# Large and small gauge transformations?

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?

• pg 23 of this reference talks a little bit about this lepp.cornell.edu/~pt267/files/documents/A_instanton.pdf Aug 2, 2013 at 13:56
• Tanks for your reply. I have read a bit about large gauge transformations in regards to instantons and understand that for instantons they can change the winding number due to quantum tunneling. However, I don't know whether or not this is the reason that this distinctions is made for solitons as well. Especially, because for (classical) solitons the winding number is gauge invariant. If anybody could give me a conclusive answer that this is indeed the reason for the distinction between small and large gauge transformations for solitons, then I will accept that. Aug 2, 2013 at 14:34

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 $\theta$.

• @Hunter The identity component $\mathcal{G}_0$ that you gauge away is almost the whole gauge group, it is an infinite dimensional group. The group of large gauge transformation is only $\mathbb{Z}$, it is discrete. Thus I think that you may consider this as a fine tuning of the gauge principle. Aug 7, 2013 at 2:43
• @Hunter Transformations that don't change the state are called gauge symmetries or redundancies. These transformations tend to the identity at spatial infinity. "Large gauge transformations" are by definition not continuously connected with the identity transformation (they don't live in the same "island" of the disconnected group as the identity). Therefore they don't tend to the identity anywhere. Therefore they do change the state (and in fact they take the state from one Hilbert space to another one). Aug 22, 2013 at 18:27
• @Hunter Therefore a better name would be " large local (space-time dependent) transformations" instead of "gauge" (this should be reserved for transformations not changing the state). David, please, correct me if I am wrong. Aug 22, 2013 at 18:28
• @drake Thank for your answer. Do you by any chance have any references where I can find more information? I couldn't find anything in Peskin and Schroeder or Lewis Ryder. I've tried to Google it, but most sources refer to changing the homotopy of instantons (including Ryder) and I don't think that is what you are talking about. Aug 22, 2013 at 22:45
• The question is about the distinction between large and small gauge transformations in the sense that small ones go to the identity at infinity. This answer discusses the difference between gauge transformations that lie in the identity component (small) and those that do not (large). These two notions of small/large gauge transformations seem to have only their names in common. Please correct me if I am wrong but as it stands I do not see any relation between the two topics. Apr 30, 2016 at 16:21

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.