December 21, 2009

GraviGUT

One of the disadvantages of having waded into the Lisi affair is that I keep getting asked about related ill-conceived ideas for “theories of everything.” For the most-part, such ideas don’t have a relentless publicity machine behind them. Nor do they receive what can only be described as a credulous reception in certain corners of the Mathematics community. But still, it is assumed that one has an opinion about them.

One such idea is the $\mathrm{Spin}\left(3,11\right)$ “GraviGUT” of Nesti and Percacci. The bosonic fields consist of a connection for a noncompact $\mathrm{Spin}\left(3,11\right)$ gauge group, and a 1-form, $\theta$, transforming in the 14-dimensional vector representation. More formally, we assume that $P\to M$ is a $\mathrm{Spin}\left(3,11\right)$ principal bundle, and $E\to M$ a vector bundle, associated associated to $P$ via the 14-dimensional vector representation. The fields consist of a connection on $P$, and a 1-form, $\theta$, with values in sections of $E$.

At least at the classical level, there’s something superficially attractive about the idea. As Percacci explains, $40=4×10$ of the components of $\theta$ can be used to Higgs $\mathrm{Spin}\left(3,11\right)$ down to $\mathrm{Spin}\left(3,1\right)×\mathrm{Spin}\left(10\right)$. The remaining $16=4×4$ components of $\theta$ form the usual vierbein. For that interpretation to hold, he requires that $\theta \left(x\right)$, viewed as a map from the tangent space at $x$ to the fiber of $E$, have rank-4. This is the higher-dimensional analogue of demanding that the vierbein be invertible.

There’s a very nice analogy, that he draws, between the resulting theory — in which the 40 broken generators of $\mathrm{Spin}\left(3,11\right)$ are nonlinearly-realized — and the universal low-energy description of the electroweak theory as a gauged nonlinear $\sigma$-model (with target space ${S}^{3}$, where we gauge an $\mathrm{SU}\left(2\right)×U\left(1\right)$ subgroup of the $\mathrm{SU}\left(2\right)×\mathrm{SU}\left(2\right)$ isometry group).

At this stage, there are a bunch of massive higher-spin bosonic fields, along with the massless fields of gravity (in the Palatini form) coupled to $\mathrm{Spin}\left(10\right)$ Yang-Mills. There are no scalar bosons, and hence no candidate for fields that Higgs $\mathrm{Spin}\left(10\right)$. But, hey, let’s not worry about that.

The analogy with the low-energy description of the electroweak theory should both please and trouble you. The gauged nonlinear $\sigma$-model can only be a low-energy description. It necessarily breaks down above some cutoff scale, $\Lambda \le 4\pi f$, where — in the case of the electroweak theory — $f=249\text{GeV}$. New degrees of freedom must enter the theory at the scale $\Lambda$. Perhaps a fundamental Higgs, perhaps something more exotic. In any case, the LHC is supposed to tell us what those new degrees of freedom are.

Surely, the same is true of the nonlinearly realized “graviGUT” symmetry? As it is a field theory of massive higher-spin bosons, it must break down above some cutoff scales, where $f$ could be as high as the Planck scale. And, indeed, you are right. The graviGUT cannot be a good Lagrangian description to arbitrarily high energies.

But, in this regard, Percacci argues, we no worse off than we were with ordinary gravity. There, too, the effective Lagrangian description breaks down above the Planck scale.

Here’s where the 21st century version of magic pixie dust comes to the rescue. Perhaps the graviGUT is asymptotically-safe. Instead of simply breaking down at high energies, perhaps the spontaneously-broken $\mathrm{Spin}\left(3,11\right)$ gauge theory is controlled by a nontrivial UV fixed-point.

Even if that were true (and there’s no reason to believe that it is), surely the indefinite signature of the Cartan form on $\mathrm{so}\left(3,11\right)$ (a generic problem, if you want play around with noncompact gauge groups, like $\mathrm{Spin}\left(3,11\right)$) renders the whole theory — even if it is controlled by a nontrivial fixed point — non-unitary. Consider, in particular, the massive 1-forms in the $\left(2,2;10\right)$. Unless I am very much mistaken, their kinetic terms have indefinite signature, as a direct consequence of the indefinite signature of the Cartan form on $\mathrm{so}\left(3,11\right)$.

Despite these obvious objections, Nesti and Percacci are excited, because it’s possible to introduce fermions in the irreducible chiral spinor representation of $\mathrm{Spin}\left(3,11\right)$. These correspond to chiral fermions (a left-handed $16$ of $\mathrm{Spin}\left(10\right)$, along with its Hermitian conjugate, a right-handed $\overline{16}$), when decomposed under the unbroken subgroup. To build a kinetic term for the fermions, they need to introduce yet another field, $\varphi$, a nowhere-vanishing section of ${\wedge }^{4}E$. Such a field is probably familiar from my previous discussion of the Plebanski action.

I suppose that, after Lisi’s “theory”, writing down a model that admits chiral fermions must feel like success. But that’s setting the bar awfully low

Posted by distler at December 21, 2009 11:23 AM

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Re: GraviGUT

The ghost issue that occurs after the effective theory breaks down is sidestepped in the paper in a way that I don’t quite understand. In the conclusions Percacci mentions:

“Here we may add that since they occur near or beyond the transition to a topological phase, the standard tree level analysis is questionable.”

Why and how would transitioning to the symmetric topological phase somehow cure a nonunitary theory (even when analyzed at tree level), especially since it is precisely in that phase where the full signature is manifest? It seems to me that you would require a lot more structure than simple asymptotic safety to ensure miraculous cancellations of that nature? Or am I completely missing something obvious.

Posted by: Haelfix on December 21, 2009 6:41 PM | Permalink | Reply to this

Re: GraviGUT

There are two ghost issues.

One stems from the indefinite signature of the Cartan form on $\mathrm{so}\left(3,11\right)$. I don’t see anything that could fix this.

The other occurs in any theory of massive, higher-spin bosons – even one where the signature is positive. I think it’s the latter that Percacci wants to evade, by invoking novel UV physics.

