The notion "inequivalent quantizations" is regularly used when topological terms are discussed.

From what I've gathered so far, "inequivalent quantizations" means that there are different quantum systems corresponding to the same classical system. For example, for a particle on ring, there are infinitely many corresponding quantum systems. These quantum systems have different magnetic fluxes through the ring. Depending on the strength of the magnetic flux an Aharonov-Bohm-type phase is induced. But such a phase (and therefore the magnetic flux) has no effect in a classical context and therefore, there are infinitely many quantum systems (each defined by a different magnetic flux) that correspond to the same classical systems.

However, on the other hand, I've learned that the corresponding gauge potential should be understood as "fictitious". While I understand how there can be many different quantum systems, characterized by different magnetic fluxes, leading to the same classical system, I'm confused how this can be the case if the corresponding gauge potential is only fictitious.

The problem becomes especially important in the context of the strong CP problem. Here, we can again argue that different values of $\theta$ characterize different "quantization", i.e. different quantum systems.

But what's the origin of these inequivalent quantizations here? Is there some "external gauge potential (analogous to the magnetic potential in the example above) we can use to understand why gluon fields pick up a phase $\theta$ as they undergo vacuum to vacuum transitions?

Or is there some way to understand the origin and meaning of "inequivalent quantizations" without referring to an "fictitious external gauge field"?

  • $\begingroup$ So the topological term doesn't impact the classical equations of motion, but does impact the quantum theory. Hence you have different quantum theories for different values of $\theta$, but their classical limits are all equivalent. In the language of geometric quantization, the (classical) symplectic form doesn't depend on $\theta$, but the Kahler (sorry no German keyboard layout) structure does, therefore the quantum theory acquires the dependence on $\theta$. Is this what you were looking for? $\endgroup$ Commented Jun 12, 2019 at 17:19
  • $\begingroup$ That's a bit more abstract than what I was hoping for but thanks! $\endgroup$
    – jak
    Commented Jun 17, 2019 at 11:55

1 Answer 1


As well known, the representations of the operator algebra of a quantum system fall into superselection sectors, which consist of sub-Hilbert spaces of a quantum state space, on which the physical observables act irreducibly. Each superselection sector may be identified with an elementary system.

There are many types of superselection sectors which are not related to the topology of the configuration space; for example, the mass superselection sectors in Galilean invariant theories, the charge superselection sectors in QED, the superselection sectors of spontaneously broken theories etc.

Inequivalent quantizations, form a special class of superselection sectors, where the quantum theory depends on additional parameters which are either absent or ineffectual in the classical theory, such as the Aharonov-Bohm fluxes.

Precisely; these sectors are in a one to one correspondence with the representations of the fundamental group $\pi_1(Q)$ of the configuration group $Q$. For this reason, they are usually referred to as the topological sectors in contrast to the other superselection sectors which are nontopological. (This is the answer to your second question).

To answer the first question (about the "fictitious" gauge fields), and sketch the proof of the above assertion, let me refer you to this question by AccidentalFourierTransform (the question is about the momentum operator as defined in Bryce DeWitt's book (chapter 11)).

DeWitt describes this type of superselection sectors as follows: When, we quantize, we need to replace classical observables not commuting with the configuration space coordinates by differential operators.

In particular, in order not to change the canonical commutation relations $ [q_i, p_j] = i \delta_{ij}$ the most general form the momentum operator can take is

$$p_i = -i \frac{\partial}{\partial q_i}-\omega_i(q)$$

(In field theory, both $q$ and $p$ are fields and the indices $i$, $j$ can ne continuous, i.e., parametrize some base space).

In order for the momentum-momentum commutation relations $ [p_i, p_j] = 0$not to change, the vector functions $\omega_i(q)$ need to satisfy the relation:

$$\frac{\partial}{\partial q_i}\omega_j(q) - \frac{\partial}{\partial q_j}\omega_i(q) = 0$$

Thus, the vector functions $\omega_i(q)$ must be vector potentials with vanishing field strength, which are called flat connections. Bundles equipped with flat connections are called flat bundles, they are classified by the character representations of the fundamental groups of the configuration space, please see, Kobayashi: Differential geometry of complex vector bundles section 1.2.

Here is a shorthand sketch of the proof. The flat vector potentials (modulo gauge transformations can be characterized by the Wilson loops: $$e^{i \oint_{\Gamma} \omega}$$ As $\Gamma$ runs over all possible loops in $Q$. However, since, the field strengths are vanishing, the result doesn't depend on the particular loop but on its homotopy class. Since the Wilson loops satisfy the group properties; we obtain that the flat connections are in a one-to-one correspondence with the character, i.e., $U(1)$ representations of the fundamental group $\pi_1(Q)$. $$\mathrm{Map}(\pi_1(Q), U(1))$$ Now, the gauge potentials $\omega_i(q)$ are not external gauge fields; they are just functions on the configuration space; however, if they are shifted by a local $U(1)$ transformations, no physical prediction of the theory is altered. This is the reason why some call them fictitious.

The $\theta$ term in Yang-Mills theory (I'll just consider the Abelian case) is an example of the above theory:

After quantization, the electric field operator has the form:

$$E_i(x) = -i \frac{\delta}{\delta A_i(x)} - \frac{\theta}{8 \pi^2} \epsilon_{ijk}F_{jk}$$ Which can be written as: $$E_i(x) = -i \frac{\delta}{\delta A_i(x)} - \Omega_i(\mathbf{A}(x))$$ with: $$\Omega_i(\mathbf{A}(x)) = \frac{\theta}{8 \pi^2} \epsilon_{ijk}F_{jk}$$ We can check that $\Omega_i$ is a flat functional connection: $$\frac{\delta\Omega_i }{\delta A_j(x)} - \frac{\delta\Omega_j }{\delta A_i(x)} =0$$ Thus, the theta vacua are in a one to one correspondence to $$\mathrm{Map}(\pi_1(\mathcal{A}/\mathcal{G}), U(1))$$ Where $\mathcal{A}/\mathcal{G}$ is the space of gauge potentials-modulo gauge transformations. Using the contractibility of the total space of gauge potentials and the long homotopy sequence, we have: $\pi_k(\mathcal{A}/\mathcal{G}) = \pi_{k-1}(\mathcal{G})$ Thus, the classification is according to: $$\mathrm{Map}(\pi_0(\mathcal{G}), U(1))$$ When the base space is $S^3$ $$\pi_0(\mathcal{G}) = \pi_0(\mathrm{Map}( S^3, U(1)) = \pi_3(U(1)) = 0$$ Which proves that the theta sectors in an Abelian theory are trivial.

Remark: The above description can be generalized to the non-Abelian case, in this case the classification of the flat bundles (thus, the nonequivalent representations) is according to the non-Abelian representations of the fundamental group. In this case, we get nontrivial inequivalent quantizations. This point was elaborated in my answer to the following question.


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