# The Fröhlich-Morchio-Strocchi mechanism and quantum gravity

### Submission summary

 As Contributors: Axel Maas Arxiv Link: https://arxiv.org/abs/1908.02140v2 Date submitted: 2020-01-15 Submitted by: Maas, Axel Submitted to: SciPost Physics Discipline: Physics Subject area: High-Energy Physics - Theory Approach: Theoretical

### Abstract

Taking manifest invariance under both gauge symmetry and diffeomorphisms as a guiding principle physical objects are constructed for Yang-Mills-Higgs theory coupled to quantum gravity. These objects are entirely classified by quantum numbers defined in the tangent space. Applying the Fr\"ohlich-Morchio-Strocchi mechanism to these objects reveals that they coincide with ordinary correlation functions in quantum-field theory, if quantum fluctuations of gravity and curvature become small. Taking these descriptions literally exhibits how quantum gravity fields need to dress quantum fields to create physical objects, i.e. giving a graviton component to ordinary observed particles. The same mechanism provides access to the physical spectrum of pure gravitational degrees of freedom.

###### Current status:
Has been resubmitted

### Submission & Refereeing History

Resubmission 1908.02140v3 on 4 February 2020
Resubmission 1908.02140v2 on 15 January 2020
Submission 1908.02140v1 on 4 September 2019

## Reports on this Submission

### Strengths

1- The work applies ideas originating from particle physics in the context of gravity. This results in a highly innovative (and partially speculative) new perspective which will certainly trigger interesting discussions in the field.

2- The construction of observables in the context of quantum gravity is a highly non-trivial problem, see e.g. arXiv:1507.07921. The work makes interesting progress in this direction.

3- The article outlines a potential connection between fundamental gravity research and a phenomenologically interesting dark matter candidate, the “geon”.

### Weaknesses

1- In some places, the work is short on details. In particular the discussion about the status of a vanishing vacuum expectation value for the spacetime metric could have taken more room.

### Report

I find the work interesting and rich of original ideas. It certainly deserves publication in SciPost Physics.

### Requested changes

I leave it up to the author whether he still would like to comment on and suitably implement the following suggestions:

1- The vanishing of the vacuum expectation value for the spacetime metric is an interesting statement. The arguments leading to this conclusion are clear. There are two potentially far-reaching consequences where it would be interesting to learn the author’s opinion.
Firstly, does the author suggest that one should focus on directly computing expectation values of the physical length $<ds^2> = < g_{\mu\nu} dx^\mu dx^\nu >$ without referring to the concept of a vacuum expectation value for $g_{\mu\nu}$?
Secondly, it is commonly accepted that one could compute a non-vanishing expectation value $<g_{\mu\nu}>$ by solving the equations of motion derived from the effective action of gravity. At this stage, it is not clear to me how to reconcile these two pictures, so it may be worthwhile to clarify this connection.

2- The role of eq. (14) is not clear. In principle $J_\mu^u$ is independent of the spin-connection. The equivalence principle would nevertheless suggest that the conservation law should be constructed from a covariant derivative including the Levi-Civita connection. Since in general $\partial_\mu$ will not commute with the metric $\partial_\mu J^\mu \not = \partial^\mu J_\mu$ which seems odd.

3- There are a few stylistic remarks which would improve the readability of the work.
E.g., the definition of the operators $O_1$ and $O_2$ could be put into an equation and then referred back to when their two-point-functions are studied.
The paragraph containing eq. (23) provided a stumbling block. My first reaction was that this cannot be true in general (the remaining paragraph explains this in detail). So perhaps one could describe the generic situation first and then use eq. (23) as a particular example.

• validity: high
• significance: high
• originality: high
• clarity: good
• formatting: good
• grammar: excellent

Author Axel Maas on 2020-02-04
(in reply to Report 1 on 2020-01-18)
Category:
remark
correction

Dear referee,

thank you for the kind report, and leaving me the option, whether I want to answer. However, the editor has decided that I have to address the issues, so my own decision is not necessary.

1- The vanishing of the vacuum expectation value for the spacetime metric is an interesting statement. The arguments leading to this conclusion are clear. There are two potentially far-reaching consequences where it would be interesting to learn the author’s opinion. Firstly, does the author suggest that one should focus on directly computing expectation values of the physical length <ds^2>=<g_μν dx_μdx_ν> without referring to the concept of a vacuum expectation value for g_μν?

Yes and no. If a gauge and coordinate system is fixed, one can talk also about the metric itself. However, just as with other quantum fields, I expect still that any expectation value <g(x)> can still either only vanish if no event is special (as long as the gauge condition treats all event equally), or as an alternative would be a metric, which is essentially x-independent, i.e. an isotropic and homogeneous one. Though to express it also a coordinate system needs to be fixed. Thus, rather correlation functions <g(x)g(y)> would make more sense.

Without fixing the gauge a similar statement would apply to <ds^2> - as long as it is evaluated at fixed events, and with an indefinite metric, I expect this to average to zero. Rather, histograms over all events would be interesting, giving average distributions of such quantities. Alternatively, integrated curvature would probably be an alternative to characterize the average geometry of space-time.

I have added these statements in section 3 when discussing expectation values.

Secondly, it is commonly accepted that one could compute a non-vanishing expectation value <g_μν> by solving the equations of motion derived from the effective action of gravity. At this stage, it is not clear to me how to reconcile these two pictures, so it may be worthwhile to clarify this connection.

If the quantum effective action is gauge-fixed and a coordinate system is chosen, the same should apply as before, and one can get a non-vanishing <g_mn>. If it is not gauge-fixed, I would expect that only zero would be such a solution. But in the latter case, a diffeomorphism-invariant formulation of the quantum effective action would likely be qualitatively similar to having Yang-Mills theory formulated in Wilson loops, and then as little can be said about the metric as can be said about the gauge fields in such a formulation of Yang-Mills theory. However, I am not sure how to formulate such a quantum effective action of either Yang-Mills theory or gravity at this stage, and can thus not provide any substantial support for my expectation.

At any rate, as before, if the quantum effective action treats all events equal, so should any such expectation value of the metric. Hence, I expect for such solutions the same as said above.

Of course, if for some reason events become special, anything can happen.

2- The role of eq. (14) is not clear. In principle J^u_μ is independent of the spin-connection. The equivalence principle would nevertheless suggest that the conservation law should be constructed from a covariant derivative including the Levi-Civita connection. Since in general ∂_μ will not commute with the metric ∂_μ J^μ≠∂^μ J_μ which seems odd.

The argument would be that J_mu is actually an expression, which involves only the scalar fields and the covariant derivative Delta_mu. On each of these D_mu should act only like partial_mu, when decomposing it into its elements. This would be like the electromagnetic current (see [18]), and comes about because the charge is actually carried by scalars, which do not involve a non-trivial coupling to the metric. I have added a corresponding statement. This could also be seen by explicitly expanding all terms, and executing the covariant derivatives explicitly. In this case, and only this case, because pd_mu J^mu=0 and pd^mu J_mu=0, there is no contradiction.

3- There are a few stylistic remarks which would improve the readability of the work. E.g., the definition of the operators O_1 and O_2 could be put into an equation and then referred back to when their two-point-functions are studied. The paragraph containing eq. (23) provided a stumbling block. My first reaction was that this cannot be true in general (the remaining paragraph explains this in detail). So perhaps one could describe the generic situation first and then use eq. (23) as a particular example.

I have followed the recommendations. I have also added a citation to the mentioned article, which I found very useful.