Posted by: Jacques Distler on December 21, 2009 7:40 PM | Permalink | PGP Sig | Reply to this

Re: GraviGUT

If SUSY is found at the LHC, then doesn’t that mean some version of supergravity must describe low energy physics? If so, then does that necessarily imply that the correct version would be N = 8 SUGRA in 4 dimensions, which lots of evidence suggests is perturbatively finite at every order? Is there a string theory which completes this theory in the UV?

How would any of this affect the prospect of asymptotic safety in gravitation? That is, if SUSY is found at the LHC, is asymptotic safety still a possibility?

Posted by: Steve on December 24, 2009 11:37 AM | Permalink | Reply to this

Re: GraviGUT

If so, then does that necessarily imply that the correct version would be $N=8$ SUGRA in 4 dimensions…

No.

$N=8$ supergravity is a fascinating theory which has absolutely nothing to do with the real world.

Is there a string theory which completes this theory in the UV?

Yes. Type-II string theory compactified on ${T}^{6}$ (or, equivalently, M-theory compactified on ${T}^{7}$).

That is, if SUSY is found at the LHC, is asymptotic safety still a possibility?

The two have absolutely nothing to do with each other.

Posted by: Jacques Distler on December 24, 2009 11:56 AM | Permalink | PGP Sig | Reply to this

Re: GraviGUT

Hello back Jacques,
I think our last email dates back to 1999, ten years, completely different stuff.

I have some comments and some replies.

First: I agree that the bar is necessarily low, exactly to avoid jumping too high… and maybe waking up after 30 years to discover that we don’t have a theory (let aside a predictive one). Therefore neither I nor Roberto spoke of a TOE.

We just spoke of fermions and it seemed nontrivial to us that a real representation could be in fact a chiral complex one of a SM+GUT group, as required by the present phenomenology. This is what is needed to unify all fermions in a _single_ multiplet, instead of the (2,16) of SO(10). So now one may try to go further.

Then, about unifications, it should be honestly noted that the more we unify, the more we need to break, and since we know just few simple symmetry breaking mechanisms (higgs, dynamical, explicit..) we usually run into more troubles.. Indeed this is evident in GUT theories, where there are many realistic different breaking chains, but they are hard to test phenomenologically (more higgses, more freedom). In the GGUT case, we commented on our hope that the requirement to have well defined low energy states will pose enough constraints to restrict the number of theories (see below).

Then, about ghosts. What we ensured to avoid is ghost states at low energy. In a broken phase, one usually does not care about ghosts if they have mass above the breaking scale. This is for example easy to understand when you integrate out some DOF: you get higher derivatives, that would seem to correspond to ghost states with mass at the breaking scale, but in fact they are not, (say the original theory is well defined) it is just an artifact of the derivative expansion (à la chiral perturbation theory).

So one should be concerned by the symmetric phase, above M_PL. You are concerned by the indefinite group metric. Yes, we all know that noncompact gauge groups leads to negative kinetic terms, but this is true when you write standard F^2 kinetic terms. Here the kinetic terms are not ordinary.

On the contrary, here as in Palatini gravity (that I recall gauges SO(1,3)) the spin connection has a different action and no bad propagating DOF arise even if the group signature is indefinite.

That kinetic terms should be different at high energy is also not difficult to imagine, since in that phase there is no metric (no _background_ metric) to write the standard action.

Actually the situation is even more curious. Since there is no background metric, the theory is like a topological one, and possible propagating states are not standard. In other words, no Lorentz symmetry of the background, no spin, no standard dispersion relations. Actually from the form of the action one can see that the action is not quadratic, but at least cubic in fields, also for fermions!
Therefore in this situation what can shed light is only a canonical analysis, able to count the number of DOF in any phase (fields-first-secondclass constraints) and the sign of the Hamiltonian. When this will be done, one could say to have a well defined _classical_ theory. I believe this will be doable.

However just as I said above and as remarked in the paper, a harder and crucial test could come from lower energy, from the need to break the GUT group, and thus from the need to extend the GUT Higgses, with all sort of bad states coming out, this time _yes_ in a phase where we have lorentz symmetry and thus where physics had better be standard.

Hope to have honestly clarified the situation. Much work is needed but I think no immediate no-go can be casted on the whole framework. On the contrary I think it is not unworthy pursuing it. Of course I’ll be very happy to receive comments/criticism.

Btw, Jacques, now that you are have ended up being one of the (few?) experts, it would be a pity if you would not go on and contribute to let this grow to a consistent, new and maybe predictive TOE :)

best,
fabrizio

Posted by: FNesti on December 24, 2009 12:29 PM | Permalink | Reply to this

Re: GraviGUT

In a broken phase, one usually does not care about ghosts if they have mass above the breaking scale. This is for example easy to understand when you integrate out some DOF: you get higher derivatives, that would seem to correspond to ghost states with mass at the breaking scale, but in fact they are not, (say the original theory is well defined) it is just an artifact of the derivative expansion (à la chiral perturbation theory).

In that case, new degrees of freedom enter, at the cutoff scale, to restore unitarity.

Your theory is the analogue of the low-energy effective theory, which breaks down above the cutoff scale. (Roberto, in his review, makes that analogy very explicitly.)

It’s not that we “don’t worry” about ghosts above the cutoff scale. Rather it’s that their apparent existence is a signal that our theory is not UV-complete, and must be replaced by something else at high energies.

Yes, we all know that noncompact gauge groups leads to negative kinetic terms, but this is true when you write standard F^2 kinetic terms.

It’s true more generally than that.

…no bad propagating DOF arise even if the group signature is indefinite.

Really? Even the massive 1-forms in the $\left(2,2;10\right)$? Those clearly have an indefinite-signature kinetic term. And they are not gauge-artifacts.

Since there is no background metric, the theory is like a topological one, and possible propagating states are not standard.

The action can be expanded about flat space. That is what I am discussing.

…a harder and crucial test could come from lower energy, from the need to break the GUT group,

That’s if you want a theory that’s half-way realistic.

To heck with realism; I’m asking about whether a theory constructed along these lines is consistent.

Posted by: Jacques Distler on December 24, 2009 5:54 PM | Permalink | PGP Sig | Reply to this

Re: GraviGUT

> In a broken phase, one usually does not care about ghosts if they have mass above the breaking scale.
>
> In that case, new degrees of freedom enter, at the cutoff scale, to restore unitarity.

