it is introduced on \(\Omega'\) rather than
on \(\Omega\). Shortly thereafter, they assert that it is not hard to
show that \(\Omega'\) is a complete nuclear space with respect to this
topology. This serves to justify a claim they make earlier in their
treatise:



> 
> ... it is precisely the consideration of the triplet of spaces
> \(\Omega \subset \Eta \subset \Omega^\*\) which give a natural basis
> for both the construction of a general theory of linear operators and
> the correct statement of certain problems of quantum field theory
> (Bogoliubov *et al.* 1975, p. 34).
> 



Note that they refer to the triplet \(\Omega \subset \Eta \subset
\Omega^\*\) as a rigged Hilbert space. In the terminology introduced
above, they refer in effect to the Gelfand triplet \((\Omega , \Eta ,
\Omega^x )\) or (equivalently) the associated rigged Hilbert space
\((\Omega , \Omega^x)\) .



Finally, it is worth mentioning that the status of the field in
algebraic QFT differs from that in Wightman's axiomatic QFT. In
both approaches, a field is an abstract system having an infinite
number of degrees of freedom. Sub-atomic quantum particles are field
effects that appear in special circumstances. In algebraic QFT, there
is a further abstraction: the most fundamental entities are the
elements of the algebra of local (and quasi-local) observables, and
the field is a derived notion. The term *local* means bounded
within a finite spacetime region, and an observable is not regarded as
a property belonging to an entity other than the spacetime region
itself. The term *quasi-local* is used to indicate that we take
the union of all bounded spacetime regions. In short, the algebraic
approach focuses on local (or quasi-local) observables and treats the
notion of a field as a derivative notion; whereas the axiomatic
approach (as characterized just above) regards the field concept as
the fundamental notion. Indeed, it is common practice for proponents
of the algebraic approach to distance themselves from the field notion
by referring to their theory as "local quantum physics".
The two approaches are mutually complementary - they have
developed in parallel and have influenced each other by analogy
(Wightman 1976). For a discussion of the close connections between
these two approaches, see Haag (1996, p. 106).


## 5 Philosophical Issues


### 5.1 Pragmatics versus Axiomatics



Most physicists use Lagrangian QFT (LQFT) to make predictions that
have been experimentally verified with extraordinary precision in some
cases. However, LQFT has been described as a "grab bag of
conflicting mathematical ideas" that has not provided a sharp
mathematical description of what counts as a QFT model (Swanson 2017,
pp. 1-2). Those criticisms motivated mathematically inclined
physicists to search for a mathematically rigorous formulation of QFT.
Axiomatic versions of QFT have been favored by mathematical physicists
and most philosophers. With greater mathematical rigor it is possible
to prove results about the theoretical structure of QFT independent of
any particular Lagrangian. Axiomatic QFT provides clear conceptual
frameworks within which precise questions and answers to
interpretational issues can be formulated. There are three main
axiomatic frameworks for QFT: Wightman QFT, Osterwalder-Schrader QFT,
and algebraic QFT. In Wightman QFT, the axioms use functional analysis
and operator algebras and is closer to LQFT since its axioms describe
covariant field operators acting on a fixed Hilbert space. The
Osterwalder-Schrader axioms use a functional integration approach to
QFT. The algebraic QFT axioms use \(C^\*\)-algebras to model local
observables. However, axiomatic QFT approaches are sorely lacking with
regards to building empirically adequate models. Unlike quantum
mechanics which has a canonical mathematical framework in terms of von
Neumann's Hilbert space formulation, QFT has no canonical
mathematical framework. Even though there is a canonical mathematical
framework for quantum mechanics, there are many interpretations of
that framework, e.g., many-worlds, GRW, Copenhagen, Bohmian, etc...
QFT has two levels that require interpretation: (1) which QFT
framework should be the focus of these foundational efforts, if any,
and (2) how that preferred framework should be interpreted. Since (1)
involves issues about mathematical rigor and pragmatic virtues, it
directly bears on the focus of this article. The lack of a canonical
formulation of QFT threatens to impede any metaphysical or
epistemological lessons that might be learned from QFT.



