## 1. The Concept of Symmetry



The term "symmetry" derives from the Greek words
*sun* (meaning 'with' or 'together')
and *metron* ('measure'), yielding
*summetria*, and originally indicated a relation of
commensurability (such is the meaning codified in Euclid's
*Elements* for example). It quickly acquired a further, more
general, meaning: that of a proportion relation, grounded on (integer)
numbers, and with the function of harmonizing the *different*
elements into a *unitary whole*. From the outset, then,
symmetry was closely related to harmony, beauty, and unity, and this
was to prove decisive for its role in theories of nature. In
Plato's *Timaeus*, for example, the regular polyhedra are
afforded a central place in the doctrine of natural elements for the
proportions they contain and the beauty of their forms: fire has the
form of the regular tetrahedron, earth the form of the cube, air the
form of the regular octahedron, water the form of the regular
icosahedron, while the regular dodecahedron is used for the form of
the entire universe. The history of science provides another
paradigmatic example of the use of these figures as basic ingredients
in physical description: Kepler's 1596 *Mysterium
Cosmographicum* presents a planetary architecture grounded on the
five regular solids.



From a modern perspective, the regular figures used in Plato's
and Kepler's physics for the mathematical proportions and
harmonies they contain (and the related properties and beauty of their
form) are symmetric in another sense that does not have to do with
proportions. In the language of modern science, the symmetry of
geometrical figures -- such as the regular polygons and polyhedra
-- is defined in terms of their invariance under specified groups
of rotations and reflections. Where does this definition stem from? In
addition to the ancient notion of symmetry used by the Greeks and
Romans (current until the end of the Renaissance), a different notion
of symmetry emerged in the seventeenth century, grounded not on
proportions but on an equality relation between elements that are
opposed, such as the left and right parts of a figure. Crucially, the
parts are *interchangeable with respect to the whole* --
they can be exchanged with one another while preserving the original
figure. This latter notion of symmetry developed, via several steps,
into the concept found today in modern science. One crucial stage was
the introduction of specific mathematical operations, such as
reflections, rotations, and translations, that are used to describe
with precision how the parts are to be exchanged. As a result, we
arrive at a definition of the symmetry of a geometrical figure in
terms of its *invariance* when equal component parts are
exchanged according to one of the specified operations. Thus, when the
two halves of a bilaterally symmetric figure are exchanged by
reflection, we recover the original figure, and that figure is said to
be invariant under left-right reflections. This is known as the
"crystallographic notion of symmetry", since it was in the
context of early developments in crystallography that symmetry was
first so defined and
 applied.[1]
 The next key step was the generalization of this notion to the
group-theoretic definition of symmetry, which arose following the
nineteenth-century development of the algebraic concept of a group,
and the fact that the symmetry operations of a figure were found to
satisfy the conditions for forming a
 group.[2]
 For example, reflection symmetry has now a precise definition in
terms of invariance under the group of reflections. Finally, we have
the resulting close connection between the notion of symmetry,
equivalence and group: a symmetry group induces a partition into
equivalence classes. The elements that are exchanged with one another
by the symmetry transformations of the figure (or whatever the
"whole" considered is) are connected by an *equivalence
relation*, thus forming an equivalence
 class.[3]



The group-theoretic notion of symmetry is the one that has proven so
successful in modern science. Note, however, that symmetry remains
linked to beauty (regularity) and unity: by means of the symmetry
transformations, distinct (but "equal" or, more generally,
"equivalent") elements are related to each other and to
the whole, thus forming a regular "unity". The way in
which the regularity of the whole emerges is dictated by the nature of
the specified transformation group. Summing up, a *unity of
different and equal elements* is always associated with symmetry,
in its ancient or modern sense; the way in which this unity is
realized, on the one hand, and how the equal and different elements
are chosen, on the other hand, determines the resulting symmetry and
in what exactly it consists.



The definition of symmetry as "invariance under a specified
group of transformations" allowed the concept to be applied much
more widely, not only to spatial figures but also to abstract objects
such as mathematical expressions -- in particular, expressions of
physical relevance such as dynamical equations. Moreover, the
technical apparatus of group theory could then be transferred and used
to great advantage within physical
 theories.[4]



When considering the role of symmetry in physics from a historical
point of view, it is worth keeping in mind two preliminary
distinctions:


* The first is between implicit and explicit uses of the notion.
Symmetry considerations have always been applied to the description
of nature, but for a long time in an implicit way only. As we
have seen, the scientific notion of symmetry (the one we are
interested in here) is a recent one. If we speak about a role of this
concept of symmetry in the ancient theories of nature, we must be
clear that it was not used explicitly in this sense at that time.
* The second is between the two main ways of using symmetry. First,
we may attribute specific symmetry properties to phenomena or to laws
(*symmetry principles*). It is the application with respect to
laws, rather than to objects or phenomena, that has become central to
modern physics, as we will see. Second, we may derive specific
consequences with regard to particular physical situations or
phenomena on the basis of their symmetry properties (*symmetry
arguments*).


## 2. Symmetry Principles



The first explicit study of the invariance properties of equations in
physics is connected with the introduction, in the first half of the
nineteenth century, of the transformational approach to the problem of
motion in the framework of analytical mechanics. Using the formulation
of the dynamical equations of mechanics due to W. R. Hamilton (known
as the Hamiltonian or canonical formulation), C. G. Jacobi developed a
procedure for arriving at the solution of the equations of motion
based on the strategy of applying transformations of the variables
that leave the Hamiltonian equations invariant, thereby transforming
step by step the original problem into new ones that are simpler but
perfectly equivalent (for further details see Lanczos
 1949).[5]
 Jacobi's canonical transformation theory, although introduced
for the "merely instrumental" purpose of solving dynamical
problems, led to a very important line of research: the general study
of physical theories in terms of their transformation properties.
Examples of this are the studies of invariants under canonical
transformations, such as Poisson brackets or Poincare's
integral invariants; the theory of continuous canonical
transformations due to S. Lie; and, finally, the connection between
the study of physical invariants and the algebraic and geometric
theory of invariants that flourished in the second half of the
nineteenth century, and which laid the foundation for the geometrical
approach to dynamical problems. The use of the mathematics of group
theory to study physical theories was central to the work, early in
the twentieth century in Gottingen, of the group whose central
figures were F. Klein (who earlier collaborated with Lie) and D.
Hilbert, and which included H. Weyl and later E. Noether. We will
return later in this section to Weyl (see
 Sections 2.1.2,
 2.2,
 2.5)
 and Noether (see
 Section 2.1.2).
 For more details on these developments see Brading and Castellani
(2007).



On the above approach, the equations or expressions of physical
interest are already given, and the strategy is to study their
symmetry properties. There is, however, an alternative way of
proceeding, namely the reverse one: start with specific symmetries and
search for dynamical equations with such properties. In other words,
we *postulate* that certain symmetries are physically
significant, rather than deriving them from prior dynamical equations.
The assumption of certain symmetries in nature is not, of course, a
novelty. Although not explicitly expressed as symmetry principles, the
homogeneity and isotropy of physical space, and the uniformity of time
(forming together with the invariance under Galilean boosts "the
older principles of invariance" -- see Wigner
 1967,[6]
 pp. 4-5), have been assumed as prerequisites in the physical
description of the world since the beginning of modern science.
Perhaps the most famous early example of the deliberate use of this
type of symmetry principle is Galileo's discussion of whether
the Earth moves in his *Dialogue concerning the two chief world
systems* of 1632. Galileo sought to neutralize the standard
arguments purporting to show that, simply by looking around us at how
things behave locally on Earth -- how stones fall, how birds fly
-- we can conclude that the Earth is at rest rather than
rotating, arguing instead that these observations do not enable us to
determine the state of motion of the Earth. His approach was to use an
analogy with a ship: he urges us to consider the behaviour of objects,
both animate and inanimate, inside the cabin of a ship, and claims
that no experiments carried out inside the cabin, without reference to
anything outside the ship, would enable us to tell whether the ship is
at rest or moving smoothly across the surface of the Earth. The
*assumption* of a symmetry between rest and a certain kind of
motion leads to the prediction of this result, without the need to
know the details of the laws governing the experiments on the ship.
The "Galilean principle of relativity" (according to which
the laws of physics are invariant under Galilean boosts, where the
states of motion considered are now those of uniform velocity) was
quickly adopted as an axiom and widely used in the seventeenth
century, notably by Huygens in his solution to the problem of
colliding bodies and by Newton in his early work on motion. Huygens
took the relativity principle as his 3rd hypothesis or axiom, but in
Newton's Principia it is demoted to a corollary to the laws of
motion, its status in Newtonian physics therefore being that of a
consequence of the laws, even though it remains, in fact, an
independent assumption.