Yes and this is exactly what can happen, see here below. But better stated, I think if decoupling happens, in a given range of energy one cares only about states of lower or equal mass.

> Your theory is the analogue of the low-energy effective theory, which breaks down above the cutoff scale. (Roberto, in his review, makes that analogy very explicitly.)

I do not exactly agree on this point. Until one writes a full theory (bosonic sector included) I can not say much more, but what we are considering here, is actually a UV completion, with fields valid above the Planck scale, gauge invariance recovered etc.

We are not just thinking to an effective theory valid below Mpl. That we hope to be just a GUT theory. We have shown that this works in the fermionic sector. A symmetric thery at high energy reduces to a well behaved (GUT) theory when considering states below the planck scale.

> It’s not that we don’t worry about ghosts above the cutoff scale. Rather it’s that their apparent existence is a signal that our theory is not UV-complete, and must be replaced by something else at high energies.

Yes. And this is what may (should) happen. With the notable difference that above planck scale, the (quantum) field theory is still to be understood, once one accepts that the vierbein background is classically zero and thus that spacetime metric, lorentz, spin are concepts that are valid only at low energy, emerging from the spontaneous breaking of lorentz+diff invariances by the background.

> …no bad propagating DOF arise even if the group signature is indefinite.
>
> Really? Even the massive 1-forms in the (2,2;10)? Those clearly have an indefinite-signature kinetic term. And they are not gauge-artifacts.

Yes but they have Planck mass, thus at low energy they are not so dangerous, while at high energy they are part of the spin connection, for which as I said one has to expand the theory around zero - probably completely different stuff.

> The action can be expanded about flat space. That is what I am discussing.

Which flat space.. I mean, one should not expand around a background that breaks the gauge symmetry, at energies above the breaking scale.. At least one can think that it is not the right background.

> To heck with realism; I’m asking about whether a theory constructed along these lines is consistent.

Yes I see. For the reasons given above, I think it is not excluded that it could be. And on the contrary I am really surprised that this framework was not studied and pursued in the past 20 years, the ingredients were there since the 70ies. Maybe it’s because it mixes gravitational and particle physics conpetences.

Let us see, in the meanwhile, merry christmas!

Fabrizio

Posted by: FNesti on December 25, 2009 6:55 AM | Permalink | Reply to this

Re: GraviGUT

I do not exactly agree on this point. Until one writes a full theory (bosonic sector included) I can not say much more, but what we are considering here, is actually a UV completion, with fields valid above the Planck scale, gauge invariance recovered etc.

The gauge symmetry is nonlinearly realized. The massive 1-forms have three polarizations, etc.

As Roberto explained, this is just like the gauged nonlinear σ-model realization of the Standard Model.

once one accepts that the vierbein background is classically zero…

Perhaps this is the source of our disagreement. My understanding (from Roberto) is that you impose that the soldering form, $\theta \left(x\right)$, has rank-4 (when viewed as a map from $\mathrm{TM}{\mid }_{x}\to E{\mid }_{x}$) by hand.

This isn’t the result of some spontaneous symmetry-breaking. Indeed, there’s nothing in your Lagrangian which forbids $\theta \left(x\right)\equiv 0$ from being a classical solution.

As such, I’ve no idea what you’re talking about when you use the phrase “symmetric theory at high energies”, nor when you talk about “expand[ing] the theory about zero.”

The gauge symetry is nonlinearly realized, at all energy scales. There are no additional degrees of freedom (as in the Standard Model, when we replace the nonlinear σ-model by a linear σ-model), and no “symmetric phase”.

Which flat space.. I mean, one should not expand around a background that breaks the gauge symmetry, at energies above the breaking scale.. At least one can think that it is not the right background.

Flat space is solution. There is absolutely no reason we cannot study the physics of fluctuations about flat space in your theory.

And, as far as I can tell, that physics is non-unitary.

And on the contrary I am really surprised that this framework was not studied and pursued in the past 20 years, the ingredients were there since the 70ies.

Lots of people, myself included, had thoughts along similar lines back in graduate school. But, since the theory looks like it’s going to be nonunitary, those thoughts didn’t seem worth pursuing….

Perhaps we were too hasty.

in the meanwhile, merry christmas!

Merry Christmas, to you! I hope you don’t spend the whole day thinking about Physics …

Posted by: Jacques Distler on December 25, 2009 11:08 AM | Permalink | PGP Sig | Reply to this

Re: GraviGUT

Merry Christmas, to you! I hope you don’t spend the whole day thinking about Physics …

Surely yes…. I also involved my wife, parents, mother and father in law.. everybody now knows about gravigut :) An interesting twist in these italian christmas lunches.. technology messing with tradition.

Perhaps this is the source of our disagreement. My understanding (from Roberto) is that you impose that the soldering form, θ(x), has rank-4 (when viewed as a map from TM∣ x→E∣ x) by hand.

Yes I think this is the point. I tend to think in terms of a sigma model only at low energy, i.e. below planck scale. It is there that we would like to ensure the absence of light ghost states.

On the other hand, at high energy, completing the analogy with the Higgs mechanism, I think one can neglect or remove the rank-4 order parameter <θ>=Mpl e, and consider that the gauge symmetry is linearly realized exactly by the full θ. Just like for the SM Higgs.. linearly realized at high energy by the full Higgs field.

There is a number of important and subtle differences though. The modulus of the Higgs is massless here (because of diffs: it is the graviton)… There is no potential triggering the SSB (not yet) and the zero or nonzero VEVs are both solutions (at least this is well known in Palatini gravity); so we are led to choose the VEV which defines the low energy phase by hand. An other difference is that since the background defines spacetime geometry, if in the symmetric phase there is zero background (just fluctuations of the full theta) then field theory is necessarily different. Just think to possible correlation functions of, say, the one-form A\mu. Without g\mu\nu one can not have nontrivial tensors.. transversality, etc. I can only imagine to go through a canonical analysis.

again,

best,

Fabrizio

Posted by: FNesti on December 25, 2009 12:31 PM | Permalink | Reply to this

Re: GraviGUT

On the other hand, at high energy, completing the analogy with the Higgs mechanism, I think one can neglect or remove the rank-4 order parameter $⟨\theta ⟩={M}_{\text{pl}}e$, and consider that the gauge symmetry is linearly realized exactly by the full $\theta$.

