U(1) gauge symmetry

For $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$, the local U(1) transformation is

A charged scalar field coupled to an Abelian gauge field through $D_{\mu }={\partial }_{\mu }-ie{a}_{\mu }$. Under $\varphi \mapsto e^{i\alpha }\varphi$ and $a_{\mu }\mapsto a_{\mu }+e^{-1}{\partial }_{\mu }\alpha$, its gauge covariant derivative transforms with the same phase. The potential determines whether the perturbative spectrum has a massless photon or exhibits the Higgs mechanism.

For unpolarized ultrarelativistic charged spin-half particles annihilating through a photon into minimally coupled unit-charge scalars of mass $m$, set $\beta =\sqrt{1-4m^2/s}$. The scalar vertex contains the difference $k_{-}-k_{+}$ of outgoing four-momenta. Contraction with the spin-averaged initial fermion current gives $\overline{|\mathcal{M}{|}^2}=e^4{\beta }^2{\sin }^2\theta /2$. The two-body final-state flux factor contributes another $\beta$, givingThe ${\sin }^2\theta$ angular dependence and threshold suppression contrast with electron-positron annihilation into a muon pair. These predictions assume the specified scalar electrodynamic coupling, rather than an arbitrary spin-zero interaction.

For $D_{\mu }={\partial }_{\mu }+iq{A}_{\mu }$, the constant phase transformation $\delta \varphi =-iqϵ\varphi$ has the displayed Noether current. Varying the gauge field gives ${\partial }_{\mu }{F}^{\mu \nu }=J^{\nu }$. In this convention $J^{\mu }=iq({\varphi }^{\ast }{\partial }^{\mu }\varphi -\varphi {\partial }^{\mu }{\varphi }^{\ast })-2q^2{A}^{\mu }|\varphi {|}^2$. The matter equations imply ${\partial }_{\mu }{J}^{\mu }=0$; the integrated electric charge additionally requires zero boundary flux. Local phase invariance gives a Noether second theorem identity, rather than independent physical charges for arbitrary functions.^[ref]

With scalar charges $e$ and $2e$, both $\psi$ and ${\varphi }^2$ transform with phase $e^{2i\alpha }$. Therefore $M^2|\psi -b{\varphi }^2{|}^2$ is a gauge-invariant potential term. It contains a nonzero direct cubic interaction $-M^2(b{\psi }^{\ast }{\varphi }^2+h.c.)$. Adding positive quadratic and quartic modulus terms gives a real bounded-below potential. All expanded interactions have field degree at most four.

The phrase large gauge transformation has two related uses, so the allowed gauge transformations and boundary conditions must be specified.

In the topological sense, a large gauge transformation lies outside the identity component of the allowed transformation group. For example, on a spatial circle of circumference $L$, the U(1) transformationshave winding numberThis integer cannot change during a homotopy through periodic functions. Hence $n\ne 0$ cannot be deformed to the identity, although $g_n(0)=g_n(L)=1$. Similarly, based SU(2) transformations on compactified three-space can have nonzero winding even while tending to the identity at infinity. ^[ref]

In discussions of asymptotic electromagnetism, transformations with nonvanishing limiting gauge parameters at infinity are also called large. After choosing the boundary phase space, their generators can carry nonzero boundary charges and yield a Ward identity relating scattering amplitudes. This usage concerns asymptotic action on physical data; a boundary value alone does not characterize the disconnected components of a general gauge-map group.

For an Abelian gauge transformation, the local electromagnetic field tensor is unchanged. Global winding or an asymptotic charge can nevertheless distinguish its action from a gauge transformation regarded as a pure redundancy.