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The QCD theta-parameter in canonical quantization

by Wen-Yuan Ai, Björn Garbrecht, Carlos Tamarit

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Authors (as registered SciPost users): Björn Garbrecht
Submission information
Preprint Link: scipost_202409_00016v1  (pdf)
Date submitted: Sept. 14, 2024, 5:29 p.m.
Submitted by: Björn Garbrecht
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
Specialties:
  • High-Energy Physics - Theory
Approach: Theoretical

Abstract

The role of the QCD theta-parameter is investigated in pure Yang--Mills theory in the spacetime given by the four-dimensional Euclidean torus. While in this setting the introduction of possibly unphysical boundary conditions is avoided, it must be specified how the sum over the topological sectors is to be carried out. To connect with observables in real time, we perceive the partition function as the trace over the canonical density matrix. The system then corresponds to one of a finite temperature on a spatial three-torus. Carrying out the trace operation requires canonical quantization and gauge fixing. Fixing the gauge and demanding that the Hermiticity of the Hamiltonian is maintained leads to a restriction of the Hilbert space of physical wave functionals that generalizes the constraints derived from imposing Gauss' law. Consequently, we find that the states in the Hilbert space are properly normalizable under an inner product that integrates over each physical configuration represented by the gauge potential one time and one time only. The observables derived from the constrained Hilbert space do not violate charge-parity symmetry. We note that an exact {hidden} symmetry of the {theory} that is present for arbitrary values of $\theta$ in the Hamiltonian is effectively promoted to parity conservation in this constrained space. These results, derived on a torus in order to avoid the introduction of boundary conditions, also carry over to Minkowski spacetime when taking account of all possible gauge transformations.

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Publication decision taken: reject

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(status: Editorial decision fixed and (if required) accepted by authors)

Reports on this Submission

Report #2 by Matthew Reece (Referee 2) on 2025-7-18 (Contributed Report)

Report

This paper claims that the theta-parameter has no physical effect in QCD. This is a surprising claim that contradicts a large body of literature over decades. For example, there is a standard derivation of the neutron EDM induced by the theta term, which is textbook material (e.g., it appears in section 94 of the textbook by Srednicki). This relies on the chiral anomaly together with chiral perturbation theory, so something rather dramatic about standard effective field theory would have to break down for it to be wrong.

It has been difficult to find referees willing to read this paper carefully and comment on precisely where the arguments go wrong, and I will not do that here. Instead, I thought it might be worthwhile to give the simplest argument I can that the theta parameter has physical effects.

First, it is well-known that the theory of a quantum particle on a circle (with coordinate $\varphi$ with period $2\pi$) has a theta term. In the Lagrangian formulation this is a total derivative term $\frac{1}{2\pi} \theta {\dot \varphi}$. In the Hamiltonian formulation, this results in a shifted kinetic term $\propto (p_\varphi - \frac{\theta}{2\pi})^2$. The energy eigenvalues, then, are proportional to $(n - \frac{\theta}{2\pi})^2$ for integer $n$, and so the spectrum depends on the physical parameter $\theta$. This example is very simple, textbook quantum mechanics of a free particle, so there should be no doubt that in this context a $\theta$ term is physical despite being a total derivative.

Now, given Yang-Mills theory in 4d with a theta term, we can dimensionally reduce on a small 2-dimensional manifold (for concreteness, say, a torus), in a state where we turn on flux through the 2-manifold. To make this as convincing as possible, we can consider SO(3) gauge theory, for which the flux through a 2-manifold is a $\mathbb{Z}_2$-valued topological invariant , so it is clear that we can make it nonzero and it cannot decay to a zero flux. By the standard logic of Kaluza-Klein reduction, this leads to a $(1+1)$-dimensional theory at low energies. The flux Higgses the gauge group SO(3) to SO(2), so the low-energy gauge theory is in fact a $(1+1)$-dimensional U(1) gauge theory. This theory inherits a $\theta\, F$ term from the $\theta \,\mathrm{tr}(F \wedge F)$ term in 4d. But it is well-known that $(1+1)$d QED, placed on a spatial circle, is just the same as the quantum mechanics of a particle on a circle. (The 0+1d degree of freedom is just the Wilson line from 1+1d.) So the $\theta$ term in this context has the same physical effect on the low-energy spectrum that we described above.

