= Large gauge transformation
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)> transformations
$$
g_n(x)=e^{2\pi inx/L},\qquad n\in\mathbb Z,
$$
have <winding number>
$$
\frac1{2\pi i}\int_0^Lg_n^{-1}\partial_xg_n\,dx=n.
$$
This integer cannot change during a <homotopy> through <periodic functions>. Hence $n\ne0$ 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. <Gauge Theory by David Tong>[Section 2.2.2, classification of based gauge maps by integer winding and theta-vacuum transformation, equation 2.41]{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.
Back to article page
