*First published Thu Jul 24, 2003; substantive revision Thu Dec 14, 2017*

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 Poincaré’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 Göttingen, 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.
Poincaré ^{[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 Poincaré 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 Poincaré 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) and Teh (2015, 2016).

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 (forthcoming), 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).

### 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. In 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 Lüders
and Pauli in the early 1950s, which states that the combination of
*C*, *P*, and *T* is a general symmetry of
physical laws. For philosophical reflections on the meaning and
grounding of the CPT theorem see Wallace (2009) and Greaves (2010). 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 β-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, β-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, Gołosz (2016) argues that
this asymmetry should be understood as applying to a physical process,
rather than to time itself. A yet more basic question is that of what
time reversal symmetry actually means. On the philosophical debate on
this point, see Roberts (2017) and references therein.

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.
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.

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 details see, for example, Teller (2000) and Martin (2003).

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. 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; 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 further detail on this topic, see Belot (1998), Nounou (2003), Weatherall (2016), and Nguyen, Teh, & Wells (forthcoming). For an approach to these questions using the theory of constrained Hamiltonian systems, see also Earman (2003b) and Castellani (2003, 2004). For an intuitive characterization of gauge symmetry, one that is more general than the Lagrangian and Hamiltonian formulations of theories using 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).

### 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 (on the various types of dualities and their significance in physics, see Castellani and Rickles (eds.), 2017). 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.

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; b) theoretical equivalence and underdetermination; and c) realism versus anti-realism.

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. 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 symétrie dans les phénomènes
physiques”, can be synthesized as follows:

- 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”.
- 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 2002), 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.)

*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 untill 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.

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. 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.

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.

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.