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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?

Hunter
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  • pg 23 of this reference talks a little bit about this http://www.lepp.cornell.edu/~pt267/files/documents/A_instanton.pdf – DJBunk Aug 02 '13 at 13:56

2 Answers2

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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 $\theta$.

David Bar Moshe
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  • Thanks for your reply. I've read your message and try to read the paper you provided a link for, but I don't really understand it to be honest. I understand that taking the quotient $\mathcal{A}/\mathcal{G}$ is mathematically correct. However, I am still not sure why large gauge transformations relate physically distinct configurations. It seems that this is contradictory to the whole idea of gauge transformations, which are by definition redundant degrees of freedom. – Hunter Aug 05 '13 at 17:20
  • If large gauge transformations do relate physically distinct configurations, then they are clearly not redundant. I don't know how to rhyme these things. – Hunter Aug 05 '13 at 17:20
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    @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. – David Bar Moshe Aug 07 '13 at 02:43
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    @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). – Diego Mazón Aug 22 '13 at 18:27
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    @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. – Diego Mazón Aug 22 '13 at 18:28
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    @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. – Hunter Aug 22 '13 at 22:45
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    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. – Friedrich Apr 30 '16 at 16:21
  • @Friedrich There is a single notion of large gauge transformations. Their description as configurations which do not tend to unity at infinity is a special case applicable to one point compactification of a flat space. On a general space there is no special point as infinity and you have to resort to the general definition. – David Bar Moshe May 01 '16 at 02:22
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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.

Philip
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