This shows that $\theta$ has a physical effect on the spectrum of the Hamiltonian theory of 4d Yang-Mills placed on a 3-torus, at least for a subset of physical states where there is a nontrivial flux through a 2-torus.

Recommendation

Reject

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Author:  Björn Garbrecht  on 2025-08-08  [id 5709]

(in reply to Report 2 by Matthew Reece on 2025-07-18)

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Referee

It has been difficult to find referees willing to read this paper carefully and comment on precisely where the arguments go wrong, and I will not do that here. Instead, I thought it might be worthwhile to give the simplest argument I can that the theta parameter has physical effects.

Reply

Let us start summarizing a few points before addressing the specific objections of the second report:

  • The main objection to the standard picture brought forward in our manuscript is that, while in the usual treatment one restricts the large gauge transformations so that they decompose into disconnected equivalence classes, these constraints are never used in the derivation of the canonical commutation relations, so that it is inconsistent to assume them. We think that this is a question that should be addressed on the basis of its own relevance, whether it is in the present correspondence or elsewhere.

  • We further find a resolution to this objection, which leaves a constraint on the Hilbert space so that the Dirac--von Neumann axiom on normalizable physical states is observed. Clarifying the status of normalization in non-Abelian gauge theory, including the question of why it would be necessary to sample in the inner product over gauge-redundant configurations, is something that, again, should not be left unresolved.

  • Concerning Report #1, we have already addressed its main points in our reply (correctness to add a total derivative to the Lagrangian, existence of the Polyakov loop in temporal gauge, existence of integer topological sectors). We find that the points brought up in Report #1 are technically incorrect. Report #2 adds indirect arguments (first, stating that strong CP violation can be established at the level of EFT alone, which is already addressed in our earlier publications; second, a dimensionally reduced model that leaves ambiguities in the quantization that we resolve in the underlying Yang--Mills theory). We shall technically address these matters below. This collection of arguments can, in our view, not be used to make the far-reaching conclusion of CP-violation in the strong interactions. More clarity could be achieved by directly addressing the main points mentioned above: Is it valid to ignore the extra gauge constraint assumed in the literature when deriving the functional Schrödinger equation? Is it acceptable to work with non-normalizable states in the gauge theory?

We think that our main points are made clear in the manuscript (though some ten months and many discussions after submission, we find some things can be expressed more concisely, which we could do in a resubmission). We have once again emphasized and summarized these in the replies to both reports. Our arguments have thus been put on the table and are there to be addressed.

Referee

This paper claims that the theta-parameter has no physical effect in QCD. This is a surprising claim that contradicts a large body of literature over decades. For example, there is a standard derivation of the neutron EDM induced by the theta term, which is textbook material (e.g., it appears in section 94 of the textbook by Srednicki). This relies on the chiral anomaly together with chiral perturbation theory, so something rather dramatic about standard effective field theory would have to break down for it to be wrong.

Reply

The existence of strong CP violation cannot be established only by an EFT analysis of the nonlinear sigma model in chiral perturbation theory. This is because in the Lagrangian of chiral perturbation theory, the CP-odd phase in the term involving the determinant of the Goldstone boson matrix, which plays a crucial role in determining the neutron dipole moment, cannot be fixed by symmetry considerations alone. While the standard assumption is that the phase is given by \(\theta\), a spurion analysis shows that an alternative phase with the same spurion transformation properties is given by the negative phase of the determinant of the quark mass matrix. The former choice leads to CP violation, while the latter does not, and the correct phase has to be fixed by matching with the UV theory. It is important to understand that this is just standard spurion analysis, rather than "something rather dramatic ... to break down".

