Symmetry (physics)

Thesymmetryof aphysical systemis a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under sometransformation.

A family of particular transformations may becontinuous(such asrotationof a circle) ordiscrete(e.g.,reflectionof 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 byLie groupswhile discrete symmetries are described byfinite groups(seeSymmetry 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 asgroup representationsand 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 inspecial relativityby a group of transformations of thespacetimeknown as thePoincaré group.Another important example is theinvarianceof the form of physical laws under arbitrary differentiable coordinate transformations, which is an important idea ingeneral 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,temperaturemay 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 isinvariantunder 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 exhibitspherical symmetry.A rotation about anyaxisof the sphere will preserve how the sphere "looks".

Invariance in force

The above ideas lead to the useful idea ofinvariancewhen 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 exhibitcylindrical symmetry,because theelectric field strengthat a given distancerfrom the wire will have the same magnitude at each point on the surface of a cylinder (whose axis is the wire) with radiusr.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 massm,starting at the origin and moving along thex-axis in opposite directions, one with speedv₁and the other with speedv₂the totalkinetic energyof the system (as calculated from an observer at the origin) is⁠1/2⁠m(v₁²+v₂²)and remains the same if the velocities are interchanged. The total kinetic energy is preserved under a reflection in they-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 ifv₁andv₂are interchanged.

Local and global

Symmetries may be broadly classified asglobalorlocal.Aglobal symmetryis one that keeps a property invariant for a transformation that is applied simultaneously at all points ofspacetime,whereas alocal symmetryis one that keeps a property invariant when a possibly different symmetry transformation is applied at each point ofspacetime;specifically a local symmetry transformation is parameterised by the spacetime co-ordinates, 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 forgauge theories.

Continuous

The two examples of rotational symmetry described above – spherical and cylindrical – are each instances ofcontinuous 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 changecontinuouslyas a function of their parameterization. An important subclass of continuous symmetries in physics are spacetime symmetries.

Spacetime

Continuousspacetime symmetriesare symmetries involving transformations ofspaceandtime.These may be further classified asspatial symmetries,involving only the spatial geometry associated with a physical system;temporal symmetries,involving only changes in time; orspatio-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 Δt;this is expressed mathematically as invariance under the transformationt→t+afor anyrealparameterstandt+ain the interval. For example, in classical mechanics, a particle solely acted upon by gravity will havegravitational potential energymghwhen suspended from a heighthabove 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 timet₀and also att₀+a,the particle's total gravitational potential energy will be preserved.
Spatial translation:These spatial symmetries are represented by transformations of the formr→→r→+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.
Spatial rotation:These spatial symmetries are classified asproper rotationsandimproper rotations.The former are just the 'ordinary' rotations; mathematically, they are represented by square matrices with unitdeterminant.The latter are represented by square matrices with determinant −1 and consist of a proper rotation combined with a spatial reflection (inversion). For example, a sphere has proper rotational symmetry. Other types of spatial rotations are described in the articleRotation symmetry.
Poincaré transformations:These are spatio-temporal symmetries which preserve distances inMinkowski spacetime,i.e. they are isometries of Minkowski space. They are studied primarily inspecial relativity.Those isometries that leave the origin fixed are calledLorentz transformationsand give rise to the symmetry known asLorentz covariance.
Projective symmetries:These are spatio-temporal symmetries which preserve thegeodesicstructure ofspacetime.They may be defined on any smooth manifold, but find many applications in the study ofexact 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 underinversion transformationsbut there is a cross-ratio on four points that is invariant.

Mathematically, spacetime symmetries are usually described bysmooth vector fieldson asmooth manifold.The underlyinglocal diffeomorphismsassociated 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 areKilling vector fieldswhich are those spacetime symmetries that preserve the underlyingmetricstructure 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 ofisometries.

Discrete

Adiscrete symmetryis 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 calledreflectionsorinterchanges.

Time reversal:Many laws of physics describe real phenomena when the direction of time is reversed. Mathematically, this is represented by the transformation, $t\,\rightarrow -t$ .For example,Newton's second law of motionstill holds if, in the equation $F\,=m{\ddot {r}}$ , $t$ is replaced by $-t$ .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 sameparabolictrajectory 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.
Spatial inversion:These are represented by transformations of the form ${\vec {r}}\,\rightarrow -{\vec {r}}$ and indicate an invariance property of a system when the coordinates are 'inverted'. Stated another way, these are symmetries between a certain object and itsmirror image.
Glide reflection:These are represented by a composition of a translation and a reflection. These symmetries occur in somecrystalsand in some planar symmetries, known aswallpaper symmetries.