Roberto, in his review, argues pretty persuasively that that’s not correct.

In any case, there is no reason (even at arbitrarily high energies) not to expand about a valid solution — like flat space.

Just like for the SM Higgs.. linearly realized at high energy by the full Higgs field.

No, it’s different. At low energies (those which have been probed to date), the Standard Model is described as a nonlinear σ-model, in which the gauge symmetry is nonlinearly-realized.

Going to a theory in which the gauge symmetry is linearly-realized involves adding a new degree of freedom to the theory, converting it from a nonlinear σ-model to a linear σ-model.

You, on the other hand, are not proposing to add new degrees of freedom. The gauge symmetry is nonlinearly-realized (always), because of the rank-4 condition on $\theta$.

I could be a stickler, and point out that this rank-4 condition is used quite extensively (and could not easily be relaxed, to allow $\theta \equiv 0$) in your treatment of fermions.

But I’m happy discussing the purely bosonic theory.

I can only imagine to go through a canonical analysis.

That would be fine …

Posted by: Jacques Distler on December 25, 2009 1:19 PM | Permalink | PGP Sig | Reply to this

Re: GraviGUT

On the other hand, at high energy, completing the analogy with the Higgs mechanism, I think one can neglect or remove the rank-4 order parameter ⟨θ⟩=M ple, and consider that the gauge symmetry is linearly realized exactly by the full θ.

Roberto, in his review, argues pretty persuasively that that’s not correct.

Well let me insist and drop any rank-4 condition on the one form θ, retaining it only on its VEV. (also, we are gauging only Lorentz, not GL4 here).

So now it is like the SM, where there is no condition on the higgs modulus. (then the fact that we still haven’t found it, is a problem, but the Standard Model has obviously the Higgs modulus and gauge symmetry is linear).

In any case, there is no reason (even at arbitrarily high energies) not to expand about a valid solution — like flat space.

I don’t see why one should expand around a wrong background at high energy. Then if the theory is not unitary and we get ill-defined states it is maybe because we should expand around an other background, or better perform a nonlinear analysis.

Going to a theory in which the gauge symmetry is linearly-realized involves adding a new degree of freedom.. You, on the other hand, are not proposing to add new degrees of freedom. The gauge symmetry is nonlinearly-realized (always), because of the rank-4 condition on θ.

When the full θ is there, it contains all the DOF and symmetry is linearly realized.

I could be a stickler, and point out that this rank-4 condition is used quite extensively (and could not easily be relaxed, to allow θ≡0) in your treatment of fermions.

I don’t think this is the case (to begin with fermions are just zero forms, spinors of the orthogonal gauge group) but let me know more precisely.

I can only imagine to go through a canonical analysis.

That would be fine …

I know.. some related interesting analysis is in 0911.3793 by K. Krasnov.

Cheers,

Fabrizio

Posted by: FNesti on December 25, 2009 1:58 PM | Permalink | Reply to this

Re: GraviGUT

So now it is like the SM, where there is no condition on the higgs modulus.

Unlike the Standard Model, the number of degrees of freedom (and the counting of propagating modes) has not changed.

(then the fact that we still haven’t found it, is a problem, but the Standard Model has obviously the Higgs modulus and gauge symmetry is linear).

It’s not obvious. In the case of the Standard Model, there could be other UV completions of the gauged nonlinear σ-model (technicolour, say). But all of them involve adding new degrees of freedom. That’s completely different from your model, where the field theory you have written down is supposed to describe physics to arbitrarily-high energies.

I don’t see why one should expand around a wrong background at high energy.

It’s not a “wrong vacuum”. It’s a perfectly fine vacuum.

The same is true in the (linear σ-model) Standard Model. There, however, for scattering processes where $E\gg \mid ⟨\varphi ⟩\mid$ the effects of the symmetry-breaking are neglible.

As Roberto showed, in his review, (and as I think you agree is evident), that’s not true here. An easy way to see the point is to note that action is horribly-degenerate, when expanded about $\theta =0$, but is perfectly-nice, when expanded about some $\theta$ of rank=4.

The difference between having a horribly-degenerate action, and having a non-degenerate one, means that you can’t neglect $⟨\theta ⟩$, even at “high energies.” (Roberto gave a more sophisticated argument, but it boils down to the same thing.)

Posted by: Jacques Distler on December 25, 2009 3:11 PM | Permalink | PGP Sig | Reply to this

Re: GraviGUT

So now it is like the SM, where there is no condition on the higgs modulus.

Unlike the Standard Model, the number of degrees of freedom (and the counting of propagating modes) has not changed.

(then the fact that we still haven’t found it, is a problem, but the Standard Model has obviously the Higgs modulus and gauge symmetry is linear).

It’s not obvious. In the case of the Standard Model, there could be other UV completions of the gauged nonlinear σ-model (technicolour, say). But all of them involve adding new degrees of freedom. That’s completely different from your model, where the field theory you have written down is supposed to describe physics to arbitrarily-high energies.

Well let us be clear, the “Standard Model” has a doublet higgs and gauge fields plus fermions. No other DOF, at any energy. You could use it up to planck scale (somebody believes this, as ugly or sad as it may be). Thus the SM higgs is a linear sigma model. Of course the world may be different, and replace the SM Higgs modulus with any exotic stuff. This goes under the name of Alternatives to or Beyond the SM :) Let LHC tell.

I don’t see why one should expand around a wrong background at high energy.

It’s not a “wrong vacuum”. It’s a perfectly fine vacuum.

Well by wrong vacuum I mean when you have ghosts, (the (2,2;10) broken gauge fields mentioned above) i.e. non positive-definite energy, i.e. most likely time or space instabilities, i.e. the vacuum will morph rapidly into an other one.

The same is true in the (linear σ-model) Standard Model. There, however, for scattering processes where E≫∣⟨ϕ⟩∣ the effects of the symmetry-breaking are neglible.