Let us give more details. In Eqs. (94.9,10) by Srednicki, all parity violation in the parameters is moved into the quark mass matrix, which can be achieved by \({\rm }U(1)_A\) transformations. However, there is another crucial assumption, which is that \(U\) and also \(U_0\) have unit determinant. Spurion analysis, however, allows \(U\) to be proportional to either minus the phase from the angle in front of the topological term (which is apparently the choice of Srednicki after transforming this angle to be zero and moving it to the quark masses) or proportional to the phase of the determinant of the quark mass matrix. This is because both transform complementarily to \(U\) under \({\rm }U(1)_A\) and therefore allow for the construction of operators that are invariant when including spurion symmetries. In other words, as mentioned before, spurion analysis on its own does not restrict the phase of \(\det U\) in the chiral Lagrangian to be aligned with \(\theta\). An equally valid choice compatible with all restrictions imposed by global anomalies is that the corresponding phase is proportional to the sum of phases from the quark masses. When choosing the latter option, the chiral perturbation theory analysis goes through as usual but with one crucial difference: The coefficient of the pseudoscalar-scalar pion-nucleon coupling becomes zero instead of being proportional to \(\theta\) and leads to a vanishing neutron dipole moment.

As spurious symmetries on their own thus do not fix the phase of \(\det U\) term in the chiral Lagrangian, the answer about the correct choice is dictated by matching with computations in the UV theory. Here, our path integral and canonical quantization calculations lead to correlators in which all phases are determined by the angles in the quark mass matrix, which matches to a chiral Lagrangian with no \(\theta\)-dependence, thus predicting a vanishing neutron dipole moment.

These points have already been discussed by us in Section 2 of [[Ai, Garbrecht, Tamarit (2021)]](https://inspirehep.net/literature/1930570) and, in view of chiral perturbation theory and the neutron EDM, in Section 7 of [[Ai, Garbrecht, Tamarit (2024)]](https://inspirehep.net/literature/2780689). It is only on the point regarding spurion analysis, and not concerning foundations of effective theory, where we deviate from the usual literature.

Referee

First, it is well-known that the theory of a quantum particle on a circle (with coordinate φ with period 2π) has a theta term. In the Lagrangian formulation this is a total derivative term 12πθ˙φ. In the Hamiltonian formulation, this results in a shifted kinetic term ∝(pφ−θ2π)2. The energy eigenvalues, then, are proportional to (n−θ2π)2 for integer n, and so the spectrum depends on the physical parameter θ. This example is very simple, textbook quantum mechanics of a free particle, so there should be no doubt that in this context a θ term is physical despite being a total derivative.

Now, given Yang-Mills theory in 4d with a theta term, we can dimensionally reduce on a small 2-dimensional manifold (for concreteness, say, a torus), in a state where we turn on flux through the 2-manifold. To make this as convincing as possible, we can consider SO(3) gauge theory, for which the flux through a 2-manifold is a Z2-valued topological invariant , so it is clear that we can make it nonzero and it cannot decay to a zero flux. By the standard logic of Kaluza-Klein reduction, this leads to a (1+1)-dimensional theory at low energies. The flux Higgses the gauge group SO(3) to SO(2), so the low-energy gauge theory is in fact a (1+1)-dimensional U(1) gauge theory. This theory inherits a θF term from the θtr(F∧F) term in 4d. But it is well-known that (1+1)d QED, placed on a spatial circle, is just the same as the quantum mechanics of a particle on a circle. (The 0+1d degree of freedom is just the Wilson line from 1+1d.) So the θ term in this context has the same physical effect on the low-energy spectrum that we described above.

This shows that θ has a physical effect on the spectrum of the Hamiltonian theory of 4d Yang-Mills placed on a 3-torus, at least for a subset of physical states where there is a nontrivial flux through a 2-torus.

Reply

First, the point that \(\theta\) can be physical in the case of quantum mechanics on a circle is not a counterexample for our results, because our conclusion on the lack of CP-violation crucially relies on the symmetry group (SU(3) for the strong interactions) being a simple, non-Abelian group which only has trivial 1D representations. In contrast with this, in the case of a particle on a circle, the underlying symmetry is discrete, given by the Abelian group of translations by \(2\pi\). This discrete symmetry group allows for nontrivial 1D representations, in contrast to SU(3), and this fact is why one can have CP violation in the circle but not in QCD, as explained in more detail below. Second, the argument about starting from a 4D torus and constructing an effective 2D theory analogue to a particle on a circle by integrating out degrees of freedom corresponding to fluctuations in a 2D manifold is wrong because it ignores the phases incurred by wave functionals in the matching between the UV and IR theories. Put differently, it ignores the fact that the theory on the circle with a \(\theta\)-term in the Lagrangian admits both CP-violating and CP-conserving spaces of states, and only when matching to the UV theory, it becomes clear which of the spaces is the relevant one, as will be elaborated upon in the following.