Although the spatial and temporal invariance of mechanical laws was
known and used for a long time in physics, and the group of the global
spacetime symmetries for electrodynamics was completely derived by H.
Poincare
 [7]
 before Einstein's famous 1905 paper setting out his special
theory of relativity, it was not until this work by Einstein that the
status of symmetries with respect to the laws was reversed. E. P.
Wigner (1967, p. 5) writes that "the significance and general
validity of these principles were recognized, however, only by
Einstein", and that Einstein's work on special relativity
marks "the reversal of a trend: until then, the principles of
invariance were derived from the laws of motion ... It is now
natural for us to derive the laws of nature and to test their validity
by means of the laws of invariance, rather than to derive the laws of
invariance from what we believe to be the laws of nature". In
postulating the universality of the global continuous spacetime
symmetries, Einstein's construction of his special theory of
relativity represents the first turning point in the application of
symmetry to twentieth-century
 physics.[8]


### 2.1 Relativity


#### 2.1.1 The special theory of relativity



Einstein's special theory of relativity (STR) is constructed on
the basis of two fundamental postulates. One is the light postulate
(that the speed of light, in the "rest frame", is
independent of the speed of the source), and the other is the
principle of relativity. The latter was adopted by Einstein explicitly
as a means of restricting the form of the laws, whatever their
detailed structure might turn out to be. Thus, we have the difference
between a "constructive" and a "principle"
theory: in the former case we build our theory based on known facts
about the constitution and behaviour of material bodies; in the latter
case we start by restricting the possible form of such a theory by
adopting certain
 principles.[9]



The principle of relativity as adopted by Einstein (1905, p. 395 of
the English translation) simply asserts that:



> 
> The laws by which the states of physical systems undergo changes are
> independent of whether these changes of states are referred to one or
> the other of two coordinate systems moving relatively to each other in
> uniform translational motion.
> 



This principle, when combined with the light postulate (and certain
other assumptions), leads to the Lorentz transformations, these being
the transformations between coordinate systems moving uniformly with
respect to one another according to STR. According to STR the laws of
physics are invariant under Lorentz transformations, and indeed under
the full Poincare group of transformations. These
transformations differ from the Galilean transformations of Newtonian
mechanics. H. Minkowski reformulated STR, showing that space and time
are part of a single four-dimensional geometry, Minkowski spacetime.
In this way, the Poincare group of symmetry transformations is
part of the structure of spacetime in STR, and for this reason these
symmetries have been labelled "geometric symmetries" by
Wigner (1967, especially pp. 15 and 17-19).



There is a debate in the literature concerning how the principle of
relativity, and more generally the global space-time symmetries,
should be understood. On one approach, the significance of space-time
symmetries is captured by considering the structure of a theory
through transformations on its models, those models consisting of
differentiable manifolds endowed with various geometric objects and
relations (see Anderson, 1967, and Norton, 1989). According to Brown
and Sypel (1995) and Budden (1997), this approach fails to recognise
the central importance of effectively isolated subsystems, the
empirical significance of symmetries resting on the possibility of
transforming such a subsystem (rather than applying the transformation
to the entire universe). For further developments in this debate,
including applications to local symmetries and to gauge theories, see
Kosso (2000), Brading and Brown (2004), Healey (2007), Healey (2009),
Greaves and Wallace (2014), Friederich (2015), Rovelli (2014), Teh
(2015, 2016) and Wallace (2022).



The global spacetime invariance principles are intended to be valid
for all the laws of nature, for all the processes that unfold in the
spacetime. This universal character is not shared by the physical
symmetries that were next introduced in physics. Most of these were of
an entirely new kind, with no roots in the history of science, and in
some cases expressly introduced to describe specific forms of
interactions -- whence the name "dynamical
symmetries" due to Wigner (1967, see especially pp. 15,
17-18, 22-27, 33).


#### 2.1.2 The general theory of relativity



Einstein's general theory of relativity (GTR) was also
constructed using a symmetry principle at its heart: the principle of
general covariance. Much ink has been spilled over the significance
and role of general covariance in GTR, including by Einstein
 himself.[10]
 For a long time he viewed the principle of general covariance as an
extension of the principle of relativity found in both classical
mechanics and STR, and this is a view that continues to provoke
vigorous debate. Norton (2003) discusses the "Kretschmann
objection" to the physical significance of general covariance.
On invariance versus covariance, see Anderson (1967), Brown and
Brading (2002), and Martin (2003, Section 2.2). What is clear is that
the mere requirement that a theory be generally covariant represents
no restriction on the form of the theory; further stipulations must be
added, such as the requirement that there be no "absolute
objects" (this itself being a problematic notion). Once some
such further requirements are added, however, the principle of general
covariance becomes a powerful tool. For a recent review and analysis
of this debate, see Pitts (2006).



In Einstein's hands the principle of general covariance was a
crucial postulate in the development of
 GTR.[11]
 The diffeomorphism freedom of GTR, i.e., the invariance of the form
of the laws under transformations of the coordinates depending
smoothly on arbitrary functions of space and time, is a
"local" spacetime symmetry, in contrast to the
"global" spacetime symmetries of STR (which depend instead
on constant parameters). For a discussion of coordinate-based
approaches to the diffeomorphism invariance of General Relativity, see
Wallace (2019), and for more on the physical interpretation of this
invariance, see Pooley (2017). Such local symmetries are
"dynamical" symmetries in Wigner's sense, since they
describe a particular interaction, in this case gravity. As is well
known, the spacetime metric in GTR is no longer a
"background" field or an "absolute object",
but instead it is a dynamical player, the gravitational field
manifesting itself as spacetime curvature.



The extension of the concept of continuous symmetry from
"global" symmetries (such as the Galilean group of
spacetime transformations) to "local" symmetries is one of
the important developments in the concept of symmetry in physics that
took place in the twentieth century. Prompted by GTR, Weyl's
1918 "unified theory of gravitation and electromagnetism"
extended the idea of local symmetries (see Ryckman, 2003, and Martin,
2003), and although this theory is generally deemed to have failed,
the theory contains the seeds of later success in the context of
quantum theory (see below,
 Section 2.5).



Meanwhile, Hilbert and Klein undertook detailed investigations
concerning the role of general covariance in theories of gravitation,
and enlisted the assistance of Noether in their debate over the status
of energy conservation in such theories. This led to Noether's
famous 1918 paper containing two theorems, the first of which leads to
a connection between global symmetries and conservation laws, and the
second of which leads to a number of results associated with local
symmetries, including a demonstration of the different status of the
conservation laws when the global symmetry group is a subgroup of some
local symmetry group of the theory in question (see Brading and Brown,
2003). A detailed, recent collective volume on the
philosophy and physics of Noether's theorems is Read
and Teh (2022). 


### 2.2 Symmetry and quantum mechanics



The application of the theory of groups and their representations for
the exploitation of symmetries in the quantum mechanics of the 1920s
undoubtedly represents the second turning point in the
twentieth-century history of physical symmetries. It is, in fact, in
the quantum context that symmetry principles are at their most
effective. Wigner and Weyl were among the first to recognize the great
relevance of symmetry groups to quantum physics and the first to
reflect on the meaning of this. As Wigner emphasized on many
occasions, one essential reason for the "increased effectiveness
of invariance principles in quantum theory" (Wigner, 1967, p.
47) is the linear nature of the state space of a quantum physical
system, corresponding to the possibility of superposing quantum
states. This gives rise to, among other things, the possibility of
defining states with particularly simple transformation properties in
the presence of symmetries.



