Local symmetry

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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[how?].[1]

Special relativity only has a global symmetry (Lorentz symmetry or more generally Poincaré symmetry).[clarification needed]

Local gauge symmetry[edit]

There are many global symmetries (such as SU(2) of isospin symmetry)[clarification needed] and local symmetries (like SU(2) of weak interactions) in particle physics.

Often, the term local symmetry is associated[why?] 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[why?], like the photon or gluon field, which induce a force in addition to requiring conservation laws.[2]


The symmetry group of Supergravity is a local symmetry, whereas supersymmetry is a global symmetry.[further explanation needed]

See also[edit]


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