View source for Symmetry (physics)

{{Short description|Feature of a system that is preserved under some transformation}}
{{Other uses|Symmetry (disambiguation)}}
{{More footnotes|date=February 2018}}
[[File:Brillouin Zone (1st, FCC).svg|thumb|right|200px|First [[Brillouin zone]] of [[FCC lattice]] showing symmetry labels]]

The '''symmetry''' of a [[physical system]] is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some [[Transformation (function)|transformation]].

A family of particular transformations may be ''continuous'' (such as [[rotation]] of a circle) or ''[[discrete space|discrete]]'' (e.g., [[Reflection (physics)|reflection]] of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by [[Lie group]]s while discrete symmetries are described by [[finite group]]s (see ''[[Symmetry group]]'').

These two concepts, Lie and finite groups, are the foundation for the fundamental theories of modern physics. Symmetries are frequently amenable to mathematical formulations such as [[representation of a Lie group|group representations]] and can, in addition, be exploited to simplify many problems.

Arguably the most important example of a symmetry in physics is that the speed of light has the same value in all frames of reference, which is described in [[special relativity]] by a group of transformations of the [[spacetime]] known as the [[Poincaré group]]. Another important example is the [[General covariance|invariance]] of the form of physical laws under arbitrary differentiable coordinate transformations, which is an important idea in [[general relativity]].

== As a kind of invariance ==
Invariance is specified mathematically by transformations that leave some property (e.g. quantity) unchanged. This idea can apply to basic real-world observations. For example, [[temperature]] may be homogeneous throughout a room. Since the temperature does not depend on the position of an observer within the room,  we say that the temperature is ''invariant'' under a shift in an observer's position within the room.

Similarly, a uniform sphere rotated about its center will appear exactly as it did before the rotation. The sphere is said to exhibit [[spherical symmetry]]. A rotation about any [[Axis of rotation|axis]] of the sphere will preserve the shape of its surface from any given vantage point.

=== Invariance in force ===
The above ideas lead to the useful idea of ''invariance'' when discussing observed physical symmetry; this can be applied to symmetries in forces as well.

For example, an electric field due to an electrically charged wire of infinite length is said to exhibit [[rotational symmetry#Rotational symmetry with respect to any angle|cylindrical symmetry]], because the [[electric field strength]] at a given distance ''r'' from the wire will have the same magnitude at each point on the surface of a cylinder (whose axis is the wire) with radius ''r''. Rotating the wire about its own axis does not change its position or charge density, hence it will preserve the field. The field strength at a rotated position is the same. This is not true in general for an arbitrary system of charges.

In Newton's theory of mechanics, given two bodies, each with mass ''m'', starting at the origin and moving along the ''x''-axis in opposite directions, one with speed ''v''<sub>1</sub> and the other with speed ''v''<sub>2</sub> the total [[kinetic energy]] of the system (as calculated from an observer at the origin) is {{nowrap|{{sfrac|1|2}}''m''(''v''<sub>1</sub><sup>2</sup> + ''v''<sub>2</sub><sup>2</sup>)}} and remains the same if the velocities are interchanged. The total kinetic energy is preserved under a reflection in the ''y''-axis.

The last example above illustrates another way of expressing symmetries, namely through the equations that describe some aspect of the physical system. The above example shows that the total kinetic energy will be the same if ''v''<sub>1</sub> and ''v''<sub>2</sub> are interchanged.

== Local and global ==
Symmetries may be broadly classified as ''global'' or ''local''. A ''global symmetry'' is one that keeps a property invariant for a transformation that is applied simultaneously at all points of [[spacetime]], whereas a ''local symmetry'' is one that keeps a property invariant when a possibly different symmetry transformation is applied at each point of [[spacetime]]; specifically a local symmetry transformation is parameterised by the spacetime coordinates, whereas a global symmetry is not. This implies that a global symmetry is also a local symmetry. Local symmetries play an important role in physics as they form the basis for [[Gauge theory|gauge theories]].

== Continuous ==
The two examples of rotational symmetry described above –&nbsp;spherical and cylindrical&nbsp;– are each instances of [[continuous symmetry]]. These are characterised by invariance following a continuous change in the geometry of the system. For example, the wire may be rotated through any angle about its axis and the field strength will be the same on a given cylinder. Mathematically, continuous symmetries are described by transformations that change [[continuous function|continuously]] as a function of their parameterization. An important subclass of continuous symmetries in physics are spacetime symmetries.

