Large and small gauge transformations?

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 $\theta$.