A field theory assigns dynamical variables to spacetime points, such as a scalar field $\phi(x)$, a vector field $A_\mu(x)$, or spinor fields $\psi(x)$. Locality demands that the Lagrangian density depends on fields and a finite number of derivatives at each point, allowing causal propagation and a relativistically consistent formulation.

For a complex scalar with global $U(1)$ symmetry, the replacement $\partial_\mu \to D_\mu = \partial_\mu + i e A_\mu$ enforces local phase invariance. The gauge field $A_\mu$ transforms so that $D_\mu\phi$ transforms covariantly. This is the prototype of interactions in electrodynamics and non-Abelian gauge theories.

For a gauge group $G$ with generators $T^a$, the gauge field becomes $A_\mu = A^a_\mu T^a$, and the field strength is $F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu + i g [A_\mu,A_\nu]$. The commutator term is the origin of self-interactions in non-Abelian gauge theories.

Continuous symmetries produce conserved currents $j^\mu$ satisfying $\partial_\mu j^\mu = 0$ on-shell. In gauge theories, some symmetries are redundancies rather than physical transformations, so not every Noether current corresponds to an independent observable. Constraint analysis becomes essential, especially in the Hamiltonian formulation.