=== Spacetime ===
{{Main|Spacetime symmetries}}
{{Lie groups}}
Continuous ''spacetime symmetries'' are symmetries involving transformations of [[space]] and [[time]]. These may be further classified as ''spatial symmetries'', involving only the spatial geometry associated with a physical system; ''temporal symmetries'', involving only changes in time; or ''spatio-temporal symmetries'', involving changes in both space and time.

* ''[[Time translation]]'': A physical system may have the same features over a certain interval of time Δ''t''; this is expressed mathematically as invariance under the transformation {{nowrap|''t'' → ''t'' + ''a''}} for any [[real number|real]] parameters ''t'' and {{nowrap|''t'' + ''a''}} in the interval. For example, in classical mechanics, a particle solely acted upon by gravity will have [[gravitational potential energy]] ''mgh'' when suspended from a height ''h'' above the Earth's surface. Assuming no change in the height of the particle, this will be the total gravitational potential energy of the particle at all times. In other words, by considering the state of the particle at some time ''t''{{sub|0}} and also at {{nowrap|''t''{{sub|0}} + ''a''}}, the particle's total gravitational potential energy will be preserved.
* ''[[Spatial translation symmetry|Spatial translation]]'': These spatial symmetries are represented by transformations of the form {{nowrap|{{vec|''r''}} → {{vec|''r''}} + {{vec|''a''}}}} and describe those situations where a property of the system does not change with a continuous change in location. For example, the temperature in a room may be independent of where the thermometer is located in the room.
* ''[[Rotational symmetry|Spatial rotation]]'': These spatial symmetries are classified as [[proper rotation]]s and [[improper rotation]]s. The former are just the 'ordinary' rotations; mathematically, they are represented by square matrices with unit [[determinant]]. The latter are represented by square matrices with determinant −1 and consist of a proper rotation combined with a spatial reflection ([[Point reflection|inversion]])<!-- odd-dimensional? -->. For example, a sphere has proper rotational symmetry. Other types of spatial rotations are described in the article ''[[Rotation symmetry]]''.
* ''Poincaré transformations'': These are spatio-temporal symmetries which preserve distances in [[Minkowski spacetime]], i.e. they are isometries of Minkowski space. They are studied primarily in [[special relativity]]. Those isometries that leave the origin fixed are called [[Lorentz transformation]]s and give rise to the symmetry known as [[Lorentz covariance]].
* ''Projective symmetries'': These are spatio-temporal symmetries which preserve the [[geodesic]] structure of [[spacetime]]. They may be defined on any smooth manifold, but find many applications in the study of [[exact solutions in general relativity]].
* ''Inversion transformations'': These are spatio-temporal symmetries which generalise Poincaré transformations to include other conformal one-to-one transformations on the space-time coordinates. Lengths are not invariant under [[inversion transformations]] but there is a cross-ratio on four points that is invariant.

Mathematically, spacetime symmetries are usually described by [[Smooth function|smooth]] [[vector field]]s on a [[smooth manifold]]. The underlying [[local diffeomorphism]]s associated with the vector fields correspond more directly to the physical symmetries, but the vector fields themselves are more often used when classifying the symmetries of the physical system.

Some of the most important vector fields are [[Killing vector field]]s which are those spacetime symmetries that preserve the underlying [[Metric tensor|metric]] structure of a manifold. In rough terms, Killing vector fields preserve the distance between any two points of the manifold and often go by the name of [[isometries]].

== Discrete ==
{{Main|Discrete symmetry}}

A ''discrete symmetry'' is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance. Discrete symmetries sometimes involve some type of 'swapping', these swaps usually being called ''reflections'' or ''interchanges''.

* ''[[T-symmetry|Time reversal]]'': Many laws of physics describe real phenomena when the direction of time is reversed. Mathematically, this is represented by the transformation, <math>t \, \rightarrow - t </math>. For example, [[Newton's second law of motion]] still holds if, in the equation <math>F \, = m \ddot {r} </math>, <math>t</math> is replaced by <math>-t</math>. This may be illustrated by recording the motion of an object thrown up vertically (neglecting air resistance) and then playing it back. The object will follow the same [[Parabola|parabolic]] trajectory through the air, whether the recording is played normally or in reverse. Thus, position is symmetric with respect to the instant that the object is at its maximum height.
* ''[[Parity (physics)|Spatial inversion]]'': These are represented by transformations of the form <math>\vec{r} \, \rightarrow - \vec{r}</math> and indicate an invariance property of a system when the coordinates are 'inverted'. Stated another way, these are symmetries between a certain object and its [[mirror image]].
*''[[Glide reflection]]'': These are represented by a composition of a translation and a reflection. These symmetries occur in some [[crystal]]s and in some planar symmetries, known as [[wallpaper group|wallpaper symmetries]].

