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

L D = ψ ¯ ( i γ μ ∂ μ − m ) ψ {\displaystyle {\mathcal {L}}_{D}={\bar {\psi }}\left(i\gamma ^{\mu }\partial _{\mu }-m\right)\psi }

Under this transformation the fermionic field changes as ψ → e i θ ψ {\displaystyle \psi \rightarrow e^{i\theta }\psi } and ψ ¯ → e − i θ ψ ¯ {\displaystyle {\bar {\psi }}\rightarrow e^{-i\theta }{\bar {\psi }}} and so:

L → L ¯ = e − i θ ψ ¯ ( i γ μ ∂ μ − m ) e i θ ψ = e − i θ e i θ ψ ¯ ( i γ μ ∂ μ − m ) ψ = L {\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

References

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


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