One view is that these two approaches to QFT, the mathematically
rigorous axiomatic approach and the pragmatic / empirically adequate
LQFT approach, are rival research programs (see David Wallace (2006,
2011) and Doreen Fraser (2009, 2011)), though Swanson (2017) argues
that they are not rival programs. Fraser (2009, 2011) argues that the
interpretation of QFT should be based on the mathematically rigorous
approach of axiomatic formulations of QFT. By contrast, Wallace (2006,
2011) argues that an interpretation of QFT should be based on LQFT.
(Wallace, in 2006, calls his preferred QFT framework conventional QFT
(CQFT), but changes his terminology to LQFT in Wallace 2011). Swanson
(2017) and Egg, Lam, and Oldofedi (2017) are good overviews of the
debate between Fraser and Wallace (for an extended analysis see James
Fraser 2016). The debate covers many different philosophical topics in
QFT, which makes it more challenging to pin down exactly what is
essential to the arguments for both sides (for one view of what is
essential for the debate, see Egg, Lam, and Oldofedi 2017). One issue
is the role of internal consistency established by mathematical rigor
versus empirical adequacy. Wallace argues that LQFT is empirically
adequate since it can describe the forces of the Standard Model. LQFT
has a collection of calculational techniques including perturbation
theory, path integrals, and renormalization group methods. One
criticism of LQFT is that the calculational techniques it uses are not
mathematically rigorous. Wallace argues that renormalization group
methods puts perturbative QFT, an approach within LQFT, on
mathematically rigorous ground and removes the main motivation for
axiomatic QFT.


#### 5.1.1 Perturbative Quantum Field Theory



What follows is a rough overview of perturbative QFT (see James Fraser
2016 for more details). Since exactly solvable free QFT models are
more mathematically tractable than interacting QFT models,
perturbative QFT treats interactions as perturbations to the free
Lagrangian assuming weak coupling. For strongly coupled theories like
quantum chromodynamics that idealization fails. Using perturbation
theory, approximate solutions for interacting QFT models can be
calculated by expanding S-matrix elements in a power series in terms
of a coupling parameter. However, the higher order terms will often
contain divergent integrals. Typically, renormalization of the higher
order terms is required to get finite predictions. Two sources of
divergent integrals are infrared (long distance, low energy) and
ultraviolet (short distance, high energy) divergences. Infrared
divergences are often handled by imposing a long distance cutoff or
putting a small non-zero lower limit for the integral over momentum. A
sharp cutoff at low momentum is equivalent to putting the theory in a
finite volume box. Imposing asymptotic boundary conditions and
restricting the observables to long distance "friendly"
observables also help with infrared divergences. Ultraviolet
divergences are often handled by imposing a momentum cutoff to remove
high momentum modes of a theory. That is equivalent to freezing out
variations in the fields at arbitrarily short length scales. Putting
the system on a lattice with some finite spacing can also help deal
with the high momentum. Dimensional regularization, where the integral
measure is redefined to range over a fractional number of dimensions,
can help with both infrared and ultraviolet divergences. The last step
in renormalization is to remove the cutoffs by taking the continuum
limit (i.e., removing the high momentum cutoff) and the infinite
volume limit (i.e., removing the low momentum cutoff). The hope is
that the limit is well-defined and there are finite expressions of the
series at each order.



James Fraser (2016) identifies three problems for perturbative QFT.
(1) *The rigor problem*: perturbative QFT is not mathematically
rigorous which makes it difficult to analyze and interpret. (2)
*The consistency problem*: perturbative calculations rest on
the interaction picture existing, but Haag's theorem seems to
show that the interaction picture does not exist. (3) *The
justification problem*: renormalization lacks physical motivation
and appears ad hoc. James Fraser argues that (1) and (2) do not pose
severe problems for perturbative QFT because it is not attempting to
build continuum QFT models. It is building approximate physical
quantities - not mathematical structures that are to be
interpreted as physical systems.