=== C, P, and T ===
{{Main|Wu experiment}}
The [[Standard Model]] of [[particle physics]] has three related natural near-symmetries. These state that the universe in which we live should be indistinguishable from one where a certain type of change is introduced.

*[[C-symmetry]] (charge symmetry), a universe where every particle is replaced with its [[antiparticle]].
*[[Parity (physics)|P-symmetry]] (parity symmetry), a universe where everything is mirrored along the three physical axes. This excludes weak interactions as demonstrated by [[Chien-Shiung Wu]].
<!-- 
This really isn't a good example, because a mirror only reflects one axis (the normal). If possible, a common mans description of inverting 3 axis really needs to be used here. The mirror is too likely to be taken literally, and lead to misunderstanding. 
-->

*[[T-symmetry]] (time reversal symmetry), a universe where the [[entropy (arrow of time)|direction of time]] is reversed. T-symmetry is counterintuitive (the future and the past are not symmetrical) but explained by the fact that the Standard Model describes local properties, not global ones like [[entropy]]. To properly reverse the direction of time, one would have to put the [[Big Bang]] and the resulting low-entropy state in the "future". Since we perceive the "past" ("future") as having lower (higher) entropy than the present, the inhabitants of this hypothetical time-reversed universe would perceive the future in the same way as we perceive the past, and vice versa.

These symmetries are near-symmetries because each is broken in the present-day universe. However, the Standard Model predicts that the combination of the three (that is, the simultaneous application of all three transformations) must be a symmetry, called [[CPT symmetry]]. [[CP violation]], the violation of the combination of C- and P-symmetry, is necessary for the presence of significant amounts of [[baryonic matter]] in the universe. CP violation is a fruitful area of current research in [[particle physics]].

{{misleading|section|date=June 2015}}

=== Supersymmetry ===
{{Main|Supersymmetry}}
A type of symmetry known as supersymmetry has been used to try to make theoretical advances in the Standard Model. Supersymmetry is based on the idea that there is another physical symmetry beyond those already developed in the Standard Model, specifically a symmetry between [[boson]]s and [[fermion]]s. Supersymmetry asserts that each type of boson has, as a supersymmetric partner, a fermion, called a superpartner, and vice versa. Supersymmetry has not yet been experimentally verified: no known particle has the correct properties to be a superpartner of any other known particle. Currently LHC is preparing for a run which tests supersymmetry.

== Generalized symmetries ==
{{See also|Non-invertible symmetry}}
Generalized symmetries encompass a number of recently recognized generalizations of the concept of a global symmetry. These include higher form symmetries, higher group symmetries, non-invertible symmetries, and subsystem symmetries.<ref name=snowmass>{{cite arXiv|last1=Cordova|first1=Clay|last2=Dumitrescu|first2=Thomas|last3=Intriligator|first3=Kenneth|last4=Shao|first4=Shu-Heng
|date=2022|title=Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond|eprint=2205.09545|class=hep-th}}</ref>

== Mathematics of physical symmetry ==
{{Main|Symmetry group}}
{{See also|Symmetry in quantum mechanics|Symmetries in general relativity}}
The transformations describing physical symmetries typically form a mathematical [[group (mathematics)|group]]. [[Group theory]] is an important area of mathematics for physicists.

Continuous symmetries are specified mathematically by ''continuous groups'' (called [[Lie group]]s). Many physical symmetries are isometries and are specified by symmetry groups. Sometimes this term is used for more general types of symmetries. The set of all proper rotations (about any angle) through any axis of a sphere form a Lie group called the [[special orthogonal group]] SO(3). (The '3' refers to the three-dimensional space of an ordinary sphere.) Thus, the symmetry group of the sphere with proper rotations is SO(3). Any rotation preserves distances on the surface of the ball. The set of all Lorentz transformations form a group called the [[Lorentz group]] (this may be generalised to the [[Poincaré group]]).

Discrete groups describe discrete symmetries. For example, the symmetries of an equilateral triangle are characterized by the [[symmetric group]] S{{sub|3}}.

