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 ${\displaystyle U(1)=e^{i\theta }}$ (for ${\displaystyle \theta }$ a constant - making it a global transformation) group on the Dirac Lagrangian:

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

Under this transformation the fermionic field changes as ${\displaystyle \psi \rightarrow e^{i\theta }\psi }$ and ${\displaystyle {\bar {\psi }}\rightarrow e^{-i\theta }{\bar {\psi }}}$[1] and so:

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

See also

• Field (physics)
• Global spacetime structure
• Local spacetime structure

References

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