True of course. But you probably would not expand around ⟨ϕ⟩ and choose a unitary gauge, to calculate high energy processes. It’s much easier and physical to use the goldstone bosons, i.e. the Higgs in place of the longitudinal gauge modes.. Well, maybe this is a technical detail, but it tells us that at high energy (temperature?) the higgs fields fluctuate around zero. You can of course maintain the higgs VEV, and it becomes irrelevant. We agree.

For gravity and GGUT I agree that things seem different if we expand around small or zero VEV. But things may turn out not to be different. The idea to impose a rank=4 constraint on theta is surely useful at low energy, to avoid us an unwanted degenerate phase (e=0) in place of gravity.

But at higher energies I do not strictly follow the point that expanding around Mpl or zero will be different. In particular I fear that if you expand around a VEV of (tiny) Mpl size, the theory behaves at larger energies as if the background were not there (could you perturb θ=Mpl e + h if h is larger than Mpl?), and that it will be much different from the theory one imagines by expanding quadratically. In other words, I believe at high energy the quadratic terms will anyway not be the dominant ones… etc. Zero or Mpl VEV, it will probably be the same. Only, with zero VEV you see it much simply that the theory is degenerate (no background metric).

Then if one would like to call these actions “horribly” degenerate, it is probably just because we can not use our (beloved) gaussian functional integration… But this is probably our limitation. On the contrary this “phase” may give us surprises in place of bad behaved physics.

But as I said, these are questions, not answers, and we are really going too far, without formulae. Thanks Jacques for the interest and this Christmas discussion.

Cheers,

Fabrizio

Posted by: FNesti on December 25, 2009 6:14 PM | Permalink | Reply to this

Re: GraviGUT

Well let us be clear, the “Standard Model” has a doublet higgs and gauge fields plus fermions. No other DOF, at any energy. You could use it up to planck scale (somebody believes this, as ugly or sad as it may be). Thus the SM higgs is a linear sigma model. Of course the world may be different, and replace the SM Higgs modulus with any exotic stuff. This goes under the name of Alternatives to or Beyond the SM :)

Physics at pre-LHC energies is described by a gauged nonlinear σ-model. The $\mathrm{SU}\left(2\right)×U\left(1\right)$ gauge symmetry is nonlinearly-realized in that theory.

I don’t care what name you give that theory. But, whatever name you give it, it cannot be good to arbitrary energies. New degrees of freedom must enter, at an energyy below $\Lambda =4\pi f$ (where $f=249\mathrm{GeV}$, in the case at hand).

Perhaps those new degrees of freedom turn the nonlinear σ-model into a linear one. Or, perhaps, they do something more complicated.

The point is that the theory in which the gauge symmetry is nonlinearly realized (and which has massive vector bosons, but no scalars) cannot be good to arbitrarily high energies. Roberto makes a strong analogy between that theory, and yours.

It’s not a “wrong vacuum”. It’s a perfectly fine vacuum.

Well by wrong vacuum I mean when you have ghosts, (the (2,2;10) broken gauge fields mentioned above) i.e. non positive-definite energy, i.e. most likely time or space instabilities, i.e. the vacuum will morph rapidly into an other one.

If flat space is not a stable vacuum in your theory, then I am probably not interested.

could you perturb $\theta ={M}_{\text{pl}}e+h$ if $h$ is larger than ${M}_{\text{pl}}$?)

I think you are conflating large excursions in field space with high energy processes. They are a-priori distinct.

Then if one would like to call these actions “horribly” degenerate, it is probably just because we can not use our (beloved) gaussian functional integration…

The action, expanded about $\theta \equiv 0$, is horribly degenerate. Whereas, expanded about $\theta$ rank-4, it’s not. Ergo, I conclude that it’s incorrect to neglect $⟨\theta ⟩$, even at “high energies”.

Thanks Jacques for the interest and this Christmas discussion.

Thank you for the stimulating discussion.

And Merry Christmas, again.

Posted by: Jacques Distler on December 25, 2009 7:39 PM | Permalink | PGP Sig | Reply to this

Re: GraviGUT

Incidentally, isn’t it a little bit premature to talk about exactly what the UV completion of this tentative theory actually is?

Whatever ‘it’ is, surely it can’t be a field theory? Aside from the UV ghosts, at the very basic level why wouldn’t this whole thing violate Coleman-Mandula?

Posted by: haelfix on December 27, 2009 2:07 AM | Permalink | Reply to this

Re: GraviGUT

Incidentally, isn’t it a little bit premature to talk about exactly what the UV completion of this tentative theory actually is?

If it is premature I think it’s only because we have little (if any) experimental probes in those energy ranges. On the other hand it is never premature to search and research.

Whatever ‘it’ is, surely it can’t be a field theory?

That’s possible but it is not evident. Then if you want to propose an alternative, you are welcome.. On the other hand in my view, these are perfectly acceptable (classical) field theories. There are examples of field theories “without a background metric” (say topological field theories, e.g. BF ones..). The transition from classical to quantum level of these theories may be hard. (e.g. this is the subject of all the LQG activity.) But let’s first think classically.

Aside from the UV ghosts, at the very basic level why wouldn’t this whole thing violate Coleman-Mandula?

This was discussed in section 7 of 0706.3307, and it boils down to this argument

Coleman-Mandula concerns the S-matrix of asymptotic states in Minkowski background. But the Minkowski metric appears only at low energy (the VEV of the vierbein) where the unified group is “broken” exactly in a direct product, as the theorem requires.

In other words, one can use the theorem in reverse: “it is necessary to live without Minkowski space to reach a unified phase, at high energy” :)

Fabrizio

Posted by: FNesti on December 27, 2009 4:32 AM | Permalink | Reply to this

Re: GraviGUT

Incidentally, isn’t it a little bit premature to talk about exactly what the UV completion of this tentative theory actually is?

I believe it is their intention that this theory is good to arbitrarily high energies.

It might be worthwhile to recall why we are interested in the aforementioned nonlinear σ-model version of the Standard Model. At very low energies, you can integrate out the massive vector bosons (their main effect being to introduce 4-fermion interactions into the low-energy effective theory). At high energies, the theory must break down.