In general, if *G* is a symmetry group of a theory describing a
physical system (that is, the dynamical equations of the theory are
invariant under the transformations of *G*), this means that
the states of the system transform into each other according to some
"representation" of the group *G*. In other words,
the group transformations are mathematically represented in the state
space by operations relating the states to each other. Roberts (2022,
2.3) characterizes this as an essential requirement for concept
like "dynamical evolution" and "spatial
translation" to have meaning in state-space theory, which
he refers to as the Representation View. In the state space of
quantum mechanics, these operations are implemented through the
operators that act on the state space and correspond to the physical
observables, and any state of a physical system can be described as a
superposition of states of elementary systems, that is, of systems the
states of which transform according to the "irreducible"
representations of the symmetry group. Quantum mechanics thus offers a
particularly favourable framework for the application of symmetry
principles. The observables representing the action of the symmetries
of the theory in the state space, and therefore commuting with the
Hamiltonian of the system, play the role of the conserved quantities;
furthermore, the eigenvalue spectra of the invariants of the symmetry
group provide the labels for classifying the irreducible
representations of the group: on this fact is grounded the possibility
of associating the values of the invariant properties characterizing
physical systems with the labels of the irreducible representations of
symmetry groups, i.e. of classifying elementary physical systems by
studying the irreducible representations of the symmetry groups.


### 2.3 Permutation symmetry



The first non-spatiotemporal symmetry to be introduced into
microphysics, and also the first symmetry to be treated with the
techniques of group theory in the context of quantum mechanics, was
*permutation symmetry* (or invariance under the transformations
of the permutation group). This symmetry, "discovered" by
W. Heisenberg in 1926 in relation to the indistinguishability of the
"identical" electrons of an atomic
 system,[12]
 is the *discrete* symmetry (i.e. based upon groups with a
discrete set of elements) at the core of the so-called quantum
statistics (the Bose-Einstein and Fermi-Dirac statistics), governing
the statistical behaviour of ensembles of certain types of
indistinguishable quantum particles (e.g. bosons and fermions). The
permutation symmetry principle states that if such an ensemble is
invariant under a permutation of its constituent particles then one
doesn't count those permutations which merely exchange
indistinguishable particles, that is the exchanged state is identified
with the original state (see French and Rickles, 2003, Section 1).



Philosophically, permutation symmetry has given rise to two main sorts
of questions. On the one side, seen as a condition of physical
indistinguishability of identical particles (i.e. particles of the
same kind in the same atomic system), it has motivated a rich debate
about the significance of the notions of identity, individuality, and
indistinguishability in the quantum domain. Does it mean that the
quantum particles are not individuals? Does the existence of entities
which are physically indistinguishable although "numerically
distinct" (the so-called problem of identical particles) imply
that the Leibniz's Principle of the Identity of Indiscernibles
should be regarded as violated in quantum physics? On the other side,
what is the theoretical and empirical status of this symmetry
principle? Should it be considered as an axiom of quantum mechanics or
should it be taken as justified empirically? It is currently taken to
explain the nature of fermionic and bosonic quantum statistics, but
why do there appear to be only bosons and fermions in the world when
the permutation symmetry group allows the possibility of many more
types? French and Rickles (2003) offer an overview of the above and
related issues, and a new twist in the tale can be found in Saunders
(2006). Saunders discusses permutation symmetry in classical physics,
and argues for indistinguishable classical particles obeying classical
statistics. He argues that the differences between quantum and
classical statistics, for certain classes of particles, therefore
cannot be accounted for solely in terms of indistinguishability. For
further discussion and references see French and Krause (2006),
Ladyman and Bigaj (2010), Caulton and Butterfield (2012), and the
related SEP entry
 identity and individuality in quantum theory.


### 2.4 *C*, *P*, *T*



Because of the specific properties of the quantum description, the
discrete symmetries of spatial reflection symmetry or *parity*
(*P*) and *time reversal* (*T*) were
"rediscovered" in the quantum context, taking on a new
significance. Parity was introduced in quantum physics in 1927 in a
paper by Wigner, where important spectroscopic results were explained
for the first time on the basis of a group-theoretic treatment of
permutation, rotation and reflection symmetries. Time reversal
invariance appeared in the quantum context, again due to Wigner, in a
1932
 paper.[13]
 To these was added the new quantum particle-antiparticle symmetry or
*charge conjugation* (*C*). Charge conjugation was
introduced in Dirac's famous 1931 paper "Quantized
singularities in the electromagnetic field".



The discrete symmetries *C*, *P* and *T* are
connected by the so-called CPT theorem, first proved by Luders
and Pauli in the early 1950s, which states that the combination of
*C*, *P*, and *T* is a general symmetry of
Lorentz-invariant physical laws. For philosophical reflections on the
meaning and grounding of the CPT theorem see Wallace (2009) and
Greaves (2010). Greaves (2010) in particular argues that the
quantization of the classical T operator leads to a CT operator, using
this in support of Feynman's famous suggestion that time reversal
has the effect of transforming particles into anti-particles. Roberts
(2022, 2.8) argues against this claim, by maintaining that the
Feynman's approach is an artifact that does not accurately capture the
description of temporal symmetry. A discussion of the proofs of
the theorem, both within the standard quantum field theory framework
and the axiomatic field theory framework, is provided in Greaves and
Thomas (2014).



The laws governing gravity, electromagnetism, and the strong
interaction are invariant with respect to *C*, *P* and
*T* independently. However, in 1956 T. D. Lee and C. N. Yang
pointed out that b-decay, governed by the weak interaction, had
not yet been tested for invariance under *P*. Soon afterwards
C. S. Wu and her colleagues performed an experiment showing that the
weak interaction violates parity. Nevertheless, b-decay respects
the combination of *C* and *P* as a symmetry. In 1964,
however, *CP* was found to be violated in weak interaction by
Cronin and Fitch in an experiment involving K-mesons. This implied, in
virtue of the CPT theorem, the violation of *T*-symmetry as
well; since then, there have been direct observations of
*T*-symmetry violation, as reported by e.g., CPLEAR
Collaboration (1998). For a careful analysis of the underlying
assumptions working in the assessment of *T*-violation see
Roberts (2015) and Ashtekar (2015). The philosophical puzzle is how we
can come to know this time asymmetry, given that we cannot truly
'reverse' time. Roberts proposes three templates for
explaining how this is possible, drawing in particular on a version of
Curie's principle. By contrast, Golosz (2016) argues that
this asymmetry should be understood as applying to a physical process,
rather than to time itself. Roberts (2022, Chapter 7) responds
that, on the Representation View, the asymmetry does indeed apply to
time itself.



This debate stems from the more basic question of what time
reversal symmetry actually means. Some have argued that the textbook
time reversal operator T does not really represent the reversal
of time alone, due to its unusual properties like conjugation of the
wavefunction and the reversal of magnetic fields, including Albert
(2000), Callender (2000), Lopez (2021, 2023). Others maintain,
instead, that the T operator is indeed the right definition of
time reversal, including Earman (2002), Malament (2004) and
Roberts (2017). An ecumenical response is given by Roberts (2022,
Chapter 2), arguing that these two approaches can be viewed as the two
sides of the same coin, corresponding, respectively, to a reversal of
time translations on spacetime and to the representation of that
reversal on state space. A response in a similar spirit for
electromagnetism is given by Struyve (forthcoming). 



The existence of parity violation in our fundamental laws has led to a
new chapter in an old philosophical debate concerning chiral or handed
objects and the nature of space. A description of a left hand and one
of a right hand will not differ so long as no appeal is made to
anything beyond the relevant hand. Yet left and right hands do
differ -- a left-handed glove will not fit on a right hand.
For a brief period, Kant saw in this reason to prefer a
substantivalist account of space over a relational one, the difference
between left and right hands lying in their relation to absolute
space. Regardless of whether this substantivalist solution succeeds,
there remains the challenge to the relationalist of accounting for the
difference between what Kant called "incongruent
counterparts" -- objects which are the mirror-image of one
another and yet cannot be made to coincide by any rigid motion. The
relationalist may respond by denying that there is any
*intrinsic* difference between a left and a right hand, and
that the incongruence is to be accounted for in terms of the relations
between the two hands (if a universe was created with only one hand in
it, it would be neither left nor right, but the second hand to be
created would be either incongruent or congruent with it). This
response becomes problematic in the face of parity violation, where
one possible experimental outcome is much more likely than its
mirror-image. Since the two possible outcomes don't differ
intrinsically, how should we account for the imbalance? This issue
continues to be discussed in the context of the
substantivalist-relationalist debate. For further details see Pooley
(2003) and Saunders (2007).


