Symmetry (physics)
Continuous and discrete transformations give rise to corresponding types of symmetries.
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 group representations and can, in addition, be exploited to simplify many problems.
A rotation about any axis of the sphere will preserve the shape of its surface from any given vantage point.
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 last example above illustrates another way of expressing symmetries, namely through the equations that describe some aspect of the physical system.
Local symmetries play an important role in physics as they form the basis for gauge theories.
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.
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 symmetries sometimes involve some type of 'swapping', these swaps usually being called reflections or interchanges.
The Standard Model of particle physics has three related natural near-symmetries.
CP violation is a fruitful area of current research in particle physics.
A type of symmetry known as supersymmetry has been used to try to make theoretical advances in the Standard Model.
Supersymmetry asserts that each type of boson has, as a supersymmetric partner, a fermion, called a superpartner, and vice versa.
[1] The transformations describing physical symmetries typically form a mathematical group.
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 unification of electromagnetism and the weak force in physical cosmology).
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.
that can be expressed (using the Einstein summation convention): Without gravity only the Poincaré symmetries are preserved which restricts
For example, local gauge transformations apply to both a vector and spinor field: where
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.
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.