But the range of energies, $gf (where $g$ is a gauge coupling, and $f=249\text{GeV}$), where the effective field theory is valid, is precisely the range of energies that we have been exploring, experimentally, for the past quarter-century.

Were it not for the indefinite signature issue, there would be a similar range of energies where the theory of Nesti and Percacci would be valid. It’s not clear that range of energies is a similarly “interesting” one.

There are examples of field theories “without a background metric” (say topological field theories, e.g. BF ones..). The transition from classical to quantum level of these theories may be hard. (e.g. this is the subject of all the LQG activity.) But let’s first think classically.

As I explained above, just because you are studying high energies, it is not legitimate to ignore $⟨\theta ⟩$, and pretend that the theory is “topological.”

Doing so leads to all sorts of mistaken conclusions about the nature of the theory. (Invoking LQG won’t get you much sympathy, here.)

Coleman-Mandula concerns the S-matrix of asymptotic states in Minkowski background.

If flat space is not a valid background for your theory, then I expect that no one is interested in it.

The right way to state things is that Coleman-Mandula tells you that the full $\mathrm{SO}\left(3,11\right)$ gauge symmetry can never be restored (even approximately). Otherwise, the scattering would be trivial (or approximately so), which it, manifestly, is not.

Unfortunately, it’s the restoration of the higher gauge symmetry that would (ordinarily) lead to the decoupling of the longitudinal modes of the massive 1-forms, at high energy. Instead, one expects that they don’t decouple, and their scattering violates the unitarity bound, at high energies (at least, that’s what would happen in a theory which was not already plagued by indefinite signature for the massive 1-forms, already at the classical level).

Posted by: Jacques Distler on December 27, 2009 10:58 AM | Permalink | PGP Sig | Reply to this

Re: GraviGUT

At very low energies, you can integrate out the massive vector bosons. … At high energies, the (effective theory of the (FN)) theory must break down.

Were it not for the indefinite signature issue, there would be a similar range of energies where the (effective phase of the (FN)) theory of Nesti and Percacci would be valid. It’s not clear that range of energies is a similarly “interesting” one.

As I explained above, just because you are studying high energies, it is not legitimate to ignore ⟨θ⟩, and pretend that the theory is “topological.” Doing so leads to all sorts of mistaken conclusions about the nature of the theory.

Well, I think you did not really prove that, you just pretended it, to continue using a perturbative approach, where quadratic terms dominate, while instead I think other terms dominate over the Planck size ones. So I thought I gave sound arguments that you will have to neglect ⟨θ⟩ at high energy. Admittedly just ‘arguments’. Will come back to this hopefully.

(Invoking LQG won’t get you much sympathy, here.)

Really? :) I was not asking for sympathy… just giving credit to what they are doing (trying to do). But this is an other story.

The right way to state things is that Coleman-Mandula tells you that the full SO(3,11) gauge symmetry can never be restored (even approximately). Otherwise, the scattering would be trivial (or approximately so), which it, manifestly, is not.

The right way is the following (from CM)

Let G be a connected symmetry group of the S matrix, and let the following five conditions hold: (1) G contains a subgroup locally isomorphic to the Poincaré group. (2) For any M>0, there are only a finite number of one-particle states with mass less than M. (3) Elastic scattering amplitudes are analytic functions of s and t, in some neighborhood of the physical region. (4) The S matrix is nontrivial in the sense that any two one-particle momentum eigenstates scatter (into something), except perhaps at isolated values of s. (5) The generators of G, written as integral operators in momentum space, have distributions for their kernels. Then, we show that G is necessarily locally isomorphic to the direct product of an internal symmetry group and the Poincaré group.

It refers to (global) symmetry of the S-matrix. Also clearly it needs the Lorentz (sub)symmetry of the S-matrix. I don’t repeat the arguments and other subtleties of 0706.3307 – just at high energy there is no background no Lorentz thus only diffs and local gauge. Don’t know about asymptotic states and S-matrix.

Probably the whole idea here is that the metric emerges to break the group and leave Lorentz symmetry, at low energy only.

Since on zero background I keep repeating myself, I’d better stop. Need to study consistency and stability, as Jacques says.

best Fabrizio

Posted by: FNesti on December 27, 2009 1:07 PM | Permalink | Reply to this

Re: GraviGUT

It refers to (global) symmetry of the S-matrix.

The “gauge transformations” that don’t go to the identity at infinity, but which preserve the boundary data (in gravity, these are the asymptotic isometries) act as global symmetries.

Well, I think you did not really prove that, you just pretended it, to continue using a perturbative approach, where quadratic terms dominate

Look.

If we neglect $⟨\theta ⟩$ (i.e., expand about $⟨\theta ⟩\equiv 0$), then arbitrarily large fluctuations of the connection are completely unsuppressed. (That’s the technical meaning of “horribly degenerate”.) On the other hand, if we do not neglect $⟨\theta ⟩$, then “large” fluctuations (except those which are pure gauge) are suppressed.

Ergo, it is incorrect to neglect $⟨\theta ⟩$.

This is not complicated. It’s a pretty standard argument.

Indeed, it’s something that crops up in topological field theories, too. Note that, for a large class of topological field theories, the action is BRST-exact. A naïve observer might say, “Why bother with a BRST-exact action? Why not just use $S=0$? The answer is that the functional integral is completely undefined, for $S\equiv 0$. In topological field theory, it doesn’t matter what BRST-exact action you use, so long as it is nondegenerate.

Since on zero background I keep repeating myself, I’d better stop.

I agree that we are going around in circles, a bit. But I do think that the issues are well-clarified.

Thanks.

Posted by: Jacques Distler on December 27, 2009 3:19 PM | Permalink | PGP Sig | Reply to this

Re: GraviGUT

The “gauge transformations” that don’t go to the identity at infinity, but which preserve the boundary data (in gravity, these are the asymptotic isometries) act as global symmetries.

There is some subtlety here, since asymptotic isometries in GR are not part of the “local lorentz” (which is buried inside the metric if you don’t introduce a vierbein) but just part of diffs. It is thus incorrect to think that they are a subgroup of the gauge group alone… In our theory on the other hand, the full invariance is a (semi)direct product of diffs and gauge. Maybe this is also a way to see the Coleman-Mandula theorem respected.