To start, let us point out that going back to [[Callan, Dashen, Gross (1976)]](https://inspirehep.net/literature/3673), it has become a standard view that \(\theta\) can be traded between the state and the Lagrangian (see however the point of view in Report #1, according to which it is not correct to have a topological term in the Lagrangian). This then implies that, given a Lagrangian only, it is not yet fixed without further ado how the state behaves under large gauge transformations. This question applies to Yang--Mills in 4D as well as for the Abelian theory on the circle, where the large gauge transformations correspond to translations by \(2\pi\) on the circle. Physics should be left unaltered by such large gauge transformations, which leaves the possibility that the latter lead to rephasings of the wave-functional/wave-function. CP-violation is related to the interplay between the \(\theta\)-parameter in the Lagrangian/Hamiltonian and the rephasings under gauge transformations. For example, if the two are correlated in a certain way, CP violation is absent. Hence, from a bottom-up perspective, just given the Lagrangian of electrodynamics on the circle and not knowing the state (i.e. wave function), we cannot make a conclusion about \(CP\) violation, because as mentioned before, the Lagrangian with nonzero \(\theta\) admits both CP-violating and CP-conserving spaces of states. A concrete example is provided by the wave functions \(\psi_n(x)=e^{i(\theta x/(2\pi)+ nx)}\), which have \(\theta\)-independent energies and lead to \(\theta\)-independent observables, canceling the phase from the Hamiltonian. Such wave functions cannot be considered in the same space of states as the CP-violating states \(\psi_n(x)=e^{i nx}\), which lead to the \(\theta\)-dependent energies mentioned by the referee, because this would lead to a Hamiltonian that is not Hermitian with respect to the standard norm defined by integrating only once over the circle. Therefore, the choice of states in the circle, and thus the conclusion about CP violation, remains ambiguous, and has to be decided on a case by case basis. When the theory on a circle arises as a low energy theory of 4D Yang-Mills on the torus, the matching of the IR and UV theories will determine which of the allowed spaces of states on the circle is selected.

Notably, and this is the main point of our manuscript, it turns out that in 4D Yang--Mills the behaviour of the wave function under large gauge transformations is not a choice. In Section 3 of our manuscript, we point out that in deriving the canonical commutation relations, no constraint has been imposed that would lead to gauge transformations falling into disconnected equivalence classes. This restriction is assumed in standard treatments, but cannot be imposed unless one changes the quantization procedure. When keeping the standard commutation relations, it thus follows that gauge transformations need not be periodic over the three-torus or go to a constant at infinity of three-dimensional space, so that all gauge transformations have to be allowed. In Section 6, going to a basis in which \(\theta\) has been redefined away from the Hamiltonian, we show that in order to have gauge-invariant observables, the wave functionals have to transform as a one-dimensional (pure-phase) representation of the non-Abelian group, which, when including all possible gauge transformations without restrictions, can only be trivial. Going back to a general basis in which \(\theta\) is present in the Hamiltonian, this leads to a correlation between \(\theta\) and the phases of the wave functionals, which cancels CP-violating effects. We thus conclude that there is no choice of vacuum states and that the theory does preserve \(CP\).

In the case of a particle on a circle, the underlying symmetry is discrete, given by the Abelian group of translations by \(2\pi\). This discrete symmetry group allows for nontrivial 1D representations, in contrast to SU(3), so that, unlike for QCD in the basis in which \(\theta\) is absent in the Lagrangian, there is no necessity for the states to rephase trivially under translations. In a general basis, this leads to no requirement of a correlation between the CP-odd parameter of the Hamiltonian and the phases of the wave functions. For this reason, \(\theta\) can be physical on the circle, while posing no contradiction with our results for QCD. Still, the choice of the space of states remains ambiguous as discussed before.

