Large and small gauge transformations?

Question

I've a questions about the difference between small and large gauge transformations (a small gauge transformation tends to the identity at spatial infinity, whereas the large transformations don't). Many sources state (without any explanation or reference) that configurations related by small gauge transformations are physically equivalent, whereas large gauge transformations relate physically distinct configurations. This seems odd to me (and some lecturers at my university even say that this is wrong), because all gauge transformations relate physically equivalent configurations.

Some of the literature that mentions the difference between small and large gauge transformations:

In Figueroa-O'Farrill's notes it is mentioned in section 3.1 (page 81-82) in http://www.maths.ed.ac.uk/~jmf/Teaching/EDC.html

In Harvey's notes, see equation (2.13) in http://arxiv.org/abs/hep-th/9603086

In Di Vecchia's notes, see the discussion above (and below) equation (5.7) http://arxiv.org/abs/hep-th/9803026

They all say that large gauge transformation relate physically distinct configurations, but they don't explain why this is true. Does anybody know why this is true?

Answer 1

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 $\theta$.

Answer 2

A answer by an example. As far as a particle in motion is in a different state that a particle at rest, a black hole in motion is in a different state that a black hole at rest. The transfaormation that maps the state of a BH at rest to the state of a BH in motion is a large gauge transformation. Hoping that this will make things clearer.