In gauge theory, physical states are often said to be characterized by equivalence classes of gauge field configurations that differ by gauge transformations. But according to Large and small gauge transformations? (among many other sources), the correct statement is that physical states are characterized by equivalence classes of gauge field configurations that differ by small gauge transformations, and that large gauge transformations (i.e. gauge transformations that are not homotopic to the identity under any homotopy that identically approaches the identity at spatial infinity) relate physically inequivalent states. A common example is Yang-Mills theory with inequivalent "topological vacua" that can be indexed by an integer $N$, where only large gauge transformations can change a state in one topological vacuum to a state in another, and instantons can tunnel between different topological sectors.

A concrete example would help illustrate why states in different topological sectors of the Hilbert space are physically inequivalent. If they are indeed inequivalent, then there should be a measurement that could distinguish them. What is a very concrete example of a physical experiment that measures which topological sector a system is in? In other words, I would like an example of (a) two gauge field configurations A and B that can only be connected by a large gauge transformation, (b) an experiment performed on gauge-invariant quantities of a particular physical system that measures which gauge field configuration describes that system, and (c) the experimental result corresponding to configuration A and the result corresponding to configuration B.

(Note that I'm not asking for an example of any experimentally observable effect of instantons or other non-perturbative phenomena. I'm asking for something more specific.)

  • $\begingroup$ I think there's an underlying premise of your question that may be misguided: It sounds as if you think that a topological sector is something a system "is in", but in fact e.g. the $\theta$-vacua of QCD are superpositions of all the instanton vacua (cf. e.g. physics.stackexchange.com/q/321867/50583, physics.stackexchange.com/q/306868/50583), that is, the "true" vacua do not correspond to the topological sectors, so it makes no sense to ask "what topological sector" a QCD system is (at least not to me right now). $\endgroup$ – ACuriousMind Nov 1 '17 at 11:24
  • $\begingroup$ Finally, when you ask about "gauge field configurations that describe the system", that is a classical turn of phrase (quantum systems don't have definite field configurations), and classically you are allowed to quotient out by the large gauge transformations, too. I think there is an interesting question here but it needs to be phrased more carefully. $\endgroup$ – ACuriousMind Nov 1 '17 at 11:24
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    $\begingroup$ @ACuriousMind Yes, I know that the vacuum is a superposition of all the topological vacua (with uniform magnitude and phase offsets determined by $\theta$). I was originally going to mention this issue in the OP but didn't want to clutter it up. By the basic postulates of quantum mechanics, measuring a state which is not an eigenstate of the measurement operator will change the system into the corresponding eigenstate, so after making the measurement the system will no longer be in the vacuum. If one starts in the vacuum, one will have an equal probability of measuring any topological ... $\endgroup$ – tparker Nov 1 '17 at 14:58
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    $\begingroup$ vacuum $n$, but I'm wondering about the phenomenology of the system after the measurement decoheres/collapses the superposition. (If you think it's unrealistic that a physical measurement could inject so much energy into a system, then you can just think of the question as asking about the different phenomenology between the two non-vacuum states $| n \rangle$ and $| m \rangle$, regardless of their applicability to the real world). $\endgroup$ – tparker Nov 1 '17 at 15:02
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    $\begingroup$ @Diracology Yes, I'm familiar with large gauge transformations - (a), (b), and (c) in the OP are not three separate questions, but three non-separable parts of a single question :-) $\endgroup$ – tparker Nov 1 '17 at 15:07

There are many examples of gauge theories with disconnected gauge groups, but as far as I know, the non-trivial topological sectors of these theories are beyond the current experimental capabilities.

In order for large gauge transformations to act nontrivially on the states, it must be a symmetry of the Hamiltonian. For example, the Maxwell Hamiltonian depends only on the electric and magnetic fields. In flat space, the electromagnetic $U(1)$ gauge group is connected, thus no large gauge transformations exists. However, if appropriate boundary conditions are provided, for example periodic boundary conditions in one direction, then large gauge transformations exist and are symmetries of the Hamiltonian.

In contrast, the Hamiltonian a charged particle moving under the influence of a background electromagnetic field is only quasi-invariant, only when the gauge transformation is performed also on the wave functions, the spectrum is conserved. In this case, large gauge transformations must be considered as gauge redundancies rather than symmetry. (However, a gas of such particles has an invariant second quantized Hamiltonian, but I don't know how to exploit this fact).

