Rotating away a constant gauge field

Question

In a few papers (see, for example, here, the bottom of the left column on the page 6, or here, the upper part of the page 5) I've met the strange calculations using the constant gauge field $$ A_{\mu}(x) = (0,0,0,A_{3} = \text{const}), \quad\text{or}\quad A_{\mu}(x) = (A_{0} = \text{const},0,0,0) $$ The authors of linked articles obtain observables expressed in terms of this gauge fields (such as chiral effects). One might think that these constant gauge fields can be gauged away, but the authors of the first linked article say that (at least about constant $A_{3}$)

One might think that a constant gauge field could be gauged away, but this is not possible by a gauge transformation satisfying the periodic boundary condition.

I don't understand this statement. Could You clarify it? Also I don't understand what is the problem with rotating away $A_{0} = \text{const}$.

Answer 1

The standard gauge transformation is $A'_\mu = A_\mu + \nabla_\mu \lambda$, where $\lambda$ is a function of $x^\mu$. If, in particular, $A^\mu$ is a constant vector field, then we could choose the gauge transformation $$ \lambda(x^\mu) = - A_\mu x^\mu $$ which implies that $\nabla_\mu \lambda = - A_\mu$ and so $A'_\mu = 0$.

However, this construction assumes that the space we're working in doesn't have any unusual topological properties. Suppose, for example, that we're looking at a space that is homeomorphic to $T^3$ (the three-torus.) This space can be obtained by taking a "cube" of $\mathbb{R}^3$ (say, $[0,L] \times [0,L] \times [0,L]$) and "gluing" pairs of opposite faces together. For any fields on this new space to be continuous, its properties on opposite faces have to "match up"; for example, we need to have $$ A^{\mu}(x,y,0) = A^{\mu}(x,y,L) $$ for all values of $x$ and $y$, and similar conditions holding for the faces points in the $x$ and $y$ directions. These conditions are often called "periodic boundary conditions".

The gauge transformation I wrote out above, however, does not satisfy these periodic conditions. In particular, if $A^\mu = (0,0,0,c)$, then $\lambda = -cz$, and $\lambda(x,y,0) = 0 \neq cL = \lambda(x,y,L)$. Thus, a field that could be gauged away in "normal" space cannot be gauged away in a topologically non-trivial space. A similar effect would be found with $A^0$ if one was working in a spacetime where time was "periodic" in this same way.

This fact, by the way, is the foundation of the field of de Rham Cohomology, which uses "gauge transformations" to explore the topological properties of various spaces.