Baker (2016) and Swanson (2017) note that LQFT makes false or unproven
assumptions such as the convergence of certain infinite sums in
perturbation theory. Dyson (1952) gives a heuristic argument that
quantum electrodynamic perturbation series do not converge. Baker and
Swanson also argue that the use of long distance cutoffs is at odds
with cosmological theory and astronomical observations which suggest
that the universe is spatially infinite. Even in the weak coupling
limit where perturbation theory can be formally applied, it is not
clear when the perturbative QFT gives an accurate approximation of the
underlying physics. In the interacting \(\phi^4\) theory, when the
dimension is less than 4 for Minkowski spacetime, the theory is
nontrivial, but when the dimension is greater than 4, the renormalized
perturbation series is asymptotic to a free field theory even though
it appears to describe nontrivial interactions. When there are 4
dimensions, the theory is also trivial if additional technical
assumptions hold (see Swanson 2017 (p. 3) for more details).


#### 5.1.2 Path Integrals in Quantum Field Theory



Another area where questions of mathematical rigor arise within
perturbative QFT is the use of path integrals. The S-matrix power
series expansion contains integrals over momentum space and this is
where path integrals / Feynman diagrams have been helpful for making
calculations. The key concept is the partition function \(Z\),
which is defined as a functional integral involving
the action, which is itself an integral of the Lagrangian. The
following details come mainly from Hancox-Li (2017). More
specifically, the action is a functional of quantum fields. The
functional integral over the action ranges over all possible
combinations of the quantum fields values over spacetime. Informally,
the sum is being taken over all possible field configurations. As
Swanson (2017) notes, the path integral requires choosing a measure
over an infinite dimensional path space, which is only mathematically
well-defined in special cases. For example, if the system is
formulated on a hypercubic lattice, then the measure can be defined
(see section 1.2 of James Fraser 2016). Another way of having a
well-defined measure is to restrict attention to a finite dimensional
subspace. But if functions are allowed to vary arbitrarily on short
length scales, then the integral ceases to be well-defined (Wallace
2006, p. 42). All of the correlation functions (i.e., vacuum state
expectation values of the fields at different spacetime points), can
be derived from the partition function \(Z\). So, given
\(Z\), all empirical quantities associated with the
Lagrangian can be calculated, e.g., scattering cross-sections. Finding
\(Z\) amounts to a solution of LQFT. \(Z\)
can be expanded in a Taylor series in the coupling
constant. When this is done, two types of divergences can occur: (1)
individual terms of the perturbation series can diverge and/or (2) the
perturbation series itself is divergent, though the series may be an
asymptotic series. To deal with (1), physicists do the following
procedures (Hancox-Li 2017, pp. 344-345): (i) regularization, which
involves reducing the number of degrees of freedom via dimensional
regularization, momentum cutoffs, or using a lattice formulation and
(ii) add counterterms to compensate for the regularization in (i). But
this construction is purely formal and not mathematically defined. The
rules used to manipulate the Lagrangian, and hence the partition
function, are not well-defined.


#### 5.1.3 Renormalization Group Techniques



Wallace (2011) argues that renormalization group techniques have
overcome the mathematical deficiencies of older renormalization
calculational techniques (for more details on the renormalization
group see Butterfield and Bouatta 2015, Fraser 2016, Hancox-Li (2015a,
2015b, 2017)). According to Wallace, the renormalization group methods
put LQFT on the same level of mathematical rigor as other areas of
theoretical physics. It provides a solid theoretical framework that is
explanatorily rich in particle physics and condensed matter physics,
so the impetus for axiomatic QFT has been resolved. Renormalization
group techniques presuppose that QFT will fail at some short length
scale, but the empirical content of LQFT is largely insensitive to the
details at such short length scales. Doreen Fraser (2011) argues that
renormalization group methods help articulate the empirical content of
QFT, but the renormalization group has no significance for the
theoretical content of QFT insofar as it does not tell us whether we
should focus on LQFT or AQFT. James Fraser (2016) and Hancox-Li
(2015b) argue that the renormalization group does more than provide
empirical predictions in QFT. The renormalization group gives us
methods for studying the behavior of physical systems at different
energy scales, namely how properties of QFT models depend or do not
depend on small scale structure. The renormalization group provides a
non-perturbative explanation of the success of perturbative QFT.
Hancox-Li (2015b) discusses how mathematicians working in constructive
QFT use non-perturbative approximations with well controlled error
bounds to prove the existence or non-existence of ultraviolet fixed
points. Hancox-Li argues that the renormalization group explains
perturbative renormalization non-perturbatively. The renormalization
group can tell us whether certain Lagrangians have an ultraviolet
limit that satisfies the axioms a QFT should satisfy. Thus, the use of
the renormalization group in constructive QFT can provide additional
dynamical information (e.g., whether a certain dynamics can occur in
continuous spacetime) that a pure axiomatic approach does not.