This is the reason why the Aharonov-Bohm system of a particle moving on a circle around a magnetic flux, does not possess large gauge symmetries in spite of the fact that the electromagnetic gauge group is disconnected because $\pi_1(U(1)) = \mathbb{Z}$.

What I am going to describe to you here is a (quite ingenious) experiment suggested by S.-R. Eric Yang. He proposed a modification of the Aharonov-Bohm setting to introduce degenerate energy eigenfunctions related by a large gauge transformation. This experiment seems feasible, but from a google scholar search, there does not seem that this experiment has yet been actually performed.

The trick is to use a spin half particle and add an electric field in the radial direction in order to generate a spin orbit interaction (proportional to: $\vec{\sigma} \cdot \left (\vec{p} – i e \vec{A} \right )$). The spin orbit term breaks the time reversal invariance, but Yang noticed is that when the Aharonov-Bohm potential is equal to a half flux quantum $A_{\phi} = \frac{1}{2}$, then both kinetic and the gauge orbit interaction terms become invariant under the large gauge transformation $e^{i\phi}$ (which shifts the gauge potential by one quantum) followed by a time reversal transformation. Thus, this transformation is a symmetry of the Hamiltonian. As a consequence there are two degenerate states of opposite spin which are related by the above transformation. These two states should be able to be distinguished between by means of their Berry phases.


I am using the term gauge group for the group of all gauge transformations (including small and large gauge transformations). In our case it is not the one dimensional group $U(1)$ of global gauge transformations, but the infinite group $\mathrm{Map}(S^1, U(1))$ of local gauge transformations. This group is disconnected. Its disconnected part modulo the connected component (which form the group of large gauge transformations) is $\pi_1(S^1)= \mathbb{Z}$ realized by means of transformations of the type $e^{i n \phi}$ for integer $n$. This is essentially the same gauge group addressed by Landsman and Wren in my answer of the attached question in the main text (in their case it is $\mathrm{Map}(S^1, U(n))$ , but since for $G$ semisimple and centerless $\pi_1(G)= 0$ , thus only the $U(1)$ part of $U(n)$ contributes to the large gauge transformations.

As I emphasized in my answer to Friedrich's comment in the attached question; the group of disconnected elements from the unit component (modulo connected component) is the basic definition of large gauge transformation. It is true that when you describe spheres as one point compactifications of flat spaces, then the large gauge transformations become those which do not approach the unit at infinity. You can do this exercise for our case by expressing $S^1$ as one point compactification of $\mathbb{R}$.

The Hamiltonian of the Aharonov-Bohm system is not invariant under large gauge transformation because the theory is only quasi-invariant and not fully invariant. This can be easily checked by direct substitution.

A symmetry of the theory is a transformation commuting with the Hamiltonian without acting on the wave functions. I only emphasized that to tell that the A-B system is not invariant under large gauge transformations.

Your last comment correctly summarises the distinction between symmetry and redundancy.

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  • $\begingroup$ I think you might be using the term "gauge group" in a slightly nonstandard way. Do you mean (a) the fiber symmetry group (the standard usage), (b) the group of all gauge transformations, or (c) the group of all gauge transformations that approach the identity at spatial infinity? You seem to be using sense (c). Under sense (a), the electromagnetic gauge group $U(1)$ is always connected and has nothing to do with the geometry or topology of spacetime. $\endgroup$ – tparker Nov 2 '17 at 18:06
  • $\begingroup$ I don't quite understand your sentence "In order for large gauge transformations to act nontrivially on the states, it must be a symmetry of the Hamiltonian." Aren't all gauge transformations symmetries of the Hamiltonian by definition? $\endgroup$ – tparker Nov 2 '17 at 18:09
  • $\begingroup$ Finally, could you be more precise in your distinction between "redundancy" and "symmetry"? Different people use those terms in different ways. Are you defining a "redundancy" to be a formal mathematically operation that leaves all physical quantities invariant, while a "symmetry" is an operation that commutes with the Hamiltonian but alters (at least some) states in physically observable ways? $\endgroup$ – tparker Nov 2 '17 at 18:13
  • $\begingroup$ I have prepared an update for your follow up questions $\endgroup$ – David Bar Moshe Nov 3 '17 at 3:52

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