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Almost every discussion of non-perturbative effects in Yang-Mills theory mentions in passing that they work in the temporal gauge. Why is this the case?

A good example is the QCD vacuum. Almost every discussion uses the temporal gauge, although the discussion in a physical gauge, like the axial gauge, is much simpler and leads directly to a non-degenerate vacuum state.

So far, I wasn't able to make sense of the "reasons" I was able to find in the literature.

For example, Shifman in his book "Advanced topics in quantum field theory" writes:

To single out the relevant degree of freedom in the infinite-dimensional space of the gluon fields, it is necessary to proceed to the Hamiltonian formulation of Yang-Mills theory. This implies, of course, that the time component of the four-potential $A_\mu$ has to be gauged away, $A_0= 0$.

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Famously $A_0$ has vanishing momentum, giving a phase space constraint. If you BRST-quantise electromagnetism to address a problem with computing the means of gauge-invariant quantities in the path-integral formulation, you introduce the Nakanishi-Lautrup field denoted $B$. Its momentum provides another phase space constraint, $\pi_B=-A^0$. If we also choose the Landau gauge, $A^\mu$ and $\pi_B$ are both conserved. The former motivates this gauge choice; the latter motivates a gauge choice $\pi_B=0$ to simplify the Hamiltonian. Then $A^0=0$.

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