### 5.2 Middle Grounds



Egg, Lam, and Oldofredi (2017) argue that the main disagreement
between Doreen Fraser and David Wallace is over the very definition of
QFT. Fraser takes QFT to be the union of quantum theory and special
relativity. If QFT = QM + SR as Fraser maintains, then LQFT fails to
satisfy that criterion since it employs cutoffs which violate
Poincare covariance. For Wallace, the violation of QFT
Poincare covariance is not as worrisome. QFT is not a truly
fundamental theory since gravity is absent. Wallace is more interested
in what QFT's approximate truth tells us about the world. LQFT
gives us an effective ontology. The renormalization group tell us that
QFT cannot be trusted in the high energy regimes where quantum gravity
can be expected to apply, i.e., the Planckian length scale where
gravitational effects cannot be ignored. The violation of
Poincare covariance via cutoffs may not amount to much if the
fundamental quantum theory of gravity imposes some real cutoff,
according to Wallace. There are, however, other options to
consider.


#### 5.2.1 Pluralistic Approaches



Some philosophers have rejected the seemingly either-or nature of the
debate between Wallace and Fraser to embrace more pluralistic views.
On these pluralistic views, different formulations of QFT might be
appropriate for different philosophical questions. Baker (2016)
advocates that AQFT or LQFT should be trusted in domains of inquiry
where their idealizations are unproblematic. For example, if the
domain to be interpreted is the Standard Model, then LQFT is the
appropriate framework. Swanson (2017) analyzes LQFT, AQFT, and
Wightman QFT and argues that all three approaches are complementary
and have no deep incompatibilities. LQFT supplies various powerful
predictive tools and explanatory schemas. It can account for gauge
theories, the Standard Model of particle physics, the weak and strong
nuclear force, and the electromagnetic force. However, the collection
of calculational techniques are not all mathematically well-defined.
LQFT provides QFT theories at only certain length scales and cannot
make use of unitarily inequivalent representations since LQFT uses
cutoffs which renders all representations finite dimensional and
unitarily equivalent by the Stone-von Neumann theorem. Axiomatic QFT
is supposed to provide a rigorous description of fundamental QFT at
all length scales, but that conflicts with the effective field theory
viewpoint where QFT is only defined for certain lengths. But if
axiomatic QFT capture what all QFTs have in common, then effective
field theories should be captured by it as well. Axiomatic QFT gives a
precise regimentation of LQFT, but it is unclear if axiomatic QFT is
fully faithful to the LQFT picture. Within the axiomatic approach,
Wightman QFT has many sophisticated tools for building concrete models
of QFT in addition to rigorously proving structural results like the
PCT theorem and the spin statistics theorem. But Wightman QFT relies
on localized gauge-dependent field operators that do not directly
represent physical properties. AQFT might provide a more physically
transparent gauge-free description of QFT. It has topological tools to
define global quantities like temperature, energy, charge, particle
number which use unitarily inequivalent representations. But AQFT has
difficulty constructing models. While LQFT is more mathematically
amorphous, there are recent algebraic constructions of low dimensional
interacting models with no known Lagrangian, which suggest that AQFT
is more general than LQFT (Swanson 2017, p. 5). However, LQFT provides
constructive QFT with guidance on correctly building models
corresponding to Lagrangians particle physicists use with great
empirical success (Hancox-Li 2017, p. 353).


