I would like to understand what is the importance of large gauge transformations. I read that these gauge transformation cannot be deformed to the identity, but why should we care about that?
[Physics] Large gauge transformations
gauge-invariancegauge-theorytopology
Related Solutions
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 $\theta$.
That large gauge transformations are not true gauge transformations (i.e. yield physically distinct states) is a purely quantum phenomenon due to a choice of quantization procedure that is present in the cases where there are large gauge transformations. Classically, large gauge transformations are always gauge transformations, i.e. trivial on the physical state space. See also this answer by David Bar Moshe.
Essentially, the special status of large gauge transformations arises from the fact that the quantization procedure for a gauge theory only imposes that applying the generators of gauge transformations to physical states must yield zero, and hence the physical states are invariant under gauge transformations generated by them. But, rather by definition, the transformations generated by the generators only yield the gauge transformations connected to the identity (the exponential map of a Lie algebra maps to the connected components of the corresponding group). Therefore, the quantization procedure by design only imposes invariance of the quantum theory under small gauge transformations.
There is no good reason to demand that the quantum theory be invariant under large gauge transformations because it is well-known that the same classical system can have different inequivalent quantizations, and the large gauge transformations simply become the transformations between these inequivalent quantizations, which seems physically reasonable - given a classical theory, its full quantum theory should be the "sum" of all possible quantizations.
Best Answer
They require a special discussion because they are different. The (defining) fact that they can't be deformed to the identity means that it is not enough to verify the invariance under infinitesimal gauge transformations: the problem is that the large gauge transformations cannot be obtained by combining many infinitesimal gauge transformations!
The modular invariance of the torus in one-loop diagrams of string theory is the canonical example.
There are two angular coordinates on the torus. Imagine that we take both of them, $x,y$, to be in the interval $(0,1)$, with both end points identified with one another. Then a theory may have gauge transformations, namely diffeomorphisms of GR. The "small ones" are those like $$ (x',y') = (x+ \sin(2\pi x)/10, y+\cos(2\pi y)/5 ),$$ something that doesn't "essentially" change the way how the torus is parameterized. Those can be obtained from infinitesimal ones and the infinitesimal ones' behavior is encoded in the behavior of the currents – in this case, the world sheet stress-energy tensor.
However, the transformation $$(x',y') = (y,-x) $$ where I chose the minus sign just to preserve the orientation, for the case that the theory on the torus isn't left-right-symmetric, is qualitatively different. You can't produce it from the infinitesimal diffeomorphisms. This is just one example of a more general class of transformations $$ (x',y') = (ax+by,cx+dy)$$ where $ad-bc=1$ and $a,b,c,d$ are integers so that the periodic identification of $(x',y')$ with the unit periodicity is the same condition as the periodic identification of $(x,y)$. They form the group $SL(2,Z)$, the so-called modular group, and it's important because one has to check the invariance of a string theory under this modular group separately. This is actually a nontrivial condition that constrains the number of degrees of freedom, imposes level-matching conditions, forces string theories on orbifold to add projections simultaneously with the twisted sectors, determines the critical dimension of string theory, enforces the duty of lattices to be even self-dual in many cases, and so on. These constraints would be missed if we ignored the large gauge transformations. We would be thinking that some theories that are actually inconsistent are consistent.
On the other hand, the modular invariance for the torus already guarantees the invariance under large gauge transformations for higher genus surfaces. I wouldn't be able to reproduce the proof but I intuitively know why it is correct and I am confident that a rigorous proof exists.
Similar issues may appear for more ordinary gauge transformations, those of the Yang-Mills type.