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. These kinds of symmetries, also known as internal symmetries, are distinguished from spacetime symmetries.

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.

Fields may have internal symmetries in addition to spacetime symmetries. In many situations, one needs fields which are a list of space-time scalars: (φ1, φ2, ... φN). For example, in weather prediction these may be temperature, pressure, humidity, etc. In particle physics, the color symmetry of the interaction of quarks is an example of an internal symmetry, that of the strong interaction. Other examples are isospin, weak isospin, charm, strangeness and any other flavour symmetry.

If there is a symmetry of the problem, not involving spacetime, under which these components transform into each other, then this set of symmetries is called an internal symmetry. One may also make a classification of the charges of the fields under internal symmetries.


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.