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Thesymmetry of aphysical system is 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 asrotation of a circle) ordiscrete (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. Continuous symmetries can be described byLie groups while 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 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 inspecial relativity by a group of transformations of thespacetime known as thePoincaré group. Another important example is theinvariance of the form of physical laws under arbitrary differentiable coordinate transformations, which is an important idea ingeneral relativity.
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 isinvariant 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 exhibitspherical symmetry. A rotation about anyaxis of the sphere will preserve the shape of its surface from any given vantage point.
The above ideas lead to the useful idea ofinvariance 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 exhibitcylindrical symmetry, because theelectric field strength at a given distancer from 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 speedv1 and the other with speedv2 the totalkinetic energy of 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 ifv1 andv2 are interchanged.
Symmetries may be broadly classified asglobal orlocal. Aglobal symmetry is one that keeps a property invariant for a transformation that is applied simultaneously at all points ofspacetime, whereas alocal symmetry is 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 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 forgauge theories.
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 changecontinuously as a function of their parameterization. An important subclass of continuous symmetries in physics are spacetime symmetries.
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Continuousspacetime symmetries are symmetries involving transformations ofspace andtime. 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.
Mathematically, spacetime symmetries are usually described bysmoothvector fields on asmooth manifold. The underlyinglocal 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 areKilling vector fields which are those spacetime symmetries that preserve the underlyingmetric 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 ofisometries.
Adiscrete 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 calledreflections orinterchanges.
TheStandard Model ofparticle 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.
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 matter in the universe. CP violation is a fruitful area of current research inparticle physics.
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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 betweenbosons andfermions. 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 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 transformations describing physical symmetries typically form a mathematicalgroup.Group theory is 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 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 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 group S3.
A type of physical theory based onlocal symmetries is called agauge theory and 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 interaction and the U(1) group describes theelectromagnetic force.)
Also, the reduction by symmetry of the energy functional under the action by a group andspontaneous symmetry breaking of transformations of symmetric groups appear to elucidate topics inparticle physics (for example, theunification ofelectromagnetism and theweak force inphysical cosmology).
The symmetry properties of a physical system are intimately related to theconservation 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 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 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 spacetimecoordinates) | 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 |
Continuous symmetries in physics preserve transformations. One can specify a symmetry by showing how a very small transformation affects various particlefields. Thecommutator of 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 (also known as adiffeomorphism) has the infinitesimal effect on ascalar,spinor orvector field that can be expressed (using theEinstein summation convention):
Without gravity only the Poincaré symmetries are preserved which restricts to be of the form:
whereM is an antisymmetricmatrix (giving the Lorentz and rotational symmetries) andP 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:
where 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 ofdifferent 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:
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:
withD generating scale transformations andK generating special conformal transformations. For example,N = 4 supersymmetric Yang–Mills theory has this symmetry whilegeneral relativity does not although other theories of gravity such asconformal gravity do. The 'action' of a field theory is aninvariant 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.
