Local gauge transformations of the nonlocal densities

The gauge-transformed nonlocal densities read [25,36,24]

$$\rho '(\vec {r},\vec {r}') = e^{i\left[\phi(\vec {r})-\phi(\vec {r}')\right]} \rho (\vec {r},\vec {r}') ,$$ (34)

$$\vec {s}'(\vec {r},\vec {r}') = e^{i\left[\phi(\vec {r})-\phi(\vec {r}')\right]} \vec {s}(\vec {r},\vec {r}') .$$ (35)

Since the local gauge transformations form a U(1) group, invariance with respect to transformations that are of the first-order in gauge angles, $\left[1+iG\right]\rho(\vec {r},\vec {r}')$, where

$$G(\vec {r},\vec {r}')=\phi(\vec {r})-\phi(\vec {r}'),$$ (36)

is enough to ensure full gauge invariance. By Taylor expanding the exponential functions in eq. 34 and 35 after they are inserted in the functional one may, of course, also prove this fact explicitly.

One specific type of gauge transformation is the Galilean transformation, for which the gauge angles depend linearly on positions, i.e., $G(\vec {r},\vec {r}')=\vec {p}\cdot(\vec {r}-\vec {r}')/\hbar$, and which corresponds to a transformation to a reference frame that moves with velocity $\vec {p}/m$. For this transformation, only first-order derivatives of $G$ survive, which makes Galilean invariance less restrictive than the full gauge invariance.