Now, when matching the 4D theory on the torus to the 2D effective theory on the circle, one has to fix from the top down which of the possible spaces of states---CP-violating or CP-conserving-—is realized on the circle. Let us consider the path integration on the 4D torus, where one desires to integrate out the heavy degrees of freedom and obtain an effective path integral for the low-energy theory. With a compact time direction, the path integral corresponds to a partition function at finite temperature. The correct path integral representation is obtained by matching to a trace of \(\exp(-\beta H)\) over the space of physical states. We have explicitly discussed this construction in Appendix A, where it is shown that the canonical trace is represented by a path integral with boundary conditions that have to be weighted by the wave functionals of the physical states (see Eq. (111)). As shown in our paper, the phases of the wave functionals cancel the \(\theta\) appearing in the Hamiltonian (see Eq. (110)), so that the resulting path integral has no \(\theta\) in the classical action. This path integral is the starting point that one must use to integrate out the higher-dimensional degrees of freedom in the manifold with a flux. As the starting path integral has no \(\theta\)-dependence, the resulting effective path integral for the 2D theory will also have no \(\theta\)-dependence. This corresponds to a canonical trace in the 2D theory over a space of states whose wave functions have phases correlated with \(\theta\) in the classical Lagrangian --that is, the CP-conserving space of states. Hence, constructing the path integral of the 2D theory with a top-down approach predicts that the space of states realized in the 2D effective theory is CP-conserving, in contrast to what was assumed by the referee.

Report #1 by Anonymous (Referee 1) on 2025-5-20 (Invited Report)

Strengths

The paper aims at addressing an important conceptual problem.

Weaknesses

Several statements are either wrong or confusing. The results claimed by the authors contradicts expectations based on locality (i.e. the independence of results on the choice of time boundary conditions).

Report

Comments are provided in the attached file

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Recommendation

Reject

  • validity: poor
  • significance: ok
  • originality: high
  • clarity: low
  • formatting: reasonable
  • grammar: good

Author:  Björn Garbrecht  on 2025-06-03  [id 5542]

(in reply to Report 1 on 2025-05-20)

Referee

Unfortunately the text is full of misconceptions or plainly wrong statements.

Response

Good, let us see about this:

Refereee

_1. Eq.(1) is not correct. A theta term in the Lagrangian is irrelevant (as it is a total derivative). The theta term only appears in the path integral (where it is not irrelevant), and is a consequence of inequivalent representations of the operator algebra or of the unbounded nature of the momentum/position operators when constructing the path integral formulation of a theory.

Response

There is nothing wrong with giving Eq. (1) as a Lagrangian for Yang--Mills theory. It is mathematically consistent and leads to widely accepted observables, up to the contested issue about CP violation. If anything, according to the prevalent view in the literature, it does not fully specify which of the possible theta-vacua is selected. Of course, the main point of the present work is to work out which. In the canonical formalism, this total derivative term can be used to generate canonical transformations in phase space, and the corresponding ambiguity in the representation of the momentum operator is actually discussed in our paper, see e.g. Eqs. (26) and (28), where it is shown that the ambiguity corresponds to a shift in the theta parameter of the Lagrangian. In either of these contexts, one can find the use of this term, e.g. Eq. (2) in [Crewther, DiVecchia, Veneziano, Witten], Eq. (1) in [Wilczek], Eq. (42) in [Jackiw], Eq.(1) in [Di Luzio, Giannotti, Nardi, Visinelli]. Just as for Eq. (1) in our paper, these papers do not go wrong on this point. So the theta term in the Lagrangian may be irrelevant (which is in fact also what we find), but it is not incorrect.

Referee

_2. In Eq.(2) the authors write the partition function of a theory in infinite volume. The authors argue that since topological quantization is only true in infinite volume, one should take the infinite volume limit at fixed topological sector. First, it is difficult to understand how one will take the infinte volume limit at fixed topological sectors if one argues that these sectors are only defined in infinte volume.

Response

The infinite volume limit can be taken by considering a sequence of field configurations that are distinct from a pure gauge only on some compact, simply connected support that is taken to infinity eventually. The normalization of the expectation values is discussed in detail in [Ai, Cruz, Garbrecht, Tamarit], [Ai, Tamarit, Garbrecht].

Referee

Second, there is a miriad of two dimensional models that have been exactly solved directly in infinite volume (i.e. the expressoin for Z in Eq.(2) is known). The results of these models do not agree with results if one takes the infinite volume at fixed topological sector.

Response

We ask the referee to please name, if not a myriad, then one or a handful of papers where such disagreement shows up. Since our arguments are stringent, we do not expect this to invalidate our conclusion, but will look into specific examples and comment on these.

