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.

For these local symmetries, 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).

General relativity has a local symmetry of diffeomorphisms (general covariance). This can be seen as generating the gravitational force.

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.

Often, the term local symmetry is associated with the local gauge symmetries in Yang–Mills theory. The Standard Model of particle physics consists of Yang-Mills Theories. In these theories, the Lagrangian is locally symmetric under some compact Lie group. Local gauge symmetries always come together with bosonic gauge fields, like the photon or gluon field, which induce a force in addition to requiring conservation laws.

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