A type of physical theory based on ''local'' symmetries is called a [[Gauge theory|''gauge'' theory]] and the symmetries natural to such a theory are called [[Gauge symmetry|gauge symmetries]]. Gauge symmetries in the [[Standard Model]], used to describe three of the [[fundamental interaction]]s, are based on the  [[SU(3) × SU(2) × U(1)]] group. (Roughly speaking, the symmetries of the SU(3) group describe the [[strong force]], the SU(2) group describes the [[weak interaction]] and the U(1) group describes the [[electromagnetic force]].)

Also, the reduction by symmetry of the energy functional under the action by a group and [[spontaneous symmetry breaking]] of transformations of symmetric groups appear to elucidate topics in [[particle physics]] (for example, the [[electroweak interaction|unification]] of [[electromagnetism]] and the [[weak force]] in [[physical cosmology]]).

=== Conservation laws and symmetry ===
{{Main|Noether's theorem}}
The symmetry properties of a physical system are intimately related to the [[conservation laws]] characterizing that system. [[Noether's theorem]] gives a precise description of this relation. The theorem states that each continuous symmetry of a physical system implies that some physical property of that system is conserved. Conversely, each conserved quantity has a corresponding symmetry. For example, spatial translation symmetry (i.e. homogeneity of space) gives rise to [[conservation of momentum|conservation of (linear) momentum]], and temporal translation symmetry (i.e. homogeneity of time) gives rise to [[conservation of energy]].

The following table summarizes some fundamental symmetries and the associated conserved quantity.

{| class="wikitable"
!Class
![[Invariant (physics)|Invariance]]
![[Conservation law (physics)|Conserved quantity]]
|-
||Proper orthochronous<br />[[Poincaré group|Poincaré symmetry]]
||[[translation in time]]<br />([[homogeneity (physics)|homogeneity]])
||[[conservation of energy|energy]] <br /> ''E''
|-
||
||[[translational invariance|translation in space]]<br />([[homogeneity (physics)|homogeneity]])
||[[conservation of momentum|linear momentum]] <br /> '''p'''
|-
||
||[[rotational invariance|rotation in space]]<br />([[isotropy]])
||[[conservation of angular momentum|angular momentum]] <br /> '''L''' = '''r''' × '''p'''
|-
||
||[[principle of relativity|Lorentz-boost]]<br />([[isotropy]])
||boost 3-vector <br />'''N''' = ''t'''''p''' − ''E'''''r'''
|-
||[[Discrete symmetry]]
||P, coordinate inversion
||[[CP-symmetry|spatial parity]]
|-
||
||C, [[charge conjugation]]
||[[CP-symmetry|charge parity]]
|-
||
||T, time reversal
||[[T-symmetry|time parity]]
|-
||
||[[CPT symmetry|CPT]]
||product of parities
|-
||[[Internal symmetry]] (independent of<br />[[spacetime]] [[coordinate]]s)
||[[U(1)]] transformation
||[[electric charge]]
|-
||
||[[U(1)]] transformation
||[[lepton number|lepton generation number]]
|-
||
||[[U(1)]] transformation
||[[hypercharge]]
|-
||
||[[U(1)]]<SUB>Y</SUB> transformation
||[[weak hypercharge]]
|-
||
||U(2) [ [[U(1)]] × [[SU(2)]] ]
||[[electroweak force]]
|-
||
||SU(2) transformation
||[[isospin]]
|-
||
||[[SU(2)]]<SUB>L</SUB> transformation
||[[weak isospin]]
|-
||
||P × SU(2)
||[[G-parity]]
|-
||
||SU(3) "winding number"
||[[baryon number]]
|-
||
||SU(3) transformation
|| [[quark color]]
|-
||
||[[SU(3)]] (approximate)
||[[flavor (physics)|quark flavor]]
|-
||
||[[U(1)]] × [[SU(2)]] × [[SU(3)]]
||[[Standard Model]]
|-
|}

== Mathematics ==
Continuous symmetries in physics preserve transformations. One can specify a symmetry by showing how a very small transformation affects various particle [[Field (physics)|field]]s. The [[commutator]] of two of these infinitesimal  transformations is equivalent to a third infinitesimal transformation of the same kind hence they form a [[Lie algebra]].