Referee

Third, we understand how to compute infinite volume quantities in a QFT since a long time. The procedure is not arbitrary, but comes from the basic relation

$\langle O(x_1 ) · · · O(x_n)\rangle_L = \langle O(x_1 ) · · · O(x_n )\rangle_{L=\infty} + e^{−mL} + . . .$

i.e. any local correlation function in a finite volume L reproduces the infinite volume result up to exponentially small corrections. This result is generic (i.e for any QFT with a mass gap), and correct no matter if there is topology quantization or not. This basic relation immediately tells us how the infinite volume limit should be taken, and for the cases where the boundary conditions in finite volume allow for topology quantization, one has to take the infinte volume limit of the sum over all topological sectors. This relation is quite straightforward

Response We disagree that this argument specifies that the infinite volume limit is to be taken after the sum over sectors. Suppose the above relation is true (as we think it is) for some finite volume system. Why would it imply in turn that there is a problem with carrying out a calculation in infinite volume in first place, as it is necessary to justify integer topological sectors?

Note that concerning Point 2 altogether, we only give a brief review of these results on functional quantization and theta because they have already been published in [Ai, Cruz, Garbrecht, Tamarit], [Ai, Tamarit, Garbrecht]. Yet, even though this is not the main concern of the present manuscript (which is canonical quantization), we are ready to address serious objections regarding the order of limits. However, as this is the salient point, criticism should answer how taking the volume to infinity after summing over the sectors does correspond to a valid deformation of the Cauchy contour in the path integral. Earlier objections, as far as we are aware of these, are discussed in Section 8 of [Ai, Tamarit, Garbrecht].

Referee

_3. "Furthermore, fixing the winding number $\Delta n$ on $T^4$ leads to a well-defined Euclidean quantum field theory for each n so that there is a priori no necessity for summing over $\Delta n$." This is just wrong. A theory at fixed topological sector is not a well defined Euclidean field theory. Fixing the topological charge breaks clustering. Formally there is not transfer matrix (Hamiltonian).

Response There may be a semantic problem here, i.e. whether the term "Euclidean field theory" requires it to emerge from Wick rotation of a Lorentzian QFT, or perhaps from evaluating it with the canonical density matrix. In fact, our meaning becomes clear when reading the subsequent sentence: "We therefore need a more detailed reasoning to determine whether we need to sum over the topological sectors and how we must weigh the contributions in such a sum, in particular in terms of phase factors." I.e. we are looking for the correct procedure so that the sum over sectors can indeed be identified with a canonical partition function. Thereby, we identify the Hamiltonian and the appropriate states in the theory, and with their behaviour under gauge transformations, implicitly also the transfer matrix. (We take that by "transfer matrix (Hamiltonian)", since the transfer matrix and the Hamiltonian are not the same, it is meant how the states transform when translated to equivalent or redundant configurations.) While this matter can be clarified, finite volume QCD at fixed topological charge indeed produces well-defined correlation functions (which is what is meant in the manuscript) and is an active subject of study, see [Aoki, Fukaya, Hashimoto, Onogi], [Kaplan, Sen]. Furthermore, while we do not agree with the use of general theta-vacua by Aoki et.al., they show that cluster decomposition emerges to arbitrary precision as the volume becomes large, which is a prerequisite for this to happen anyway. A similar reasoning is provided in Section S5.2 of [Ai, Cruz, Garbrecht, Tamarit], where it is shown that the argument of Weinberg, vol.2, Chapter 23.6, on cluster decomposition generalizes when partitioning large volumes with fixed topology into subvolumes.

Referee

_4. The use of Hilbert space along the text is incorrect. Sentences like " . . the Hilbert space is too large. . . " just makes no sense. The Hilbert space is always very big because by definition a Hilbert space includes all functions whose inner product is finite. There is not even the requirement of continuity on a Hilbert space (i.e. the sentence the Hilbert space of continuos functions makes no sense). When doing canonical quantization one have to focus on the operators acting on the Hilbert space, and in which cases the symmetric operators define QM observables. This is how the theta dependence appears, unfortunately this discussion is not found in the paper.