#### 5.2.2 Constructive Quantum Field Theory



Constructive QFT is an attempt to mediate between LQFT and axiomatic
QFT by rigorously constructing specific interacting models of QFT. The
nontrivial solutions it constructs are supposed to correspond to
Lagrangians that particle physicists use. This ensures that various
axiomatic systems have a physical connection to the world via the
empirical success of LQFT. While constructive QFT has done this for
some models with dimensions less than 4, it has not yet been
accomplished for a 4 dimensional Lagrangian that particle physicists
use. Any model that satisfies the Osterwalder-Schrader axioms will
automatically satisfy the Wightman axioms. Constructive QFT tries to
construct the functional integral measures for path integrals by
shifting from Minkowski spacetime to Euclidean spacetime via a Wick
rotation (what follows is based on section four of Hancox-Li (2017)).
In Euclidean field theory, the Schwinger functions, which are defined
in terms of \(Z\), must satisfy the
Osterwalder-Schrader axioms. The measure of \(Z\) is a
Gaussian measure on the Schwartz space of rapidly decreasing
functions. The Osterwalder-Schrader axioms are related to the Wightman
axioms by the Osterwalder-Schrader Reconstruction Theorem which states
that any set of functions satisfying the Osterwalder-Schrader axioms
determines a unique Wightman model whose Schwinger functions form that
set. It allows the constructive field theorists to use the advantages
of Euclidean space for defining a measure while ensuring that they are
constructing models that exist in Minkowski spacetime. It still has to
be verified that the solution corresponds to a renormalized
perturbation series that physicists derive for the corresponding
Lagrangian in LQFT. The challenge is how to translate something not
mathematically well-defined into something that is while showing that
the "solutions" in LQFT can be reproduced by something
that is consistent with a set of axioms. This is crucial since, as
Swanson (2017) points out, it is unclear whether perturbation theory
is an accurate guide for the underlying physics described by LQFT.
This leads Hancox-Li (2017) to argue that mathematically unrigorous
LQFT is relevant to the rigorous program of constructive QFT in
building rigorous interacting models of QFT. Those models correspond
to the Lagrangians of interest to particle physicists. Hence, LQFT can
inform the theoretical content of QFT.



Another tool of constructive QFT is the use of asymptotic series,
which can tell us which function the perturbative series is asymptotic
to, which perturbative QFT does not. Constructive QFT tries to
determine some properties of non-perturbative solutions to the
equations of motion which guarantee that certain methods of summing
asymptotic expansions will lead to a unique solution (see Hancox-Li
2017, pp. 349-350, for more details). Is the rigorously defined
partition function \(Z\) asymptotic to the renormalized
perturbative series? Roughly, a function is asymptotic to a series
expansion when successive terms of the series provide an increasingly
accurate description of how quickly the function grows. The difference
between the function and each order of the perturbation series is
approximately small. But there are many different functions that have
the same asymptotic expansion. Ideally, we want there to be a unique
function because then there is a unique non-perturbative solution. The
concept of strong asymptoticity requires that the difference between
the function and each order of the series is smaller than what was
required by asymptoticity. A strongly asymptotic series uniquely
determines a function. If there is a strong asymptotic series, then
the function can be uniquely reconstructed from the series by Borel
summation. The Borel transform of the series is given by dividing the
coefficients each term in the series by a factorial of the order of
that term and then integrating to recover the exact function. In
constructive QFT, the goal is to associate a unique function with a
renormalized perturbation series and some kind of Borel summability is
the main candidate so far, though the Borel transform cannot remove
large-order divergences. The asymptotic behavior of the renormalized
perturbation series can be extremely sensitive to the choice of
regularization and render it asymptotic to a free field theory even if
it appears to describe nontrivial perturbations (see Swanson 2017, p.
11, for more details).