But let me clarify the ⟨θ⟩ issue and the meaning of a phase with no background:

If we neglect ⟨θ⟩ (i.e., expand about ⟨θ⟩≡0), then arbitrarily large fluctuations of the connection are completely unsuppressed. (That’s the technical meaning of “horribly degenerate”.) On the other hand, if we do not neglect ⟨θ⟩, then “large” fluctuations (except those which are pure gauge) are suppressed.

Ok, here is an (the) important point: yes, fluctuations of A are not suppressed if ⟨θ⟩=0, but I argued that they couldn’t anyway be suppressed by a VEV so tiny with respect to the characteristic energies. But more importantly you should not think that A are completely free as they would be with S=0! So in this respect, my term “topological” phase is misleading, I was just meaning that there is no background metric - maybe I should say “background independent”…

To see what happens concretely, let us review the Cartan-Palatini action, θθF (with F=dA+AA).

With nonzero ⟨θ⟩, we see a mass term for A, from the AA part.

With zero ⟨θ⟩, the gauge fields A are still determined (classically) by their own EOM Dθ=dθ+Aθ=0. This equation is algebraic and can be solved exactly also if θ is made of “fluctuations” only. This means that A is anyway lagrange multipler for itself (an auxiliary field) and enforces constraints. (You can also see this by adding a total derivative.)

Thus, the action is not topological in the sense that it has no DOF, and it is not even degenerate in the sense of having dS/dA=0. It just has more fields than the propagating ones. Indeed, the counting from the canonical analysis shows that two standard degrees of freedom are present at full nonlinear level, without expanding around a background.

However, the crucial difference with the expansion around nonzero ⟨θ⟩, is that these DOF propagate nonlinearly. (There was a nice review on all similar formulations of gravity).

Now at quantum level: one worries that A is not constrained, having no kinetic or mass term… and the functional integral will be illdefined. But the real question to be posed is: how do we quantize auxiliary fields that by definition have no momentum? The answer is the Dirac quantization procedure: one first eliminates constraints, and finds a set of canonically conjugated fields and their hamiltonian. At this point everything is straightforward, poisson brackets go into commutators etc. (with caveats from time definition, recovering diff invariance etc..)

Only, now we have a quantum system with nonlinear propagating DOF, nothing weird in this, but there is no preferred background, so it is hard to even think about a possible propagator.. (calling for gauge invariant states etc..)

(And things could be so simple if one could directly solve the constraints and write the Hamiltonian. But this is (maybe) just a technical obstruction. My limited understanding).

So summarizing, a canonical analysis should be carried out first (classically) and this will give information about the true DOFs, their Hamiltonian and stability.

Then one may even go back to a lagrangian and functionally integrate it.

Hope this is clear although simplified!

Fabrizio

Posted by: FNesti on December 28, 2009 12:03 PM | Permalink | Reply to this

Re: GraviGUT

Ok, here is an (the) important point: yes, fluctuations of A are not suppressed if ⟨θ⟩=0, but I argued that they couldn’t anyway be suppressed by a VEV so tiny with respect to the characteristic energies.

Sorry, Fabrizio, but that’s just wrong.

Since the same issue arises in pure gravity, written in Palatini form (without the pesky –and presumably fatal – additional complication of the negative-signature modes), I suggest you think the matter through there.

Expanding about ${e}_{\mu }^{a}$ invertible is completely different from expanding about ${e}_{\mu }^{a}\equiv 0$.

It is just wrong to think that relaxing the invertibility constraint is harmless “at high energies”. It is, in fact, completely fatal.

But more importantly you should not think that A are completely free as they would be with S=0!

“Free” is a misnomer. The functional integral over $A$ is not “free”; it is completely undefined.

maybe I should say “background independent”…

You could say that, but it would just be a meaningless buzzphrase. Imposing the condition that ${e}_{\mu }^{a}$ be invertible does not single out a particular background. So the theory with that condition imposed is no more “background dependent” than the theory without it.

With zero $⟨\theta ⟩$, the gauge fields $A$ are still determined (classically) by their own EOM $D\theta =d\theta +A\theta =0$

No, Fabrizio, that’s not the equation of motion for $A$. The equation of motion for $A$ is

$\theta \wedge \left(d\theta +A\wedge \theta \right)=0$

The two are equivalent, if and only if $\theta$ is invertible. Which is the whole point!

(Moreover, even your equation only allow one to solve algebraically for $A$, when $\theta$ is invertible.)

Hope this is clear although simplified!

It’s clear, and clearly wrong.

Sorry that I seem impatient, but this is basic stuff, and I’m astounded that we should be having a discussion about your more complicated theory, if we can’t even get the basics of Palatini gravity correct.

Posted by: Jacques Distler on December 28, 2009 2:14 PM | Permalink | PGP Sig | Reply to this

Re: GraviGUT

The equation of motion for A is

θ∧(dθ+A∧θ)=0

The two are equivalent, if and only if θ is invertible.

Yes of course, I said simplified.

Which is the whole point!

No it isn’t. It would be if there you had ⟨θ⟩ in front!

Yes of course there are branches, and of course the solution for A will be singular (indeterminate) for vanishing (zero) θ.

But to be clear we are not discussing here the first orders of either expansion (of course different), you are concerned with two different “functional” integrals right? It is not a classical issue, it arises when we integrate on fields – measure diverging, ok?

So I suppose you will like anyway to integrate on all θ=⟨θ⟩+δθ, regardless of what you expand about, right? (otherwise where do you stop in |δθ|?)

You immediately see where I am going, but if you want to be precise and dwell into this quantum issue, let me divide the argument in two:

1. Without expansion: I have a problem at singular θ where the measure on A diverges. We may be tempted to avoid such configurations by requiring rank=4. I’m fine with that afterall (but it is a strange non-constraint).

The only other alternative is that I leave everything free but observe that the (bad) noninvertible subset has zero relative measure in the space of θs, and if you pass me this, infinite in the space of A. I let you guess the result, I really don’t know. This is why I think one should first try to solve canonical constraints to reach good variables. This I think is why people tried other actions with other fields where this problem is absent but there are others. But my understanding here is limited, admittedly.