A general coordinate transformation described as the general field <math>h(x)</math> (also known as a [[diffeomorphism]]) has the infinitesimal effect on a [[scalar field|scalar]] <math>\phi(x)</math>, [[spinor field|spinor]] <math>\psi(x)</math> or [[vector field]] <math>A(x)</math> that can be expressed (using the [[Einstein summation convention]]):

:<math>
\delta\phi(x) = h^{\mu}(x)\partial_{\mu}\phi(x) 
</math>
:<math>
\delta\psi^\alpha(x) = h^{\mu}(x)\partial_{\mu}\psi^\alpha(x) +  \partial_\mu h_\nu(x) \sigma_{\mu\nu}^{\alpha \beta} \psi^{\beta}(x)
</math>
:<math>
\delta A_\mu(x) = h^{\nu}(x)\partial_{\nu}A_\mu(x) + A_\nu(x)\partial_\mu h^{\nu}(x) 
</math>

Without gravity only the Poincaré symmetries are preserved which restricts <math>h(x)</math> to be of the form:

:<math>
h^{\mu}(x) = M^{\mu \nu}x_\nu + P^\mu 
</math>

where '''M''' is an antisymmetric [[Matrix (mathematics)|matrix]] (giving the Lorentz and rotational symmetries) and '''P''' is a general vector (giving the translational symmetries). Other symmetries affect multiple fields simultaneously. For example, local gauge transformations apply to both a vector and spinor field:

:<math>
\delta\psi^\alpha(x) = \lambda(x).\tau^{\alpha\beta}\psi^\beta(x) 
</math>
:<math>
\delta A_\mu(x) = \partial_\mu \lambda(x)
,</math>

where <math>\tau</math> are generators of a particular [[Lie group]]. So far the transformations on the right have only included fields of the same type. Supersymmetries are defined according to how the mix fields of ''different'' types.

Another symmetry which is part of some theories of physics and not in others is scale invariance which involve Weyl transformations of the following kind:

:<math>
\delta \phi(x) = \Omega(x) \phi(x) 
</math>

If the fields have this symmetry then it can be shown that the field theory is almost certainly conformally invariant also. This means that in the absence of gravity h(x) would restricted to the form:

:<math>
h^{\mu}(x) = M^{\mu \nu}x_\nu + P^\mu + D x_\mu + K^{\mu} |x|^2 - 2 K^\nu x_\nu x_\mu 
,</math>

with '''D''' generating scale transformations and '''K''' generating special conformal transformations. For example, [[N = 4 supersymmetric Yang–Mills theory]] has this symmetry while [[general relativity]] does not although other theories of gravity such as [[conformal gravity]] do. The 'action' of a field theory is an [[invariant (physics)|invariant]] under all the symmetries of the theory. Much of modern theoretical physics is to do with speculating on the various symmetries the Universe may have and finding the invariants to construct field theories as models.

In string theories, since a string can be decomposed into an infinite number of particle fields, the symmetries on the string world sheet is equivalent to special transformations which mix an infinite number of fields.

== See also ==
{{cols|colwidth=21em}}
*[[Conserved current]] & [[Charge (physics)|Charge]]
*[[Coordinate-free]]
*[[Covariance and contravariance of vectors|Covariance and contravariance]]
*[[Fictitious force]]
*[[Galilean invariance]]
*[[Principle of covariance]]
*[[General covariance]]
*[[Harmonic coordinate condition]]
*[[Inertial frame of reference]]
*[[List of mathematical topics in relativity]]
*[[Standard Model (mathematical formulation)]]
*[[Wheeler–Feynman absorber theory]]
{{colend}}

== References ==
{{Reflist}}

=== General readers ===
{{refbegin}}
*{{cite book |author1-link=Leon Lederman |first1=L. |last1=Lederman |author2-link=Christopher T. Hill |first2=C.T. |last2=Hill |title=Symmetry and the Beautiful Universe |publisher=Prometheus Books |date=2011 |orig-date=2005 |isbn=9781615920419 |url={{GBurl|e9AQ5zH13qQC|p=7}}}}
*{{cite book |first=B. |last=Schumm |title= Deep Down Things: The Breathtaking Beauty of Particle Physics |publisher=Johns Hopkins University Press |date=2004 |isbn=978-0-8018-7971-5 |url={{GBurl|IxwAEAAAQBAJ|pg=PR7}}}}
*{{cite book |author-link=Victor J. Stenger |first=V.J. |last=Stenger |title=Timeless Reality: Symmetry, Simplicity, and Multiple Universes |publisher=Prometheus Books |date=2000 |isbn=9781573928595 }} Chapter 12 is a gentle introduction to symmetry, invariance, and conservation laws.
*{{cite book |author-link=Anthony Zee |first=A. |last=Zee |title=Fearful Symmetry: The search for beauty in modern physics |publisher=Princeton University Press |edition=2nd |date=2007 |isbn=978-0-691-00946-9 |url=http://press.princeton.edu/titles/8509.html}}
{{refend}}

