Local symmetry

In physics, a local symmetry is symmetry of some physical quantity, which smoothly depends on the point of the base manifold. Such quantities can be for example an observable, a tensor or the Lagrangian of a theory. If a symmetry is local in this sense, then one can apply a local transformation (resp. local gauge transformation), which means that the representation of the symmetry group is a function of the manifold and can thus be taken to act differently on different points of spacetime.

The diffeomorphism group is a local symmetry and thus every geometrical or generally covariant theory (i.e. a theory whose equations are tensor equations, for example general relativity) has local symmetries.

Often the term local symmetry is specifically associated with local gauge symmetries in Yang–Mills theory (see also standard model) where the Lagrangian is locally symmetric under some compact Lie group. Local gauge symmetries always come together with some bosonic gauge fields, like the photon or gluon field, which induce a force in addition to requiring conservation laws.^[1]

Examples

• General relativity has a local symmetry (general covariance, diffeomorphisms) which can be seen as generating the gravitational force.^[2] Special relativity only has a global symmetry (Lorentz symmetry or more generally Poincaré symmetry)
• There are many global symmetries (such as SU(2) of isospin symmetry) and local symmetries (like SU(2) of weak interactions) in particle physics. The standard model of particle physics consists of Yang-Mills Theories
• The symmetry group of Supergravity is a local symmetry, whereas supersymmetry is a global symmetry.

See also

• Field (physics)
• Global spacetime structure
• Local spacetime structure
• Gauge theory
• Gravitation (book)

References

1. Kaku, Michio (1993). Quantum Field Theory: A Modern Introduction. New York: Oxford University Press. ISBN 0-19-507652-4.
2. Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (1973-09-15). "Gravitation". San Francisco: W. H. Freeman. ISBN 978-0-7167-0344-0.