Abelian gauge theory

An Abelian gauge theory has a commutative gauge group. Its gauge field strength has no commutator term, so the pure gauge Lagrangian has no cubic or quartic gauge-boson self-interactions.

For

D_{μ} = \partial_{μ} + i e A_{μ}

, the local U(1) transformation is

ψ \mapsto e^{i eχ} ψ, A_{μ} \mapsto A_{μ} - \partial_{μ} χ .

(1)

Large gauge transformation

 1  0

A large gauge transformation does not approach the identity at the relevant spatial or asymptotic boundary. Although it leaves the local field strength unchanged, its boundary charge can act nontrivially on physical states and constrain soft limits through a Ward identity.