### 2.5 Gauge symmetry



The starting point for the idea of *continuous internal
symmetries* was the interpretation of the presence of particles
with (approximately) the same value of mass as the components
(*states*) of a single physical system, connected to each other
by the transformations of an underlying symmetry group. This idea
emerged by analogy with what happened in the case of permutation
symmetry, and was in fact due to Heisenberg (the discoverer of
permutation symmetry), who in a 1932 paper introduced the SU(2)
symmetry connecting the proton and the neutron (interpreted as the two
states of a single system). This symmetry was further studied by
Wigner, who in 1937 introduced the term *isotopic spin* (later
contracted to *isospin*). The various internal symmetries are
invariances under phase transformations of the quantum states and are
described in terms of the unitary groups SU(*N*). The term
"gauge" is sometimes used for all continuous internal
symmetries, and is sometimes reserved for the *local* versions
(these being at the core of the Standard Model for elementary
 particles).[14] 



The phase of the quantum wavefunction encodes internal degrees of
freedom. With the requirement that a theory be invariant under
*local gauge transformations* involving the phase of the
wavefunction, Weyl's ideas of 1918 found a successful home in
quantum theory (see O'Raifeartaigh, 1997). Weyl's new 1929
theory was a theory of electromagnetism coupled to matter. The history
of gauge theory is surveyed briefly by Martin (2003), who highlights
various issues surrounding gauge symmetry, in particular the status of
the so-called "gauge principle", first proposed by Weyl. A
recent account of the conceptual structure of the gauge
argument is provided by Gomes, Roberts and Butterfield
(2022). The main steps in development of gauge theory are the
Yang and Mills non-Abelian gauge theory of 1954, and the problems and
solutions associated with the successful development of gauge theories
for the short-range weak and strong interactions. In subsequent years,
we have come to understand that "local" gauge symmetries are not just
a feature of electromagnetism, but also of the majority of our
best theories, including various gravitational theories and
models within condensed matter physics. 



The main philosophical questions raised by gauge theory all hinge upon
how we should understand the relationship between mathematics and
physics. There are two broad categories of discussion. The first
concerns the gauge principle, already mentioned, and the issue here is
the extent to which the requirement that we write our theories in
locally-symmetric form enables us to derive new physics. The analysis
concerns listing what premises constitute the gauge principle,
examining the status of these premises and what motivation might be
given for them, determining precisely what can be obtained on the
basis of these premises, and what more needs to be added in order to
arrive at a (successful) physical theory. For discussion of this
argument, see Teller (2000), Martin (2003) and Gomes et al
(2022). In Teh (forthcoming), the case is made that the debate
surrounding the gauge principle is related to the debate about
the difference between substantive and non-substantive forms of
"general covariance", a term that is used to
pick out the "local gauge (diffeomorphism) symmetry" of
general relativity.



The second category concerns the question of which quantities in a
gauge theory represent the "physically real" properties.
This question arises acutely in gauge theories because of the apparent
failure of determinism if certain redundancies of description are not
taken into account in formulating the initial value problem. The
problem was first encountered in GTR (which in this respect is a gauge
theory), and for further details the best place to begin is with the
literature on Einstein's "hole argument" (see Earman
and Norton, 1987; Earman, 1989, Chapter 9; and more recently Norton,
1993; Rynasiewicz, 1999; Saunders, 2002; Pooley, 2021; and the
references therein). In practice, we find that only gauge-invariant
quantities are observables, and this seems to rescue us. However, this
is not the end of the story. The other canonical example is the
Aharanov-Bohm effect, and we can use this to illustrate the
interpretational problem associated with gauge theories, sometimes
characterized as a dilemma: failure of determinism or
action-at-a-distance (see Healey, 2001). Restoring determinism depends
on only gauge-invariant quantities being taken as representing
"physically real" quantities, but accepting this solution
apparently leaves us with some form of non-locality between causes and
effects.



Furthermore, we face the question of how to understand the role of the
non-gauge-invariant quantities appearing in the theory, and the
problem of how to interpret what M. Redhead calls "surplus
structure" (see Redhead, 2003). For an early discussion of
"surplus structure" in the philosophical literature, see Belot
(1998) and Nounou (2003). More recently, the topic of "surplus
structure" in gauge theories has been discussed in Weatherall
(2016), Murray, Nguyen & Teh (2020), Gomes (2021),
Murgueitio-Ramirez and Teh (forthcoming), Dougherty (2020) and Wallace
(2022): the more recent discussion draws interesting connections
between the discussion of gauge theory and cutting edge topics in
theoretical physics, e.g. the application of category theory to
physics, and the use of the covariant phase space formalism. For
an approach to these questions using the theory of constrained
Hamiltonian systems, see also Earman (2003b), Castellani (2003, 2004),
and Pitts (2014). For an intuitive characterization of gauge symmetry,
one that is more general than the Lagrangian and Hamiltonian
formulations of theories in which gauge symmetry is usually
expressed, see Belot (2008). How best to interpret gauge theories is
an open issue in the philosophy of physics. Healey (2007) discusses
the conceptual foundations of gauge theories, arguing in favour of a
non-separable holonomy interpretation of classical Yang-Mills gauge
theories of fundamental interactions. Catren (2008) tackles the
ontological implications of Yang-Mills theory by means of the fiber
bundle formalism. Useful references are the Metascience review
symposium on Healey (2007) (Rickles, Smeenk, Lyre and Healey, 2009),
and the "Synopsis and Discussion" of the workshop
"Philosophy of Gauge Theory," Center for Philosophy of
Science, University of Pittsburgh, 18-19 April 2009 (available
online). A recent, detailed overview and discussion of the
various issues raised by the question of the status (both in
physics and in philosophy of physics) of gauge symmetries is provided
in Berghofer, Francois, Friederich, Gomes, Hetzroni, Maas, and
Sondenheimer (2023).


### 2.6 Dualities



Thus far, we have been discussing symmetries which act on the space of
states of a physical theory. In recent years, much discussion within
physics and philosophy has centered on certain kinds of symmetries
that act on the space of theories. When such symmetries are
interpreted as realizing an "equivalence" (which sense is
itself something that requires philosophical work to explicate)
between two theories, the theories are typically said to be related by
a "duality symmetry" (and if we are speaking of
"symmetry" in the strict sense of an automorphism, then
such dualities are called "self-dualities").



The dualities playing a central role in contemporary physics are of
various types: dualities between quantum field theories (like the
generalized electric-magnetic duality), between string theories (such
as T-duality and S-duality), and between physical descriptions which
are, respectively, a quantum field theory and a string theory, as in
the case of gauge/gravity dualities. Historically, the first
relevant dualities to be used in physics were electric-magnetic
duality, momentum-position duality (via Fourier transform) -
wave-particle duality in the QM context - and the
Kramers-Wannier duality of the two-dimensional Ising model in
statistical physics. On the various types of dualities and their
significance in physics, see in particular Castellani and Rickles
(2017). An introductory review of gauge/gravity
duality, emphasising the conceptual aspects, is De
Haro, Mayerson, and Butterfield (2016). 



In general, because dualities are transformations between theories,
their implications are more radical than those of symmetry
transformations. While symmetries are mapping between the solutions of
the same theory, different theoretical descriptions can have very
different interpretations in terms of objects, properties, degrees of
freedom, and spacetime frameworks. Thus, dualities naturally offer a
new and interesting viewpoint on many traditional issues in the
philosophy of science, such as a) reduction, emergence, and
fundamentality (for a discussion of duality and emergence see, for
example, Rickles, 2013, and Castellani and De Haro, 2020); b)
theoretical equivalence and underdetermination (see Read
and Moller-Nielsen, 2020, on the import of duality
for these issues); and c) realism versus anti-realism (Le Bihan and
Read, 2018, provide a survey of the landscape of possible
ontological interpretations of duality-related theories; a specific
example, is the case of the "particle
democracy" related to weak/strong dualities, as
discussed in Castellani, 2017). A comprehensive discussion of the
many issues related to the physics and philosophy of
dualities is De Haro and Butterfield (forthcoming).