=== Technical readers ===
{{refbegin}}
*{{cite book |last1=Brading |first1=K.|author1-link=Katherine Brading |last2=Castellani |first2=E. |title=Symmetries in Physics: Philosophical Reflections |publisher=Cambridge University Press |date=2003 |isbn=978-1-139-44202-2 |url={{GBurl|fB38Q9D-WdwC|pg=PR5}}}}
*{{cite book |last1=Brading |first1=K.|author1-link=Katherine Brading |last2=Castellani |first2=E. |chapter=Symmetries and Invariances in Classical Physics |editor1-last=Butterfield |editor1-first=J. |editor2-first=J. |editor2-last=Earman |editor2-link=John Earman |title=Philosophy of Physic Part B |publisher=North Holland |date=2007 |isbn=978-0-08-046665-1 |pages=1331–68 |url={{GBurl|UGpJEFA5zXcC|pg=PR10}} }}
*{{cite book |last1=Debs |first1=T. |last2=Redhead |first2=M. |title=Objectivity, Invariance, and Convention: Symmetry in Physical Science |publisher=Harvard University Press |date=2007 |isbn=978-0-674-03413-6 |url={{GBurl|lx0XQHtvdjoC|pg=PP1}} }}
*{{citation |first=J. |last=Earman |author-link=John Earman |date=2002 |url=http://philsci-archive.pitt.edu/878/1/PSA2002.pdf |title=Laws, Symmetry, and Symmetry Breaking: Invariance, Conservations Principles, and Objectivity.}} Address to the 2002 meeting of the [[Philosophy of Science Association]].
<!-- *{{cite book |first=G. |last=Kalmbach H.E. |title=Quantum Mathematics: WIGRIS |publisher=RGN Publications |date=2014 |isbn= |pages= |url=}} Not found -->
*{{cite book |first=K. |last=Mainzer |title=Symmetries of Nature: A Handbook for Philosophy of Nature and Science |publisher=de Gruyter |date=1996 |isbn=978-3-11-088693-1 |url={{GBurl|QekhAAAAQBAJ|pg=PP11}}}}
*{{cite journal |first=A. |last=Mouchet |title=Reflections on the four facets of symmetry: how physics exemplifies rational thinking |journal=European Physical Journal H |volume=38 |issue= 5|pages=661–702 |date=2013 |doi=10.1140/epjh/e2013-40018-4 |arxiv=1111.0658 |bibcode=2013EPJH...38..661M |url=http://hal.archives-ouvertes.fr/hal-00637572 |citeseerx=10.1.1.400.2867|s2cid=14475702 }}
*{{cite book |first=William J. |last=Thompson |title=Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems |publisher=Wiley |date=1994 |isbn=0-471-55264-X }}
*{{cite book |author-link=Bas Van Fraassen |first=B. |last=Van Fraassen |title=Laws and symmetry |publisher=Oxford University Press |date=1989 |isbn=978-0-19-151999-4 |url={{GBurl|6QdREAAAQBAJ|pg=PR11}} }}
*{{cite book |author-link=Eugene Wigner |first=E. |last=Wigner |title=Symmetries and Reflections |publisher=M.I.T. Press |date=1970 |isbn=978-0-262-73021-1 |orig-year=1967 |url=}}
{{refend}}

== External links ==
*[https://feynmanlectures.caltech.edu/I_52.html The Feynman Lectures on Physics Vol. I Ch. 52: Symmetry in Physical Laws]
*[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/symmetry-breaking/ Symmetry]"—by [[Katherine Brading|K. Brading]] and E. Castellani.
*[http://www.quantumfieldtheory.info Pedagogic Aids to Quantum Field Theory] Click on link to Chapter 6: Symmetry, Invariance, and Conservation for a simplified, step-by-step introduction to symmetry in physics.

{{C, P and T}}
{{Authority control}}

{{DEFAULTSORT:Symmetry In Physics}}
[[Category:Concepts in physics]]
[[Category:Conservation laws]]
[[Category:Diffeomorphisms]]
[[Category:Differential geometry]]
[[Category:Symmetry]]