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The Coulomb gauge imposeson the magnetic vector potential. A gauge transformation $A\mapsto A+\nabla \chi$ reaches this gauge when $\chi$ solves

A first-class constraint has Poisson brackets with all constraints that vanish on the constraint surface. For a regular constrained Hamiltonian system, first-class constraints generate gauge transformations; each independent constraint and its gauge direction remove one canonical pair.

For electric charge $q$ and electromagnetic four-potential $A_a$, the derivative ${\mathcal{D}}_a={\nabla }_a-iq{A}_a$ transforms covariantly under an electromagnetic gauge transformation.

For the Maxwell field, the variation $\delta A_{\nu }=\alpha F_{0\nu }=\alpha ({\partial }_0{A}_{\nu }-{\partial }_{\nu }{A}_0)$ is time translation combined with a gauge transformation. With constant $\alpha$, $\delta F_{\mu \nu }=\alpha {\partial }_0{F}_{\mu \nu }$, and the Lagrangian changes by $\alpha {\partial }_0\mathcal{L}$. Its Noether current is directly the time component of the gauge-invariant Maxwell stress-energy tensor, providing the physical energy density without a potential-dependent improvement step.

A classical field theory is a gauge theory when its description has a local redundancy represented by gauge transformations. A gauge field defines the covariant derivative that compares matter fields at different points, and its gauge field strength measures the curvature of that comparison. Electromagnetism is the familiar Abelian example; Yang-Mills theory generalizes it to a non-Abelian internal group. Both classical equations and their quantum field theory can have this structure. Physically equivalent gauge descriptions must give the same observables, even though the gauge potential changes.

Multiplication by $e^{ia\psi (x)}$ is a gauge transform that preserves pointwise modulus and mass while shifting the gradient by $ia(\nabla \psi )u$.

A time-independent gauge transformation $\mathbf{A}\mapsto \mathbf{A}+\nabla \chi$ in an Eisenhart-Duval lift is the coordinate change $u\mapsto u+2\chi$. It preserves the metric one-form $du-2\mathbf{A}\cdot d\mathbf{x}-Udt$ and adds only $-2d\chi /dt$ to the reduced Lagrangian.

An isorotation is a global transformation generated by isospin. In the pion Skyrme model it acts as $U\mapsto AU{A}^{†}$, with constant $A\in SU(2)$, rotating the three pion components while preserving the scalar component. The matrices $A$ and $-A$ induce the same classical rotation, so this action factors through $SO(3)$; quantum states may carry half-integer isospin on its $SU(2)$ cover. This global internal rotation is distinct from a spatial rotation or a local gauge transformation.

The kinetic momentum $\mathbf{\Pi }=\mathbf{p}-q\mathbf{A}$ equals $m\mathbf{v}$ for a nonrelativistic charged particle and is invariant under an electromagnetic gauge transformation.

The source-free electromagnetic Lagrangian density is quadratic in the electromagnetic field tensor. Its Euler-Lagrange field equation is ${\partial }_{\mu }{F}^{\mu \nu }=0$. Antisymmetry of $F$ makes the density invariant under gauge transformations, and contraction of its indices makes it a Lorentz scalar.