Within the philosophical literature, work relating symmetries to
dualities generally responds to one of the following three
questions:


* What is an appropriate formal framework for understanding the
inter-theoretic relationship realized by duality symmetries? Within
the context of quantum gauge/gravity duality, a rudimentary framework
for doing so was sketched in De Haro, Teh, and Butterfield (2017) and
then further developed in De Haro and Butterfield (2017) for the case
of bosonization dualities. A comprehensive, detailed survey is
provided in De Haro and Butterfield (2018). And within the far simpler
context of classical mechanics, Teh and Tsementzis (2017) explore the
use of the quintessential symmetry of classical phase space (viz.
symplectomorphisms) to relate Hamiltonian and Lagrangian theories, and
proceed to discuss how can use this framework as a toy model for
thinking about more sophisticated dualities.
* What is the relationship between the local symmetry of a theory
and the symmetries of its dual theory? One reason that this question
is pressing is that (as we mentioned in Section 2.5), local symmetries
are typically regard as "surplus" or
"redundant" structure of a theory; thus one might hope to
be able to construct a dual theory that does not contain such surplus
structure. Indeed, it was conjectured in Polchinski and Horowitz
(2009) that for gravity/gauge duality, the local symmetries (i.e. the
diffeomorphisms) of the bulk theory will always be
"invisible" from the perspective of its dual boundary
theory. However, De Haro, Teh, and Butterfield (2017) argue that this
conjecture is not true in full generality: there is a special class of
diffeomorphisms which do not vanish in the boundary theory, but
instead correspond to conformal transformations of the boundary
CFT.
* Should duality symmetries themselves by understood as a certain
kind of "gauge symmetry", thus explaining why we should
treat dual theories as "physically equivalent"? On this
question, Read (2016) undertakes the task of using the "hole
argument" as a lens with which to compare string-theoretic
dualities with (intra-theory) gauge symmetries.


## 3. Symmetry Arguments



Consider the following cases.


* Buridan's ass: situated between what are, for him, two
completely equivalent bundles of hay, he has no reason to choose the
one located to his left over the one located to his right, and so he
is not able to choose and dies of starvation.
* Archimedes's equilibrium law for the balance: if equal
weights are hung at equal distances along the arms of a balance, then
it will remain in equilibrium since there is no reason for it to
rotate one way or the other about the balance point.
* Anaximander's argument for the immobility of the Earth as
reported by Aristotle: the Earth remains at rest since, being at the
centre of the spherical cosmos (and in the same relation to the
boundary of the cosmos in every direction), there is no reason why it
should move in one direction rather than another.



What do they have in common?



First, these can all be understood as examples of the application of
the Leibnizean Principle of Sufficient Reason (PSR): if there is no
sufficient reason for one thing to happen instead of another, the
principle says that nothing happens (the initial situation does not
change). But there is something more that the above cases have in
common: in each of them PSR is applied on the grounds that the initial
situation has a given symmetry: in the first two cases, bilateral
symmetry; in the third, rotational symmetry. The symmetry of the
initial situation implies the complete equivalence between the
existing alternatives (the left bundle of hay with respect to the
right one, and so on). If the alternatives are completely equivalent,
then there is no sufficient reason for choosing between them and the
initial situation remains unchanged.



Arguments of the above kind -- that is, arguments leading to
definite conclusions on the basis of an initial symmetry of the
situation plus PSR -- have been used in science since antiquity
(as Anaximander's argument testifies). The form they most
frequently take is the following: a situation with a certain symmetry
evolves in such a way that, in the absence of an asymmetric cause, the
initial symmetry is preserved. In other words, a breaking of the
initial symmetry cannot happen without a reason, or *an asymmetry
cannot originate spontaneously*. Van Fraassen (1989) devotes a
chapter to considering the way these kinds of symmetry arguments can
be used in general problem-solving.



Historically, the first explicit formulation of this kind of argument
in terms of symmetry is due to the physicist Pierre Curie towards the
end of nineteenth century. Curie was led to reflect on the question of
the relationship between *physical properties* and *symmetry
properties* of a physical system by his studies on the thermal,
electric and magnetic properties of crystals, these properties being
directly related to the structure, and hence the symmetry, of the
crystals studied. More precisely, the question he addressed was the
following: in a given physical medium (for example, a crystalline
medium) having specified symmetry properties, which physical phenomena
(for example, which electric and magnetic phenomena) are allowed to
happen? His conclusions, systematically presented in his 1894 work
"Sur la symetrie dans les phenomenes
physiques", can be synthesized as follows:


1. A phenomenon can exist in a medium possessing its characteristic
symmetry or that of one of its subgroups. What is needed for its
occurrence (i.e. for something rather than nothing to happen) is not
the presence, but rather the absence, of certain symmetries:
"Asymmetry is what creates a phenomenon".
2. The symmetry elements of the causes must be found in their
effects, but the converse is not true; that is, the effects can be
more symmetric than the causes.



Conclusion (a) clearly indicates that Curie recognized the important
function played by the concept of *symmetry breaking* in
physics (he was indeed one of the first to recognize it). Conclusion
(b) is what is usually called "Curie's principle" in
the literature, although notice that (a) and (b) are not independent
of one another.



In order for Curie's principle to be applicable, various
conditions need to be satisfied: the causal connection must be valid,
the cause and effect must be well-defined, and the symmetries of both
the cause and the effect must also be well-defined (this involves both
the physical and the geometrical properties of the physical systems
considered). Curie's principle then furnishes a *necessary
condition* for given phenomena to happen: only those phenomena can
happen that are compatible with the symmetry conditions established by
the principle.



Curie's principle has thus an important methodological function:
on the one side, it furnishes a kind of selection rule (given an
initial situation with a specified symmetry, only certain phenomena
are allowed to happen); on the other side, it offers a falsification
criterion for physical theories (a violation of Curie's
principle may indicate that something is wrong in the physical
 description).[15]



Such applications of Curie's principle depend, of course, on our
accepting its validity, and this is something that has been questioned
in the literature, especially in relation to spontaneous symmetry
breaking (see below, next section). Different proposals have been
offered for justifying the principle. We have presented it here as an
example of symmetry considerations based on Leibniz's PSR, while
Curie himself seems to have regarded it as a form of the causality
principle. In current literature, it has become standard to understand
the principle as following from the invariance properties of
deterministic physical laws. According to this "received
view", as first formulated in Chalmers (1970) and then developed
in more recent literature (Ismael 1997, Belot 2003, Earman 2004),
Curie's principle is expressed in terms of the relationship
between the symmetries of earlier and later states of a system, and
the laws connecting these states. In fact, this is a misrepresentation
of Curie's original principle: Curie's focus was on the
case of co-existing, functionally related features of a system's
state, rather than temporally ordered cause and effect pairs. For a
discussion on such questions as to whether there is more that one
"Curie's principle" and, independently of how it is
formulated, what are the aspects that make it so scientically
fruitful, see Castellani and Ismael (2016). As regards the status
of the principle, Norton (2016) argues that it is a
"truism", on the grounds that whether the principle
succeeds or fails depends on how one chooses to attach causal labels
to the scientific description. For Roberts (2013), Curie's
principle fails when the symmetry is time reversal. Roberts (2016)
claims that the truth of the principle is contingent on special
physical facts, and attributes to its failure an important role in the
detection of parity violation and CP violation (on this point, see
also Roberts, 2015, and Ashtekar, 2015).


## 4. Symmetry Breaking



A symmetry can be exact, approximate, or broken. Exact means
unconditionally valid; approximate means valid under certain
conditions; broken can mean different things, depending on the object
considered and its context.