C, P, and T

TheStandard Modelofparticle physicshas 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 itsantiparticle.
P-symmetry(parity symmetry), a universe where everything is mirrored along the three physical axes. This excludes weak interactions as demonstrated byChien-Shiung Wu.

T-symmetry(time reversal symmetry), a universe where thedirection of timeis 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 likeentropy.To properly reverse the direction of time, one would have to put theBig Bangand 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, calledCPT symmetry.CP violation,the violation of the combination of C- and P-symmetry, is necessary for the presence of significant amounts ofbaryonic matterin the universe. CP violation is a fruitful area of current research inparticle physics.

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 betweenbosonsandfermions.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

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.^[1]

Mathematics of physical symmetry

The transformations describing physical symmetries typically form a mathematicalgroup.Group theoryis an important area of mathematics for physicists.

Continuous symmetries are specified mathematically bycontinuous groups(calledLie groups). 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 thespecial orthogonal groupSO(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 theLorentz group(this may be generalised to thePoincaré group).

Discrete groups describe discrete symmetries. For example, the symmetries of an equilateral triangle are characterized by thesymmetric groupS₃.

A type of physical theory based onlocalsymmetries is called agaugetheoryand the symmetries natural to such a theory are calledgauge symmetries.Gauge symmetries in theStandard Model,used to describe three of thefundamental interactions,are based on theSU(3) × SU(2) × U(1)group. (Roughly speaking, the symmetries of the SU(3) group describe thestrong force,the SU(2) group describes theweak interactionand the U(1) group describes theelectromagnetic force.)

Also, the reduction by symmetry of the energy functional under the action by a group andspontaneous symmetry breakingof transformations of symmetric groups appear to elucidate topics inparticle physics(for example, theunificationofelectromagnetismand theweak forceinphysical cosmology).

Conservation laws and symmetry

The symmetry properties of a physical system are intimately related to theconservation lawscharacterizing that system.Noether's theoremgives 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 toconservation of (linear) momentum,and temporal translation symmetry (i.e. homogeneity of time) gives rise toconservation of energy.

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

Class	Invariance	Conserved quantity
Proper orthochronous Lorentz symmetry	translation in time (homogeneity)	energy E
	translation in space (homogeneity)	linear momentum p
	rotation in space (isotropy)	angular momentum L=r×p
	Lorentz-boost (isotropy)	boost 3-vector N=tp−Er
Discrete symmetry	P, coordinate inversion	spatial parity
	C,charge conjugation	charge parity
	T, time reversal	time parity
	CPT	product of parities
Internal symmetry(independent of spacetime coordinates)	U(1)gauge transformation	electric charge
	U(1)gauge transformation	lepton generation number
	U(1)gauge transformation	hypercharge
	U(1)_Ygauge transformation	weak hypercharge
	U(2) [U(1)×SU(2)]	electroweak force
	SU(2) gauge transformation	isospin
	SU(2)_Lgauge transformation	weak isospin
	P × SU(2)	G-parity
	SU(3) "winding number"	baryon number
	SU(3) gauge transformation	quark color
	SU(3)(approximate)	quark flavor
	S(U(2) × U(3)) [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 particlefields.Thecommutatorof two of these infinitesimal transformations is equivalent to a third infinitesimal transformation of the same kind hence they form aLie algebra.

A general coordinate transformation described as the general field $h(x)$ (also known as adiffeomorphism) has the infinitesimal effect on ascalar $\phi (x)$ ,spinor $\psi (x)$ orvector field $A(x)$ that can be expressed (using theEinstein summation convention):

\delta \phi (x)=h^{\mu }(x)\partial _{\mu }\phi (x)

\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)

\delta A_{\mu }(x)=h^{\nu }(x)\partial _{\nu }A_{\mu }(x)+A_{\nu }(x)\partial _{\mu }h^{\nu }(x)

Without gravity only the Poincaré symmetries are preserved which restricts $h(x)$ to be of the form:

h^{\mu }(x)=M^{\mu \nu }x_{\nu }+P^{\mu }

whereMis an antisymmetricmatrix(giving the Lorentz and rotational symmetries) andPis 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:

\delta \psi ^{\ Alpha }(x)=\lambda (x).\tau ^{\ Alpha \beta }\psi ^{\beta }(x)

\delta A_{\mu }(x)=\partial _{\mu }\lambda (x),

where $\tau$ are generators of a particularLie 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 ofdifferenttypes.