Response In the present form of the manuscript, we have followed the terminology of [Jackiw] and [Fradkin's textbook] who somewhat loosely refer to constructions that do not satisfy the above requirements as Hilbert space. In particular, we have taken over Fradkin's "This Hilbert space is actually much too large.", found after Eq. (9.39). While colloquial, we did not find this to make no sense. In an eventual revision, we shall nonetheless be more precise in that we do not refer to the function space before specifying an inner product and making sure that all elements have a well-defined norm as a Hilbert space. We do not find an instance where we introduce a "Hilbert space of continuos functions" or require something to the same effect. We therefore kindly ask the referee to point out where we do so, and we will remove corresponding ambiguities and inaccuracies. Also, we emphasize that the standard theta vacua, together with an inner product summing over large gauge transformations, do not have a finite inner product, cf. Eq. (1.b) in [Okubo, Marshak], and therefore are not, without further ado, part of a Hilbert space. We hence ask whether there is any instance in the previous literature where such a finite inner product for theta-vacua is constructed? The consistent choice of inner product and construction of the pertaining Hilbert space really is the key point that the present manuscript resolves and what should be commented on when addressing the core issues.

Referee

_5. The correct general constructions on a Torus leadsto the topological charge to be $\Delta n\in Z+1/N$. In other words, particular choices of the boundary conditions (matrices t) can lead to the charge in SU (2) to be fractional: only $\pm n+1/2$ are possible values, and in particular there is not configuration with zero topological charge.

Response We disagree. There are indeed configurations with zero topological charge, namely the trivial one of vanishing four-potential and all configurations emerging through continuous deformations. Fractional charges arise when generalizing the cocycle condition to be valid modulo elements of the centre of the group. Then, conserved fluxes permeate the torus. [`t Hooft], [van Baal]. The formula presented by the referee only applies for certain configurations of twists but not in general. When we are assuming spatial isotropy (reduced to the symmetries of the torus), configurations with fluxes through the torus do not contribute, and there are only integer sectors left. The issue also has bearing on the dilute instanton gas calculation on tori, where integer topologies are assumed [Borsanyi et.al.]. If the referee were right, such results would also have to be rectified. Anyway, we find that there is no no-go that would preclude integer sectors. It would therefore be useful if the referee could provide a reference or explain the reasoning here. We also emphasize that no construction of the partition function and hence no sum over sectors is required in canonical quantization, the core objective of the present manuscript. This only comes up when deriving a path integral from this, as we indeed discuss as an application.

Referee

_6. Although chapter 2 is basically a review of well known facts, it lacks rigor. The condition of the temporal gauge Eq.(12) is not a gauge fixing condition on the Torus (the zero momentum mode of $A_\mu$ is physical, and determines the valaue of the gauge invariant Polyakov loops). Imposing this condition, again, breaks locality and formally there is no Hamiltonian. In other words, the crucial condition for the cancellation of the phases in the following chapters, is in fact not a gauge choice, but has physical consequences and the theory with such condition does not have a Hamiltonian. It is difficult to follow the rest of the paper after this problem.

Response We do not think that the torus geometry precludes the use of holonomic gauge conditions such as temporal gauge, in particular when there are textbooks [Fradkin] in which the ordinary Lorentz-invariant path integral with a local Lagrangian is recovered from transition amplitudes defined by means of canonical quantization in the temporal gauge. In this construction, any nonlocalities in the individual transition amplitudes/transfer matrix elements are circumvented by the introduction of auxiliary integrations over paths $A_0(x)$. Again, it would be useful if the referee could back the claim up with a reference or a conclusive argument. We note that gauge conditions (and, more generally, nontrivial topologies) generally enforce the use of nontrivial transition functions, which is discussed in Section~2. As for the Polyakov loop, it can only be used in its standard form when it is not piercing a surface with nontrivial transition function, see Section II of [Herbst, Luecker, Pawlowski], where it is explicitly pointed out that the definition of the Polyakov loop in the temporal gauge has to be generalized to include the dependence on the transition functions. More generally, temporal gauge is assumed in the usual construction of theta-vacua [Callan, Dashen, Gross],[Jackiw]. If there were a problem with temporal gauge on a torus, it would therefore imply that the theory as quantized in above references (leading to standard theta-vacua) as well as in the present manuscript (with a properly normalizable vacuum) cannot be evaluated with periodic temporal boundary conditions corresponding to finite temperature. We do not think that this is the case. More generally, we are not aware that specific topologies can exclude gauges in the general principal bundle construction. The referee should therefore please point out the references that say otherwise.