The study of symmetry breaking also goes back to Pierre Curie.
According to Curie, symmetry breaking has the following role: for the
occurrence of a phenomenon in a medium, the original symmetry group of
the medium must be lowered (broken, in today's terminology) to
the symmetry group of the phenomenon (or to a subgroup of the
phenomenon's symmetry group) by the action of some cause. In
this sense symmetry breaking is what "creates the
phenomenon". Generally, the breaking of a certain symmetry does
not imply that no symmetry is present, but rather that the situation
where this symmetry is broken is characterized by a lower symmetry
than the original one. In group-theoretic terms, this means that the
initial symmetry group is broken to one of its subgroups. It is
therefore possible to describe symmetry breaking in terms of relations
between transformation groups, in particular between a group (the
unbroken symmetry group) and its subgroup(s). As is clearly
illustrated in the 1992 volume by I. Stewart and M. Golubitsky,
starting from this point of view a general theory of symmetry breaking
can be developed by tackling such questions as "which subgroups
can occur?", "when does a given subgroup occur?"



Symmetry breaking was first explicitly studied in physics with respect
to physical objects and phenomena. This is not surprising, since the
theory of symmetry originated with the visible symmetry properties of
familiar spatial figures and every day objects. However, it is with
respect to the laws that symmetry breaking has acquired special
significance in physics. There are two different types of symmetry
breaking of the laws: "explicit" and
"spontaneous", the case of spontaneous symmetry breaking
being the more interesting from a physical as well as a philosophical
point of view.


### 4.1 Explicit symmetry breaking



Explicit symmetry breaking indicates a situation where the dynamical
equations are not manifestly invariant under the symmetry group
considered. This means, in the Lagrangian (Hamiltonian) formulation,
that the Lagrangian (Hamiltonian) of the system contains one or more
terms explicitly breaking the symmetry. Such terms can have different
origins:



(a) Symmetry-breaking terms may be introduced into the theory by hand
on the basis of theoretical/experimental results, as in the case of
the quantum field theory of the weak interactions, which is expressly
constructed in a way that manifestly violates mirror symmetry or
parity. The underlying result, in this case, is parity
non-conservation in the case of the weak interaction, first predicted
in the famous (Nobel-prize winning) 1956 paper by T. D. Lee and C.N.
Yang.



(b) Symmetry-breaking terms may appear in the theory because of
quantum-mechanical effects. One reason for the presence of such terms
-- known as "anomalies" -- is that in passing
from the classical to the quantum level, because of possible operator
ordering ambiguities for composite quantities such as Noether charges
and currents, it may be that the classical symmetry algebra (generated
through the Poisson bracket structure) is no longer realized in terms
of the commutation relations of the Noether charges. Moreover, the use
of a "regulator" (or "cut-off") required in
the renormalization procedure to achieve actual calculations may
itself be a source of anomalies. It may violate a symmetry of the
theory, and traces of this symmetry breaking may remain even after the
regulator is removed at the end of the calculations. Historically, the
first example of an anomaly arising from renormalization is the
so-called chiral anomaly, that is the anomaly violating the chiral
symmetry of the strong interaction (see Weinberg, 1996, Chapter
22).



(c) Finally, symmetry-breaking terms may appear because of
non-renormalizable effects. Physicists now have good reasons for
viewing current renormalizable field theories as *effective field
theories*, that is low-energy approximations to a deeper theory
(each effective theory explicitly referring only to those particles
that are of importance at the range of energies considered). The
effects of non-renormalizable interactions (due to the heavy particles
not included in the theory) are small and can therefore be ignored at
the low-energy regime. It may then happen that the coarse-grained
description thus obtained possesses more symmetries than the deeper
theory. That is, the effective Lagrangian obeys symmetries that are
not symmetries of the underlying theory. These
"accidental" symmetries, as Weinberg has called them, may
then be violated by the non-renormalizable terms arising from higher
mass scales and suppressed in the effective Lagrangian (see Weinberg,
1995, pp. 529-531).


### 4.2 Spontaneous symmetry breaking



Spontaneous symmetry breaking (SSB) occurs in a situation where, given
a symmetry of the equations of motion, solutions exist which are not
invariant under the action of this symmetry without any explicit
asymmetric input (whence the attribute
 "spontaneous").[16]
 A situation of this type can be first illustrated by means of simple
cases taken from classical physics. Consider for example the case of a
linear vertical stick with a compression force applied on the top and
directed along its axis. The physical description is obviously
invariant for all rotations around this axis. As long as the applied
force is mild enough, the stick does not bend and the equilibrium
configuration (the lowest energy configuration) is invariant under
this symmetry. When the force reaches a critical value, the symmetric
equilibrium configuration becomes unstable and an infinite number of
equivalent lowest energy stable states appear, which are no longer
rotationally symmetric but are related to each other by a rotation.
The actual breaking of the symmetry may then easily occur by effect of
a (however small) external asymmetric cause, and the stick bends until
it reaches one of the infinite possible stable asymmetric equilibrium
 configurations.[17]
 In substance, what happens in the above kind of situation is the
following: when some parameter reaches a critical value, the lowest
energy solution respecting the symmetry of the theory ceases to be
stable under small perturbations and new asymmetric (but stable)
lowest energy solutions appear. The new lowest energy solutions are
asymmetric but are all related through the action of the symmetry
transformations. In other words, there is a degeneracy (infinite or
finite depending on whether the symmetry is continuous or discrete) of
distinct asymmetric solutions of identical (lowest) energy, the whole
set of which maintains the symmetry of the theory.



In *quantum physics* SSB actually does not occur in the case of
finite systems: tunnelling takes place between the various degenerate
states, and the true lowest energy state or "ground state"
turns out to be a unique linear superposition of the degenerate
states. In fact, SSB is applicable only to infinite systems --
many-body systems (such as ferromagnets, superfluids and
superconductors) and fields -- the alternative degenerate ground
states being all orthogonal to each other in the infinite volume limit
and therefore separated by a "superselection rule" (see
for example Weinberg, 1996, pp. 164-165).



Historically, the concept of SSB first emerged in condensed matter
physics. The prototype case is the 1928 Heisenberg theory of the
ferromagnet as an infinite array of spin 1/2 magnetic dipoles, with
spin-spin interactions between nearest neighbours such that
neighbouring dipoles tend to align. Although the theory is
rotationally invariant, below the critical Curie temperature
*T**c* the actual ground state of the
ferromagnet has the spin all aligned in some particular direction
(i.e. a magnetization pointing in that direction), thus not respecting
the rotational symmetry. What happens is that below
*T**c* there exists an infinitely degenerate
set of ground states, in each of which the spins are all aligned in a
given direction. A complete set of quantum states can be built upon
each ground state. We thus have many different "possible
worlds" (sets of solutions to the same equations), each one
built on one of the possible orthogonal (in the infinite volume limit)
ground states. To use a famous image by S. Coleman, a little man
living inside one of these possible asymmetric worlds would have a
hard time detecting the rotational symmetry of the laws of nature (all
his experiments being under the effect of the background magnetic
field). The symmetry is still there -- the Hamiltonian being
rotationally invariant -- but "hidden" to the little
man. Besides, there would be no way for the little man to detect
directly that the ground state of his world is part of an infinitely
degenerate multiplet. To go from one ground state of the infinite
ferromagnet to another would require changing the directions of an
infinite number of dipoles, an impossible task for the finite little
man (Coleman, 1975, pp. 141-142). As said, in the infinite
volume limit all ground states are separated by a superselection rule.
Ruetsche (2006) discusses symmetry breaking and ferromagnetism from
the algebraic perspective. Liu and Emch (2005) address the
interpretative problems of explaining SSB in nonrelativistic quantum
statistical mechanics. Fraser (2016) discusses SBB in finite systems,
arguing against the indispensability of the thermodynamic limit in the
characterization of SSB in statistical mechanics.



The same picture can be generalized to *quantum field theory*
(QFT), the ground state becoming the *vacuum state*, and the
role of the little man being played by ourselves. This means that
there may exist symmetries of the laws of nature which are not
manifest to us because the physical world in which we live is built on
a vacuum state which is not invariant under them. In other words, the
physical world of our experience can appear to us very asymmetric, but
this does not necessarily mean that this asymmetry belongs to the
fundamental laws of nature. SSB offers a key for understanding (and
utilizing) this physical possiblity.



