# The Fröhlich-Morchio-Strocchi mechanism and quantum gravity

### Submission summary

 As Contributors: Axel Maas Arxiv Link: https://arxiv.org/abs/1908.02140v1 Date submitted: 2019-09-04 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.

### Ontology / Topics

See full Ontology or Topics database.

###### 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

### Anonymous Report 1 on 2019-12-9 Invited Report

• Cite as: Anonymous, Report on arXiv:1908.02140v1, delivered 2019-12-09, doi: 10.21468/SciPost.Report.1382

### Report

The author applies ideas that have been formulated by Frohlich et al. (FMS) for the construction of gauge-invariant states in the electroweak sector of the standard model to quantum gravity. The author concludes that this implies a generic gravitational dressing for all states of a particle theory. Finally, the FMS construction is discussed in the context of pure gravitational degrees of freedom, lending support to a speculative picture of massive gravity states (gravi-balls or geons) that may be a candidate for dark matter.

The research ideas presented in the manuscript are highly original as they represent an unconventional new line of thought, transferring concepts from gauge theories of particle physics to quantum gravity. I am convinced that the ideas presented here will trigger some discussion within the quantum gravity community. Before I can recommend publication, I would appreciate a response by the author to the following set of questions:

### Requested changes

(1) I find the discussion of local vs. global Lorentz symmetry slightly confusing: In the text, the author seems to consider global Lorentz transformations (which appear to be called "event-independent" transformations in the m/s). However, the explicit transformation rules, e.g., for the connection in the 2nd equation on p. 4, clearly exhibit that the formulation used by the author is locally Lorentz invariant (as it should). This distinction is particularly emphasized below Eq. (11), where the author points out the contradistinction between his preceding discussion and Kibble-Sciama gravity (which is locally Lorentz invariant). As this difference appears to be rather important (e.g., vanishing of correlators for spin?), a clarifying discussion may be needed.

(2) The author suggests an analogy between the vacuum expectation value of a scalar field in a theory with a Higgs mechanism and the split of the quantum metric into a classical metric and fluctuations. I wonder whether the analogy also implies that a classical background metric $g_{\mu\nu}\neq 0$ should also be viewed as a broken symmetry (in a gauge-fixed formulation)? Maybe the author could comment on this.

(3) The author advocates to use the expectation value of the invariant length (Eq. on p. 8) as the relevant argument for distance-type arguments in correlation functions. This seems to imply that a correlation function cannot be defined in terms of a single functional integral, since both expectation values (or $r$ and of the correlator) have to be computed. I wonder whether this should be viewed as a problem or a feature of quantum gravity as presented in the author's discussion.

(4) As may already be clear from my clumsy report, I suggest that all equations should be numbered.

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

Author Axel Maas on 2020-01-15
(in reply to Report 1 on 2019-12-09)

Dear referee,

I am grateful for the constructive and helpful report. I report here my answers and changes to the manuscript.

1. I find the discussion of local vs. global Lorentz symmetry slightly confusing: In the text, the author seems to consider global Lorentz transformations (which appear to be called "event-independent" transformations in the m/s). However, the explicit transformation rules, e.g., for the connection in the 2nd equation on p. 4, clearly exhibit that the formulation used by the author is locally Lorentz invariant (as it should). This distinction is particularly emphasized below Eq. (11), where the author points out the contradistinction between his preceding discussion and Kibble-Sciama gravity (which is locally Lorentz invariant). As this difference appears to be rather important (e.g., vanishing of correlators for spin?), a clarifying discussion may be needed.

2. The author suggests an analogy between the vacuum expectation value of a scalar field in a theory with a Higgs mechanism and the split of the quantum metric into a classical metric and fluctuations. I wonder whether the analogy also implies that a classical background metric gμν≠0 should also be viewed as a broken symmetry (in a gauge-fixed formulation)? Maybe the author could comment on this.

3. The author advocates to use the expectation value of the invariant length (Eq. on p. 8) as the relevant argument for distance-type arguments in correlation functions. This seems to imply that a correlation function cannot be defined in terms of a single functional integral, since both expectation values (or r and of the correlator) have to be computed. I wonder whether this should be viewed as a problem or a feature of quantum gravity as presented in the author's discussion.

---

@1: The referee is correct: There has been indeed a misunderstanding on my side, and as it was written, it was not adequate.

The important step was to identify that the local Lorentz symmetry is not a gauge symmetry, but still a local reparametrization symmetry, in the sense that introducing the vierbein it allows at every event to do it differently. I have substantially expanded section 2 to discuss this issue, as it is indeed important for how to observe spin.

In section 5, it is now discussed that it is necessary in order to have non-vanishing correlator with a non-trivial spin, this local reparametrization invariance needs to be taken into account. Three different options are discussed. It is emphasized that this is different from a gauge symmetry, as would be obtained when making the spin connection a dynamical field as proposed by Kibble-Sciama. As a fourth option also a second FMS-split is discussed in terms of a dynamical spin connection.

