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 $U(1)=e^{i\theta }$ (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 fermionic field changes as $\psi \rightarrow e^{i\theta }\psi$ and ${\bar {\psi }}\rightarrow e^{-i\theta }{\bar {\psi }}$ ^[1] and so:

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

References

↑ http://www.damtp.cam.ac.uk/user/tong/qft.html

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[1] ttp://www.damtp.cam.ac.uk/user/tong/qft.html

Global symmetry

See also

References