Gauge fixing choice for the gauge field $A_0$

In many situations, I have seen that the the author makes a gauge choice $A_0=0$, e.g. Manton in his paper on the force between the 't Hooft Polyakov monopole.

Please can you provide me a mathematical justification of this? How can I always make a gauge transformation such that $A_0=0$?

Under a gauge transformation $A_i$ transforms as

$$A_i \to g A_i g^{-1} - \partial_i g g^{-1},$$

where $g$ is in the gauge group.

Answer

The "gauge fixing" condition $A_0=0$ called the temporal gauge or the Weyl-gauge please see the following Wikipedia page). This condition is only a partial gauge fixing condition because, the Yang-Mills Lagrangian remains gauge invariant under time independent gauge transformations:

$A_i \to g A_i g^{-1} - \partial_i g g^{-1}, i=1,2,3$

with $g$ time independent.

However, this is not the whole story: The time derivative of $A_0$ doesn't appear in the Yang-Mills Lagrangian.Thus it is not a dynamical variable. It is just Lagrange multiplier. It's equation of motion is just the Gauss law:

$\nabla.E=0$.

One cannot obtain this equation after setting $A_0 = 0$. So it must be added as a constraint and it must be required to vanish in canonical quantization on the physical states. (This is the reason that it is called the Gauss law constraint).