Global symmetry

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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:

${\displaystyle {{ℒ}}_D=\overline{\psi }(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi }$

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

${\displaystyle {ℒ}\to \overline{{ℒ}}=e^{-i\theta }\overline{\psi }(i{\gamma }^{\mu }{\partial }_{\mu }-m)e^{i\theta }\psi =e^{-i\theta }{e}^{i\theta }\overline{\psi }(i{\gamma }^{\mu }{\partial }_{\mu }-m)\psi ={ℒ}}$

• Field
• Global spacetime structure
• Local spacetime structure

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