In the end, I cannot (yet) completely solve the issue, and highlight this. This touches upon the question whether spin, like energy, should become a low-energy effective observable, or not. I argue for the former, but this is my personal bias. I hope this is now clearer, and conveys the message. I hope not being able to provide a final answer is acceptable at the current state of affairs.

---

@2: In a background-field formulation, the background field enjoys full background symmetries, and thus I would not consider it as an analogy to the BEH mechanism.

In the FMS approach presented, however, this is indeed different. But just as in the BEH case, the superficial breaking emerges from a two-step process: The first is to fix a gauge/coordinate system. This already breaks explicitly any such symmetry. The second afterwards is then a split of the coordinate/gauge-fixed fields into a classical part (vev or classical metric) and a fluctuation field. As the vev part is no longer invariant under arbitrary coordinate/gauge transformation, this can be considered as a manifestation of breaking the symmetry by the fixing.

But as in the BEH case, I want to stress that the actual breaking occurs at the level of the gauge/coordinate system fixing, not by the split. And it is this fixing which makes the split meaningful and possible. And the split-off part is no longer symmetric under gauge/coordinate transformations.

The non-trivial part is that dynamically the fluctuation field is small in the BEH case. Thus, it appears as if the physics is dominated by a quantity which does not posses the original symmetry, which gives rise to the idea of that the symmetry is spontaneously broken, as the most important contribution does not have the symmetry. But, in fact, it is really just the clever choice of gauge which allows a split such that one part dominates. The symmetry is actual well and intact for any observable, though not for gauge-dependent quantities.

Provided also in quantum gravity such a split with a dominating part is possible, I would state that the same happens as in BEH physics, and, if you like, call this spontaneous breaking of diffeomorphism symmetry. However, personally I think this way of speaking is misleading, and I would prefer to avoid this.

At any rate, I have added a corresponding discussion in the text between equations (23) and (24), and expanded footnote 5.

---

@3: The correlation function itself is a single quantity. However, it depends on the events at which the fields are evaluated, and thus on objects, which are defined without a coordinate system. Especially,if the propagator is itself diffeomorphism invariant, this yields a diffeomorphism invariant result. The problem is thus to have a diffeomorphism invariant characterization of the two events. However, if the two events are not special in any way, which is assumed here by the quantization eq. (18), the only thing on which the propagator can depend are diffeomorphism-invariant characterizations of relative properties between both events. Still, the correlator is fully calculated at this point. It is just not particularly useful without somehow characterizing the two events on which it depends, if one wishes to, say, plot the propagator as a function of something. Of course, in intermediate steps on any fixed configuration the propagator can be calculated using coordinates just as usual. But this choice will become irrelevant when performing the path integral.

In any given manifold, one such relative characterization between two events is their geodesic distance. However, when integrating over all manifolds, viz. metrics, this geodesic distance is also an expectation value, and thus to obtain this information, (21) is used. Thus, this is not needed for the correlator itself. But if one wants to think of the correlator not as an event-dependent quantity, but as a quantity depending on a diffeomorphism-invariant characterization of the relation between both events, this is necessary.

And then this becomes a feature of quantum gravity, as was first, to my knowledge, remarked by Schaden in [23].

I do not think that this poses more than a (probably very involved) technical problem: Due to events being the underlying quantities on which the fields are defined, rather than on vectors in Minkowski space-time in QFT, this requires naturally a diffeomorphism-invariant characterization of this. It may well be that there is a better solution to this than eq. (21), but for the moment this appears a reasonable starting point.

I have substantially expanded the discussion at the end of section 3 to include these points, as well as the notion of the fields to depend on events rather than vectors in the beginning of section 3.

---

All equations are now numbered.

Author Axel Maas on 2020-01-15
(in reply to Report 1 on 2019-12-09)
Category:
correction

Dear referee,

I am grateful for the constructive and helpful report. I report here my answers and changes to the manuscript.

1. I find the discussion of local vs. global Lorentz symmetry slightly confusing: In the text, the author seems to consider global Lorentz transformations (which appear to be called "event-independent" transformations in the m/s). However, the explicit transformation rules, e.g., for the connection in the 2nd equation on p. 4, clearly exhibit that the formulation used by the author is locally Lorentz invariant (as it should). This distinction is particularly emphasized below Eq. (11), where the author points out the contradistinction between his preceding discussion and Kibble-Sciama gravity (which is locally Lorentz invariant). As this difference appears to be rather important (e.g., vanishing of correlators for spin?), a clarifying discussion may be needed.

2. The author suggests an analogy between the vacuum expectation value of a scalar field in a theory with a Higgs mechanism and the split of the quantum metric into a classical metric and fluctuations. I wonder whether the analogy also implies that a classical background metric gμν≠0 should also be viewed as a broken symmetry (in a gauge-fixed formulation)? Maybe the author could comment on this.