The concept of SSB was transferred from condensed matter physics to
QFT in the early 1960s, thanks especially to works by Y. Nambu and G.
Jona-Lasinio. Jona-Lasinio (2003) offers a first-hand account of how
the idea of SSB was introduced and formalized in particle physics on
the grounds of an analogy with the breaking of (electromagnetic) gauge
symmetry in the 1957 theory of superconductivity by J. Bardeen, L. N.
Cooper and J. R. Schrieffer (the so-called BCS theory). The
application of SSB to particle physics in the 1960s and successive
years led to profound physical consequences and played a fundamental
role in the edification of the current Standard Model of elementary
particles. In particular, let us mention the following main results
that obtain in the case of the spontaneous breaking of a continous
internal symmetry in QFT.



*Goldstone theorem*. In the case of a global continuous
symmetry, massless bosons (known as "Goldstone bosons")
appear with the spontaneous breakdown of the symmetry according to a
theorem first stated by J. Goldstone in 1960. The presence of these
massless bosons, first seen as a serious problem since no particles of
the sort had been observed in the context considered, was in fact the
basis for the solution -- by means of the so-called Higgs
mechanism (see the next point) -- of another similar problem,
that is the fact that the 1954 Yang-Mills theory of non-Abelian gauge
fields predicted unobservable massless particles, the gauge
bosons.



*Higgs mechanism*. According to a "mechanism"
established in a general way in 1964 independently by (i) P. Higgs,
(ii) R. Brout and F. Englert, and (iii) G. S. Guralnik, C. R. Hagen
and T. W. B. Kibble, in the case that the internal symmetry is
promoted to a local one, the Goldstone bosons "disappear"
and the gauge bosons acquire a mass. The Goldstone bosons are
"eaten up" to give mass to the gauge bosons, and this
happens without (explicitly) breaking the gauge invariance of the
theory. Note that this mechanism for the mass generation for the gauge
fields is also what ensures the *renormalizability* of theories
involving massive gauge fields (such as the Glashow-Weinberg-Salam
electroweak theory developed in the second half of the 1960s), as
first generally demonstrated by M. Veltman and G. 't Hooft in
the early 1970s. The Higgs mechanism it at the center of a lively
debate among philosophers of physics: see, for example, Smeenk
(2006), Lyre (2008), Struyve (2011), Friederich (2013).
For a historical-philosophical analysis, see also Borrelli (2012). A
detailed discussion of the foundational and interpretational issues
raised by the Higgs mechanism is provided in Berghofer,
Francois, Friederich, Gomes, Hetzroni, Maas, and Sondenheimer
(2023, 3.3).



*Dynamical symmetry breaking* (DSB). In such theories as the
unified model of electroweak interactions, the SSB responsible (via
the Higgs mechanism) for the masses of the gauge vector bosons is
because of the symmetry-violating vacuum expectation values of scalar
fields (the so-called Higgs fields) introduced *ad hoc* in the
theory. For different reasons -- first of all, the initially
*ad hoc* character of these scalar fields for which there was
no experimental evidence until the results obtained in July 2012 at
the LHC -- some attention has been drawn to the possibility that
the Higgs fields could be phenomenological rather than fundamental,
that is bound states resulting from a specified dynamical mechanism.
SSB realized in this way has been called
 "DSB".[18]



Symmetry breaking raises a number of *philosophical* issues.
Some of them relate only to the breaking of specific types of
symmetries, such as the issue of the significance of parity violation
for the problem of the nature of space (see
 Section 2.4,
 above). Others, for example the connection between symmetry breaking
and observability, are particular aspects of the general issue
concerning the status and significance of physical symmetries, but in
the case of SSB they take on a stronger force: what is the
epistemological status of a theory based on "hidden"
symmetries and SSB? Given that what we directly observe -- the
physical situation, the phenomenon -- is asymmetric, what is the
evidence for the "underlying" symmetry? On this point, see
for example Morrison (2003) and Kosso (2000). In the absence of direct
empirical evidence, the above question then becomes whether and how
far the predictive and explanatory power of theories based on SSB
provides good reasons for believing in the existence of the hidden
symmetries. Finally, there are issues raised by the motivation for,
and role of, SSB. See for example Earman (2003a), using the algebraic
formulation of QFT to explain SSB; for further philosophical
discussions on SBB in QFT in the algebraic approach, see Ruetsche
(2011), Fraser (2012), and references therein. Landsman (2013)
discusses the issue whether SBB in infinite quantum systems can be
seen as an example of asymptotic emergence in physics. Roberts (2022,
4.5.4) argues that SBB provides an example of spacetime symmetries
that fail to be dynamical symmetries, *contra* a famous
principle of Earman (1989, 3.4). 



SSB allows symmetric theories to describe asymmetric reality. In
short, SSB provides a way of understanding the complexity of nature
without renouncing fundamental symmetries. But why should we prefer
symmetric to asymmetric fundamental laws? In other words, why assume
that an observed asymmetry requires a cause, which can be an explicit
breaking of the symmetry of the laws, asymmetric initial conditions,
or SSB? Note that this assumption is very similar to the one expressed
by Curie in his famous 1894 paper. Curie's principle (the
symmetries of the causes must be found in the effects; or,
equivalently, the asymmetries of the effects must be found in the
causes), when extended to include the case of SSB, is equivalent to a
methodological principle according to which an asymmetry of the
phenomena must come from the breaking (explicit or spontaneous) of the
symmetry of the fundamental laws. What the real nature of this
principle is remains an open issue, at the centre of a developing
debate (see
 Section 3,
 above).



Finally, let us mention the argument that is sometimes made in the
literature that SSB implies that Curie's principle is violated
because a symmetry is broken "spontaneously", that is
without the presence of any asymmetric cause. Now it is true that SSB
indicates a situation where solutions exist that are not invariant
under the symmetry of the law (dynamical equation) without any
explicit breaking of this symmetry. But, as we have seen, the symmetry
of the "cause" is not lost, it is conserved in the
ensemble of the solutions (the whole
 "effect").[19]


## 5. General Philosophical Questions



Much of the recent philosophical literature on symmetries in physics
discusses specific symmetries and the intepretational questions they
lead to. The rich variety of symmetries in modern physics means that
questions concerning the status and significance of symmetries in
physics in general are not easily addressed. However, something can be
said in more general terms and we offer a few remarks in that
direction here, starting with the main roles that symmetry plays in
physics.



One of the most important roles played by symmetry is that of
*classification* -- for example, the classification of
crystals using their remarkable and varied symmetry properties. In
contemporary physics, the best example of this role of symmetry is the
classification of elementary particles by means of the irreducible
representations of the fundamental physical symmetry groups, a result
first obtained by Wigner in his famous paper of 1939 on the unitary
representations of the inhomogeneous Lorentz group. When a symmetry
classification includes all the necessary properties for
characterizing a given type of physical object (for example, all
necessary quantum numbers for characterizing a given type of
particle), we have the possibility of defining types of entities on
the basis of their transformation properties. This has led
philosophers of science to explore a structuralist approach to the
entities of modern physics, in particular a group-theoretical account
of objects (see the contributions in Castellani, 1998, Part II). The
relation between symmetry, group and structure and its exploitation in
structuralist approaches, especially in the framework of structural
realism, is a much debated issue. See, for example, Roberts (2011),
French (2014) and Caulton (2015), and the SEP entry
 Structural Realism.



Symmetries also have a *normative* role, being used as
constraints on physical theories. The requirement of invariance with
respect to a transformation group imposes severe restrictions on the
form that a theory may take, limiting the types of quantities that may
appear in the theory as well as the form of its fundamental equations.
A famous case is Einstein's use of general covariance when
searching for his gravitational equations.



The group-theoretical treatment of physical symmetries, with the
resulting possibility of unifying different types of symmetries by
means of a unification of the corresponding transformation groups, has
provided the technical resources for symmetry to play a powerful role
in theoretical *unification*. This is best illustrated by the
current dominant research programme in theoretical physics aimed at
arriving at a unified description of all the fundamental forces of
nature (gravitational, weak, electromagnetic and strong) in terms of
underlying local symmetry groups.



