The gauge symmetry in classical pure Yang-Mills theory with a gauge field $A_{\mu}$ requires an action $S$ to be invariant under continuous transformations
$$
A_{\mu}(g) \to g(A_{\mu} + i\partial_{\mu})g^{-1}
$$
When we talk about quantized theory, we're dealing with the Hilbert space of rays $|\Psi (A_{\mu})\rangle$, which must be invariant under unitary transformation $U(g)$:
$$
\tag 0 |\Psi(A_{\mu})\rangle \to U(g)|\Psi (A_{\mu})\rangle = |\Psi(A_{\mu})\rangle
$$
Equivalently, for infinitesimal transformation with generator $G(x)$ one has to require
$$
G(x)|\Psi (A_{\mu})\rangle = 0
$$
This reduces the Hilbert space by projecting it on a space with only physical gauge field polarizations. That's why the gauge symmetry is called a do-nothing transformation.
Next, suppose the "large" gauge transformation whose element $g_{(n)}$ carries a non-zero winding number $n$. We have that for the vacuum state on zero winding number configuration $|0\rangle$
$$
\tag 1 U(g_{(n)})|0\rangle = |n\rangle
$$
One can introduce a $\theta$-vacuum defined as
$$
|\theta\rangle =\sum_{n}e^{in\theta}|n\rangle,
$$
so
$$
\tag 2 U(g_{(n)})|\theta\rangle = e^{in\theta}|\theta\rangle
$$
So we have that "large" gauge transformations are not do-nothing transformations; moreover, even after introducing the new vacuum it still acts non-trivially!
My questions are:
-
$(0)$ corresponds to invariance of classical gauge theory action under local gauge transformations. To which corresponds $(1)$? Naively I think that large gauge transformations change the gauge field strength tensor, but I would like to formalize this.
-
Finally, whether a classical analog of $(2)$ exists? Is this correspondence completely determined by classical gauge fields topology?
Best Answer
That large gauge transformations are not true gauge transformations (i.e. yield physically distinct states) is a purely quantum phenomenon due to a choice of quantization procedure that is present in the cases where there are large gauge transformations. Classically, large gauge transformations are always gauge transformations, i.e. trivial on the physical state space. See also this answer by David Bar Moshe.
Essentially, the special status of large gauge transformations arises from the fact that the quantization procedure for a gauge theory only imposes that applying the generators of gauge transformations to physical states must yield zero, and hence the physical states are invariant under gauge transformations generated by them. But, rather by definition, the transformations generated by the generators only yield the gauge transformations connected to the identity (the exponential map of a Lie algebra maps to the connected components of the corresponding group). Therefore, the quantization procedure by design only imposes invariance of the quantum theory under small gauge transformations.
There is no good reason to demand that the quantum theory be invariant under large gauge transformations because it is well-known that the same classical system can have different inequivalent quantizations, and the large gauge transformations simply become the transformations between these inequivalent quantizations, which seems physically reasonable - given a classical theory, its full quantum theory should be the "sum" of all possible quantizations.