3. The author advocates to use the expectation value of the invariant length (Eq. on p. 8) as the relevant argument for distance-type arguments in correlation functions. This seems to imply that a correlation function cannot be defined in terms of a single functional integral, since both expectation values (or r and of the correlator) have to be computed. I wonder whether this should be viewed as a problem or a feature of quantum gravity as presented in the author's discussion.

---

@1: The referee is correct: There has been indeed a misunderstanding on my side, and as it was written, it was not adequate.

The important step was to identify that the local Lorentz symmetry is not a gauge symmetry, but still a local reparametrization symmetry, in the sense that introducing the vierbein it allows at every event to do it differently. I have substantially expanded section 2 to discuss this issue, as it is indeed important for how to observe spin.

In section 5, it is now discussed that it is necessary in order to have non-vanishing correlator with a non-trivial spin, this local reparametrization invariance needs to be taken into account. Three different options are discussed. It is emphasized that this is different from a gauge symmetry, as would be obtained when making the spin connection a dynamical field as proposed by Kibble-Sciama. As a fourth option also a second FMS-split is discussed in terms of a dynamical spin connection.

In the end, I cannot (yet) completely solve the issue, and highlight this. This touches upon the question whether spin, like energy, should become a low-energy effective observable, or not. I argue for the former, but this is my personal bias. I hope this is now clearer, and conveys the message. I hope not being able to provide a final answer is acceptable at the current state of affairs.

---

@2: In a background-field formulation, the background field enjoys full background symmetries, and thus I would not consider it as an analogy to the BEH mechanism.

In the FMS approach presented, however, this is indeed different. But just as in the BEH case, the superficial breaking emerges from a two-step process: The first is to fix a gauge/coordinate system. This already breaks explicitly any such symmetry. The second afterwards is then a split of the coordinate/gauge-fixed fields into a classical part (vev or classical metric) and a fluctuation field. As the vev part is no longer invariant under arbitrary coordinate/gauge transformation, this can be considered as a manifestation of breaking the symmetry by the fixing.

But as in the BEH case, I want to stress that the actual breaking occurs at the level of the gauge/coordinate system fixing, not by the split. And it is this fixing which makes the split meaningful and possible. And the split-off part is no longer symmetric under gauge/coordinate transformations.

The non-trivial part is that dynamically the fluctuation field is small in the BEH case. Thus, it appears as if the physics is dominated by a quantity which does not posses the original symmetry, which gives rise to the idea of that the symmetry is spontaneously broken, as the most important contribution does not have the symmetry. But, in fact, it is really just the clever choice of gauge which allows a split such that one part dominates. The symmetry is actual well and intact for any observable, though not for gauge-dependent quantities.

Provided also in quantum gravity such a split with a dominating part is possible, I would state that the same happens as in BEH physics, and, if you like, call this spontaneous breaking of diffeomorphism symmetry. However, personally I think this way of speaking is misleading, and I would prefer to avoid this.

At any rate, I have added a corresponding discussion in the text between equations (23) and (24), and expanded footnote 5.

---

@3: The correlation function itself is a single quantity. However, it depends on the events at which the fields are evaluated, and thus on objects, which are defined without a coordinate system. Especially,if the propagator is itself diffeomorphism invariant, this yields a diffeomorphism invariant result. The problem is thus to have a diffeomorphism invariant characterization of the two events. However, if the two events are not special in any way, which is assumed here by the quantization eq. (18), the only thing on which the propagator can depend are diffeomorphism-invariant characterizations of relative properties between both events. Still, the correlator is fully calculated at this point. It is just not particularly useful without somehow characterizing the two events on which it depends, if one wishes to, say, plot the propagator as a function of something. Of course, in intermediate steps on any fixed configuration the propagator can be calculated using coordinates just as usual. But this choice will become irrelevant when performing the path integral.

In any given manifold, one such relative characterization between two events is their geodesic distance. However, when integrating over all manifolds, viz. metrics, this geodesic distance is also an expectation value, and thus to obtain this information, (21) is used. Thus, this is not needed for the correlator itself. But if one wants to think of the correlator not as an event-dependent quantity, but as a quantity depending on a diffeomorphism-invariant characterization of the relation between both events, this is necessary.

And then this becomes a feature of quantum gravity, as was first, to my knowledge, remarked by Schaden in [23].

I do not think that this poses more than a (probably very involved) technical problem: Due to events being the underlying quantities on which the fields are defined, rather than on vectors in Minkowski space-time in QFT, this requires naturally a diffeomorphism-invariant characterization of this. It may well be that there is a better solution to this than eq. (21), but for the moment this appears a reasonable starting point.

I have substantially expanded the discussion at the end of section 3 to include these points, as well as the notion of the fields to depend on events rather than vectors in the beginning of section 3.

---

All equations are now numbered.