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^{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}

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