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:

\delta \phi (x)=\Omega (x)\phi (x)

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:

h^{\mu }(x)=M^{\mu \nu }x_{\nu }+P^{\mu }+Dx_{\mu }+K^{\mu }|x|^{2}-2K^{\nu }x_{\nu }x_{\mu },

withDgenerating scale transformations andKgenerating special conformal transformations. For example,N= 4super-Yang–Millstheory has this symmetry whilegeneral relativitydoes not although other theories of gravity such asconformal gravitydo. The 'action' of a field theory is aninvariantunder 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.

References

^Cordova, Clay; Dumitrescu, Thomas; Intriligator, Kenneth; Shao, Shu-Heng (2022). "Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond".arXiv:2205.09545[hep-th].

General readers

Lederman, L.;Hill, C.T.(2011) [2005].Symmetry and the Beautiful Universe.Prometheus Books.ISBN 9781615920419.
Schumm, B. (2004).Deep Down Things: The Breathtaking Beauty of Particle Physics.Johns Hopkins University Press.ISBN 978-0-8018-7971-5.
Stenger, V.J.(2000).Timeless Reality: Symmetry, Simplicity, and Multiple Universes.Prometheus Books.ISBN 9781573928595.Chapter 12 is a gentle introduction to symmetry, invariance, and conservation laws.
Zee, A.(2007).Fearful Symmetry: The search for beauty in modern physics(2nd ed.). Princeton University Press.ISBN 978-0-691-00946-9.

Technical readers

Brading, K.; Castellani, E. (2003).Symmetries in Physics: Philosophical Reflections.Cambridge University Press.ISBN 978-1-139-44202-2.
Brading, K.; Castellani, E. (2007). "Symmetries and Invariances in Classical Physics". In Butterfield, J.;Earman, J.(eds.).Philosophy of Physic Part B.North Holland. pp. 1331–68.ISBN 978-0-08-046665-1.
Debs, T.; Redhead, M. (2007).Objectivity, Invariance, and Convention: Symmetry in Physical Science.Harvard University Press.ISBN 978-0-674-03413-6.
Earman, J.(2002),Laws, Symmetry, and Symmetry Breaking: Invariance, Conservations Principles, and Objectivity.(PDF)Address to the 2002 meeting of thePhilosophy of Science Association.
Mainzer, K. (1996).Symmetries of Nature: A Handbook for Philosophy of Nature and Science.de Gruyter.ISBN 978-3-11-088693-1.
Mouchet, A. (2013)."Reflections on the four facets of symmetry: how physics exemplifies rational thinking".European Physical Journal H.38(5): 661–702.arXiv:1111.0658.Bibcode:2013EPJH...38..661M.CiteSeerX10.1.1.400.2867.doi:10.1140/epjh/e2013-40018-4.S2CID 14475702.
Thompson, William J. (1994).Angular Momentum: An Illustrated Guide to Rotational Symmetries for Physical Systems.Wiley.ISBN 0-471-55264-X.
Van Fraassen, B.(1989).Laws and symmetry.Oxford University Press.ISBN 978-0-19-151999-4.
Wigner, E.(1970) [1967].Symmetries and Reflections.M.I.T. Press.ISBN 978-0-262-73021-1.

External links

The Feynman Lectures on Physics Vol. I Ch. 52: Symmetry in Physical Laws
Stanford Encyclopedia of Philosophy:"Symmetry"—by K. Brading and E. Castellani.
Pedagogic Aids to Quantum Field TheoryClick on link to Chapter 6: Symmetry, Invariance, and Conservation for a simplified, step-by-step introduction to symmetry in physics.

[snowmass-1] Cordova, Clay; Dumitrescu, Thomas; Intriligator, Kenneth; Shao, Shu-Heng (2022). "Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond".arXiv:2205.09545[hep-th].

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