Global symmetry in field theory

A global symmetry has transformation parameters constant across spacetime and preserves the action. It can relate physically distinct states. Continuous global symmetries give Noether currents and conserved charges when boundary fluxes vanish; discrete symmetries do not receive currents from the same infinitesimal theorem. A gauge redundancy instead uses local parameters to relate descriptions of the same physical state, except that nontrivial boundary transformations may carry physical charges. The global phase symmetry of a complex scalar field is an important internal example.

Adding a free real scalar field $\varphi$ to covariantly gauge-fixed QED gives an infinitesimal global invariance $\delta A_{\mu }=\epsilon {\partial }_{\mu }\varphi$, $\delta \psi =-ie\epsilon \varphi \psi$, $\delta \varphi =-\epsilon (\partial \cdot A)/\xi$. The matter terms transform as an ordinary gauge transformation. The scalar kinetic variation combines with the gauge fixing variation into $\delta \mathcal{L}=-(\epsilon /\xi ){\partial }_{\mu }[(\partial \cdot A){\partial }^{\mu }\varphi ]$. Consequently the action is invariant under boundary conditions removing this total derivative, without using field equations.

An internal circle group action with spacetime-independent parameter $\alpha$. For canonical complex scalar fields, Noether's theorem gives $j^{\mu }=i{\sum }_j{q}_j{\phi }_j^{†}\overleftrightarrow{{\partial }^{\mu }}{\phi }_j$. The Noether charge of the particle annihilated by ${\phi }_j$ is $q_j$, and its antiparticle has charge $-q_j$.

The interaction $-\frac{1}{2}(h{\varphi }^2\Phi +h^{\ast }{\varphi }^{†2}{\Phi }^{†})$ is invariant under a global phase symmetry precisely when the displayed charge relation holds. With $q_{\varphi }=1$, a heavy particle annihilated by $\Phi$ has charge $-2$, so its allowed decay channel is $\Phi \to \bar{\varphi }\bar{\varphi }$. The reverse heavy antiparticle decays into $\varphi \varphi$. Crossing only one heavy external state changes the exchange channel of mixed-species quantum scattering.

For a conserved charge $Q$ and invariant vacuum, its global symmetry commutes with the S-matrix. Matrix elements between charge eigenstates give the displayed selection rule. Constant electromagnetic phases cancel at every QED vertex, and nonzero physical amplitudes conserve total charge. Local gauge invariance strengthens this global rule to the Ward-Takahashi identity.