Referee

Finally, let me end this report with some general comments. Although the paper is presented as a reconciliation between a proposed order of limits and the finite volume formalism, one has to note that the different order of limits affects not only the theta dependence of quantities, but many other observables. Should we take the other order of limits now?, or we can just drop theta and do the usual ordering of limits?. If the other order of limits is still required (and theta was not visible), why is this work needed? I do not want to use any argument of authority and I hope that I have given compelling evidence that the manuscript has fundamental problems. Nevertheless the authors should be aware of the extense literature in topological field theories (starting with Witten seminal papers), and the vast literature in critical phenomena where two dimensional systems have been exactly solved (in particular all U (N)/SU (N) pure gauge theories in two dimensions). Some of these exact solutions are in direct contradiction with the order of limits advocated by the authors and we find no explanation on what is the mistake that has been done in more than 40 years of scientific literature. It is well known that time boundary conditions can be chosen at will without affecting the Hamiltonian. The authors choose periodic so that topological sectors are well defined in finite volume. But obviously the same spectrum of the Hamiltonian can be obtained using different time boundary conditions (i.e. open), where no topology quantization is present. Their argument(s) seem to break when topology is not well defined in finite volume.

Response Again, we kindly ask to provide particular references on the 2D theories so that we can comment on these and whether these are relevant analogies. Open boundary conditions systematically obtained from integrating out the fields in the volume complement are discussed in Sections S5.1 and S5.2 of [Ai, Cruz, Garbrecht, Tamarit], finding agreement with the infinite-volume calculation. It should also be clear that the canonical construction of the Hamiltonian and the pertaining Hilbert space, which is the subject of the present manuscript, does not make assumptions about the length of the time interval on which one aims to evaluate amplitudes. Also, we do not fix the field configurations on which the wave-functional is defined (up to temporal gauge) and therefore do not impose quantized topology in the full spacetime volume under consideration. So it is not true that the arguments we present do not apply to finite times without well-defined topology.

Summary Of the objections raised in the report, Points 1 (use of an incorrect Lagrangian for Yang--Mills theory), 5 (no temporal gauge allowed on the torus) and 6 (no integer topological sectors on the torus) would most seriously affect the chain of argument in the manuscript. Yet, we explain here why these objections are invalid and point to literature that aligns with our reasoning on this. This should give a basis for assessing the validity of these objections. Since these, as pointed out above, also have bearing on the validity of some substantial existing literature beyond the present manuscript, they should ideally be sorted out through the present communication. For these reasons, we do not intend to substantially back down here, but we can still add more explanations to the manuscript.

Points 3 and 4 appear to be more about the phrasing, which we are happy to clarify. Point 2 concerns the finite/infinite volume issue, which is not the main subject of the present paper, which is about canonical quantization. This should be kept in mind, while we are, of course, ready also to defend this point. Generally, inasmuch reference to the literature and some assertions are made in the referee report, it would perhaps correspond to good practice to also provide the specific references. We have pointed out the specific instances above.

We do not think the points emphasized in the report will lead to a resolution on the presence/absence of strong CP because the arguments in favour of it appear to align with our view: These also often start with the Lagrangian (1), theta-vacua constructions are based on temporal gauge and summations are done over integer (not fractional) topological sectors. To really decide the question, one may need to address: --Are periodic boundary conditions for permissible gauge transformations $U$ on the torus (or $U$ going to a constant at spatial infinity in $R^3$) necessary (Section 5)? If one wants to work with these constraints, why are they not accounted for in canonical quantization, e.g. enforced through Lagrange multipliers? This is crucial because without these constraints there are no large gauge transformations on spatial surfaces that are not continuously connected with the identity. --Why may the standard theta-vacua only be improperly normalizable? According to the Dirac--von Neumann axioms, these are then not physical states. (At best, improperly normalizable states can be used as a continuous basis, but the ground state of QCD should be normalizable in order to have well defined probabilities). We find that this issue requires the definition of a gauge-fixed inner product and construction of the Hilbert space as carried out in the present manuscript.

While one should always watch out for technical inaccuracies, we think these are the two main conceptual points where the present manuscript differs from the prevalent view.

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