In the cases when the gauge group is disconnected, both choices of
defining the physical space as a the quotient of the field space by the
whole gauge group $\mathcal{A}_{physical} = \frac{\mathcal{A} }{ \mathcal{G}}$ or by
its connected to the identity component $\mathcal{A}_{physical} = \frac{\mathcal{A} }{ \mathcal{G}_0}$ are mathematically sound. In the second case,
the large gauge transformations are not included in the reduction, thus
they transform between physically distinct configurations., and in
quantum theory between physically distinct states.

However, as N.P. Landsman reasons, the first choice overlooks inequivalent quantizations that correspond to the same classical theory.
In the case of the magnetic monopoles these distinct quantizations
correspond to monopoles with fractional electric charge (Dyons). This phenomenon was discovered by Witten (the Witten effect). If the whole gauge
group including the large gauge transformations is quotiened by, no such states would be present in the quantum theory.

In the monopole theory, the inequivalent quantizations can be obtained
by adding a theta term to the Lagrangian (just as the case of instantons). Landsman offers an explanation of this term in the quantum Hamiltonian picture: Assuming $\pi_0(\mathcal{G})$ is Abelian, then when the gauge group is not connected, then a gauge invariant inner product can be defined as:

$\langle \psi| \phi \rangle_{phys} = \sum_{n \in \pi_0(\mathcal{G})} \int_{g\in \mathcal{G_0}} e^{i \pi \theta n} \langle \psi| U(g) |\phi \rangle$

Where the original states belong to the (big) gauge noninvariant Hilbert space. This inner product is $\mathcal{G}_0$ invariant for all values of $\theta$.

This post imported from StackExchange Physics at 2014-04-05 17:25 (UCT), posted by SE-user David Bar Moshe