Global symmetry

In physics, a global symmetry is a symmetry that holds at all points in the spacetime under consideration, as opposed to a local symmetry which varies from point to point.

Global symmetries require conservation laws, but not forces, in physics.

An example of a global symmetry is the action of the $U(1)=e^{iq\theta }$ (for $\theta$ a constant - making it a global transformation) group on the Dirac Lagrangian:

{\mathcal {L}}_{D}={\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi

Under this transformation the wavefunction changes as $\psi \rightarrow e^{iq\theta }\psi$ and ${\bar {\psi }}\rightarrow e^{-iq\theta }{\bar {\psi }}$ and so:

{\mathcal {L}}\rightarrow {\bar {\mathcal {L}}}=e^{-iq\theta }{\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }-m\right)e^{iq\theta }\psi =e^{-iq\theta }e^{iq\theta }{\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi ={\mathcal {L}}

See also