Symmetry (physics)

First Brillouin zone of FCC lattice showing symmetry labels

A 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.

A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Mathematically, symmetries are described by symmetry groups; continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups, lattice groups or other discrete groups. Symmetries are frequently amenable to mathematical formulations such as group representations.

Symmetries can be exploited to simplify many problems in physics and explain physical phenomena such as conservation laws, but symmetry principles also play a foundational role in all of the fundamental theories of modern physics.

One of the most important examples of 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. Other important examples include the invariance of the form of physical laws under arbitrary differentiable coordinate transformations in general relativity, and the gauge invariances of electromagnetism and the standard model of particle physics.

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 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 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 v₁ and the other with speed v₂ the total kinetic energy of the system (as calculated from an observer at the origin) is ⁠1/2⁠m(v₁² + v₂²) 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 v₁ and v₂ 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. Local symmetries play an important role in physics as they form the basis for gauge theories.

Continuous

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

Spacetime

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 Δt; this is expressed mathematically as invariance under the transformation t → t + a for any real parameters t and 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 t₀ and also at t₀ + a, the particle's total gravitational potential energy will be preserved.
Spatial translation: These spatial symmetries are represented by transformations of the form r→ → 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 as proper rotations and improper rotations. The former are just the 'ordinary' rotations; mathematically, they are represented by orthogonal matrices with unit determinant. The latter are represented by orthogonal matrices with determinant −1 and consist of a proper rotation combined with a spatial reflection. 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 transformations and give rise to the symmetry known as Lorentz covariance.
Homotheties (also known as dilatations) and special conformal transformations: These are transformations which do not preserve ordinary spacial distances or those of Minkowski spacetime, so are not isometries, but belong to a more general class of spacetime symmetries called conformal symmetries which preserve orientations and angles. The Poincaré transformations are also conformal.

Mathematically, spacetime symmetries are usually described by smooth vector fields on a smooth manifold. The underlying local diffeomorphisms 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 fields which are those spacetime symmetries that preserve the underlying 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

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.

Discrete spacetime symmetries include: the following.

Time reversal or T-symmetry: 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 motion still 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 same 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 inversion, parity or P-symmetry: 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'. This is sometimes described as a symmetry between an object or system and its mirror image, but care must be taken; a mirror image is a reflection through a single plane, while parity is simultaneous reflection across all three coordinate planes.
Glide reflection: These are represented by a composition of a translation and a reflection. These symmetries occur in some crystals and in some planar symmetries, known as wallpaper symmetries.
Inversion transformation: These are anti-conformal discrete transformations, meaning that they preserve angles but do not preserve orientation.

There are also discrete internal symmetries, including the following.

Charge conjugation or C-symmetry: Some models in particle physics are invariant under the interchange of each particle with its antiparticle.

C, P, and T in fundamental physics

The C, P and T symmetries are particularly important in particle physics. Many of the laws of physics are invariant under these symmetries, but as demonstrated by the Wu experiment, the weak nuclear force violates P-symmetry. It was formerly believed that the combination CP (i.e. simultaneous application of the C- and P-transformations) may nonetheless be a true symmetry, but it was later discovered this is also violated in certain interactions. The Standard Model of particle physics accounts for this "CP violation" but is invariant under the combination CPT in accordance with the CPT Theorem.

More general notions of symmetry

In addition to the classical symmetries covered by most of this article, some more general notions of symmetry have been proposed which are not described by the action of groups, but by some more general mathematical objects.

Supersymmetry

Supersymmetry is a hypothetical symmetry between the two different kinds of particles which appear in particle physics, the bosons and the fermions. Motivated by major unsolved problems in physics such as dark matter, the hierarchy problem and the unification of the microscopic fundamental forces, a supersymmetric extension of the Standard Model has been proposed in which each type of boson has a fermionic partner, called a superpartner, and vice versa. Supersymmetry is also a major ingredient of String theory. However, supersymmetry has not yet been experimentally verified: no known particle has the correct properties to be a superpartner of any other known particle. Mathematically, supersymmetry is described by supergroups and Lie superalgebras.

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]

The role of Lie groups

The transformations describing physical symmetries typically form a mathematical group. Group theory is an important area of mathematics for physicists.

Continuous symmetries are specified mathematically by continuous groups (called Lie 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 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).

A type of physical theory based on local symmetries is called a gauge theory and the symmetries natural to such a theory are called gauge symmetries. The theory of electromagnetism has the gauge group U(1), which corresponds to the invariance of the electrical and magnetic fields under particular transformations of the potentials. Gauge symmetries in the Standard Model, used to describe three of the fundamental interactions, are based on the SU(3) × SU(2) × U(1) gauge group with the SU(3) group being the gauge group of the strong force and the SU(2) × U(1) being that of the unified electroweak force. Early in the history of the universe, the latter symmetry was spontaneously broken to the U(1) of electromagnetism by the Higgs mechanism, causing the separation of the weak force and the electromagnetic force.

Conservation laws and symmetry

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 (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	Invariance	Conserved quantity
Proper orthochronous Poincaré 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) transformation	electric charge
	U(1) transformation	lepton generation number
	U(1) transformation	hypercharge
	U(1)_Y transformation	weak hypercharge
	U(2) [ U(1) × SU(2) ]	electroweak force
	SU(2) transformation	isospin
	SU(2)_L transformation	weak isospin
	P × SU(2)	G-parity
	SU(3) "winding number"	baryon number
	SU(3) transformation	quark color
	SU(3) (approximate)	quark flavor
	U(1) × SU(2) × SU(3)	Standard Model

Symmetry transformations of fields

Continuous symmetries in physics preserve transformations. One can specify a symmetry by showing how a very small transformation affects various particle fields. 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 $h(x)$ (also known as a diffeomorphism) has the infinitesimal effect on a scalar $\phi (x)$ , spinor $\psi (x)$ or vector field $A(x)$ that can be expressed (using the Einstein 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 }

where M is an antisymmetric 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:

\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 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:

\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 },

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 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.

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 the Philosophy 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. CiteSeerX 10.1.1.400.2867. doi:10.1140/epjh/e2013-40018-4. S2CID 14475702. {{cite journal}}: Cite uses deprecated parameter |citeseerx= (help)
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 Theory Click 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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