It is often said that many physical phenomena can be explained as
(more or less direct) consequences of symmetry principles or symmetry
arguments. In the case of symmetry principles, the
*explanatory* role of symmetries arises from their place in the
hierarchy of the structure of physical theory, which in turn derives
from their generality. As Wigner (1967, pp. 28ff) describes the
hierarchy, symmetries are seen as properties of the laws. Symmetries
may be used to explain (i) the form of the laws, and (ii) the
occurrence (or non-occurrence) of certain events (this latter in a
manner analogous to the way in which the laws explain why certain
events occur and not others). In the case of symmetry arguments, we
may, for example, appeal to Curie's principle to explain the
occurrence of certain phenomena on the basis of the symmetries (or
asymmetries) of the situation, as discussed in section 3, above.
Furthermore, insofar as explanatory power may be derived from
unification, the unifying role of symmetries also results in an
explanatory role.



From these different roles we can draw some preliminary conclusions
about the status of symmetries. It is immediately apparent that
symmetries have an important heuristic function, indicating a strong
methodological status. Is this methodological power connected to an
ontological or epistemological status for symmetries?



According to an *ontological* viewpoint, symmetries are seen as
a substantial part of the physical world: the symmetries of theories
represent properties existing in nature, or characterize the structure
of the physical world. It might be claimed that the ontological status
of symmetries provides the reason for the methodological success of
symmetries in physics. A concrete example is the use of symmetries to
predict the existence of new particles. This can happen via the
classificatory role, on the grounds of vacant places in symmetry
classification schemes, as in the famous case of the 1962 prediction
of the particle Omega- in the context of the hadronic
classification scheme known as the "Eightfold Way". (See
Bangu, 2008, for a critical analysis of the reasoning leading to this
prediction.) Or, as in more recent cases, via the unificatory role:
the paradigmatic example is the prediction of the *W* and
*Z* particles (experimentally found in 1983) in the context of
the Glashow-Weinberg-Salam gauge theory proposed in 1967 for the
unification of the weak and electromagnetic interactions. These
impressive cases of the prediction of new phenomena might then be used
to argue for an ontological status for symmetries, via an inference to
the best explanation.



Another reason for attributing symmetries to nature is the so-called
geometrical interpretation of spatiotemporal symmetries, according to
which the spatiotemporal symmetries of physical laws are interpreted
as symmetries of spacetime itself, the "geometrical
structure" of the physical world. Moreover, this way of seeing
symmetries can be extended to non-external symmetries, by considering
them as properties of other kinds of spaces, usually known as
"internal spaces". The question of exactly what a realist
would be committed to on such a view of internal spaces remains open,
and an interesting topic for discussion.



One approach to investigating the limits of an ontological stance with
respect to symmetries would be to investigate their empirical or
observational status: can the symmetries in question be directly
observed? We first have to address what it means for a symmetry to be
observable, and indeed whether all symmetries have the same
observational status. Kosso (2000) arrives at the conclusion that
there are important differences in the empirical status of the
different kinds of symmetries. In particular, while global continuous
symmetries can be directly observed -- via such experiments as
the Galilean ship experiment -- a local continuous symmetry can
have only indirect empirical evidence. Brading and Brown (2004)
argue for a different interpretation of Kosso's
 examples,[20]
 but agree with Kosso's assessment that local symmetry
transformations cannot exhibit an analogue of the Galileo's Ship
Experiment. Against this view, Greaves and Wallace (2014) and Teh
(2016) have recently argued that when suitably understood, local
symmetries can indeed have direct empirical significance (their
argument turns on the possibility of using asymptotically non-trivial
local symmetries to construct Galileo's Ship scenarios for
certain systems). In particular, Teh (2016) argues that this
phenomenon is responsible for the empirically significant symmetries
of topological soliton solutions. In a similar spirit, Gomes (2021)
argues that a certain class of gauge transformations are on a par with
Galileo's Ship, under special conditions, and that those conditions
imply a certain kind of holism in the sense that the state of the
Universe is not uniquely determined by the state of its isolated
subsystems. On the other hand, Friederich (2015) contends that on
one plausible axiomatization of the schema introduced by Greaves and
Wallace, it is possible to deduce that local symmetries do not have
direct empirical significance. This debate was recently expanded by
Roberts (2022, 4.3) to include discrete symmetry transformations like
time reversal. Roberts argues that these symmetries have direct
empirical significance, but at a
"higher-order" level, that derives from the direct
empirical significance of spacetime translations.



The direct observational status of the familiar global spacetime
symmetries leads us to an *epistemological* aspect of
symmetries. According to Wigner, the spatiotemporal invariance
principles play the role of a prerequisite for the very possibility of
discovering the laws of nature: "if the correlations between
events changed from day to day, and would be different for different
points of space, it would be impossible to discover them"
(Wigner, 1967, p. 29). For Wigner, this conception of symmetry
principles is essentially related to our ignorance (if we could
directly know all the laws of nature, we would not need to use
symmetry principles in our search for them). Others, on the contrary,
have arrived at a view according to which symmetry principles function
as "transcendental principles" in the Kantian sense (see
for instance Mainzer, 1996). It should be noted in this regard that
Wigner's starting point, as quoted above, does not imply exact
symmetries -- all that is needed epistemologically is that the
global symmetries hold approximately, for suitable spatiotemporal
regions, such that there is sufficient stability and regularity in the
events for the laws of nature to be discovered.



There is another reason why symmetries might be seen as being
primarily epistemological. As we have mentioned, there is a close
connection between the notions of symmetry and equivalence, and this
leads also to a notion of irrelevance: the equivalence of space points
(translational symmetry), for example, may be understood in the sense
of the irrelevance of an absolute position to the physical
description. There are two ways that one might interpret the
epistemological significance of this: on the one hand, we might say
that symmetries are associated with unavoidable redundancy in our
*descriptions* of the world, while on the other hand we might
maintain that symmetries indicate a limitation of our epistemic access
-- there are certain properties of objects, such as their
absolute positions, that are not observable. The view that symmetries
are connected with the presence of non-observable quantities in the
physical description, with the corollary that the empirical violation
of a symmetry is intended in the sense that what was thought to be a
non-observable turns out to be actually an observable, was
particularly defended by one of the discover of parity violation, T.D.
Lee (see section 2.4). According to Lee (1981), "the root of all
symmetry principles lies in the assumption that it is impossible to
observe certain basic quantities" (p. 178). See on this (and,
more generally, on the relation between symmetry, equivalence and
irrelevance) Castellani (2003). Dasgupta (2016) defends an epistemic
interpretation of symmetry on a similar basis as Lee, that he calls
the "symmetry to reality inference". In contrast,
Roberts (2022, 4.4) argues that the symmetry to reality inference is
not generally valid.



Finally, we would like to mention an aspect of symmetry that might
very naturally be used to support either an ontological or an
epistemological account. It is widely agreed that there is a close
connection between symmetry and *objectivity*, the starting
point once again being provided by spacetime symmetries: the laws by
means of which we describe the evolution of physical systems have an
objective validity because they are the same for all observers. The
old and natural idea that what is objective should not depend upon the
particular perspective under which it is taken into consideration is
thus reformulated in the following group-theoretical terms: what is
objective is what is invariant with respect to the transformation
group of reference frames, or, quoting Hermann Weyl (1952, p. 132),
"objectivity means invariance with respect to the group of
automorphisms [of
 space-time]".[21]
 Debs and Redhead (2007) label as "invariantism" the view
that "invariance under a specified group of automorphisms is
both a necessary and sufficient condition for objectivity" (p.
60). They point out (p. 73, and see also p. 66) that there is a
natural connection between "invariantism" and structural
realism.



Growing interest, recently, in the metaphysics of physics includes
interest in symmetries. Baker (2010) offers an accessible
introduction, and Livanios (2010), connecting discussions of
symmetries to dispositions and essences, is an example of this
work.



To conclude: symmetries in physics offer many interpretational
possibilities, and how to understand the status and significance of
physical symmetries clearly presents a challenge to both physicists
and philosophers. A recent, comprehensive survey of the
philosophy of symmetry is Teh forthcoming.