2. Expansion around ⟨θ⟩: nice, you may think to have no problem, but you want to stay far from θ=⟨θ⟩+δθ~0, or otherwise where θ=⟨θ⟩+δθ is degenerate there’s a flat direction for A again. I don’t see what you are going to do, assume a limited integration domain for θ (which? I miss a a motivation for a perturbation theory) or maybe just consider again that θ~0 has ~zero relative measure and there A fluctuates a lot. As above.

So the problem of a difficult (full) functional integral is there even if you expand around some ⟨θ⟩.

So as I hope I could explain, there is not much difference in having zero or nonzero background and quantum theories will be the same at high energy. (there would be however a huge difference for the low energy ~classical theory, i.e. GR).

My whole point was and is that anyway you have to solve constraints (maybe with branches) before thinking of fluctuations and quantization and functional integral.

Posted by: FNesti on December 28, 2009 5:27 PM | Permalink | Reply to this

Re: GraviGUT

Sigh.

Let us not discuss the ill-definedness of the functional integral (on which, I believe you are confused). Instead, let us discuss the analysis you “really” want to do, namely the canonical quantization.

As you recall, the process of quantization proceeds as follows

1. Find the classical phase space, a symplectic manifold $ℳ$, and its symplectic form, $\omega$.
2. Choose a polarization, i.e, a foliation of $ℳ$, which is “Lagrangian” with respect to $\omega$.

There are two complications.

First, we are dealing with a gauge theory. That is, we have constraints. Conceptually, it is simplest to imagine solving the contraints in the classical system (prior to quantization). Technically, that may be difficult, but this is entirely independent of the issues we are interested in, here, so let us ignore the whole issue of constraints.

Second, we are doing field theory, which means that $ℳ$ is an infinite-dimensional symplectic manifold. Which means that regularization is important, and therein come all of the usual subtleties of field theory. This, again, is irrelevant to the point at hand.

Anyway, let’s just proceed with step 1. As is well-known the invariant description of the phase space of a classical dynamical system is as the space of solutions to the classical equations of motion. (Equivalently, we can label solutions by the corresponding initial-value data, which gives, perhaps, a more familiar description of the phase space.)

So let’s do that. As you noted, on the locus where $\theta$ is invertible, $A$ can be solved for algebraically. But, on the locus where $\theta$ is degenerate (in the worst case, $\theta =0$), the dimension of the “space of solutions” jumps dramatically (by an infinite amount). In other words, we are about as far from having a symplectic manifold (even an infinite-dimensional one) as we could possibly be.

Conventionally, we remove that locus, considering only invertible $\theta$s. And then we get a nice (infinite-dimensional) symplectic manifold*. If, instead, we try to include the locus $\theta =0$, we see that the canonical analysis goes just as bad as the functional integral analysis did.

In any case, the real point has nothing to do with whether we try to proceed by perturbing about some background $⟨\theta ⟩$, or whether we are doing path-integrals or canonical quantization.

The real point is that, in constructing the quantum field theory on ${ℝ}^{4}$, we need to fix the asymptotic behaviour of the fields (boundary conditions).

There is a huge difference between fixing

(1)${\theta }_{\mu }^{a}\left(x\right)\stackrel{{x}^{\mu }\to \infty }{\to }{\delta }_{\mu }^{a}$

versus

(2)${\theta }_{\mu }^{a}\left(x\right)\stackrel{{x}^{\mu }\to \infty }{\to }0$

You would like to pretend that the latter is the correct thing to do, “at high energies” (whatever the heck that could possibly mean, in this context). But, unlike the familiar context of spontaneously-broken symmetries, where the analogue of (2) becomes an arbitrarily good approximation to the analogue of (1), that’s manifestly not true here.

(Indeed, the fact that (2) is, in no sense, a good approximation to (1) is precisely why you prefer it. Following (1) would lead us to the inexorable conclusion that your theory is non-unitary, a conclusion you would like to avoid.)

* Actually, in canonical quantization, we can do a little better than starting with a symplectic manifold. We can start with a symplectic *orbifold*. I believe this is actually necessary, when we study gauge theories, which is the real case of interest.

Posted by: Jacques Distler on December 29, 2009 12:08 AM | Permalink | PGP Sig | Reply to this

Re: GraviGUT

Hello Jacques,

thanks, as far as I understand all the difficulties you point out are correct. So I agree on those, but on the important point:

The real point is that, in constructing the quantum field theory on ℝ 4, we need to fix the asymptotic behaviour of the fields (boundary conditions).

There is a huge difference between fixing θ(x)→Mpl δ versus θ(x)→0.

You would like to pretend that the latter is the correct thing to do, …

No. I argued that θ fluctuates (in an inherently nonperturbative way). But as i said above, I do not know about asymptotic states. In fact, I never claimed that one should consider the quanta of θ around zero, exactly for the problems we described. The asymptotic states will probably be some different combination of θ, A and other fields (this is what constraints with all their caveats from classical to quantum level ask us for) possibly gauge and diff invariant, etc.

While here my (our?) understanding stops, we are already accustomed to changes in asymptotic states from one phase to an other. (and here the analogy with the electroweak higgs mechanism will cease, one of the differences I mentioned above).

best, f

Posted by: FNesti on December 29, 2009 5:30 AM | Permalink | Reply to this

Re: GraviGUT

…we are already accustomed to changes in asymptotic states from one phase to an other.

I’m still not sure what you mean by different “phases” here.

In the electroweak theory, there is a phase transition, as a function of temperature, or as a function of the parameters in the scalar potential. So it makes sense to talk about two phases, one in which the gauge symmetry is spontaneously-broken, and one in which is it unbroken. And there’s a gauge-invariant order-parameter, $⟨{\mid \varphi \mid }^{2}⟩$, which distinguishes the two phases.

I don’t see any such (set of) adjustable parameter(s) in your theory, nor any such phase transition.

To be precise, here, I presume the order-parameter for the phase-transition is supposed to be something like $⟨\theta ⟩$. But I don’t see how to make sense of that intuition, nor how to show that there’s a phase transition, as a function either of the parameters in your Lagrangian (what parameters?), or of temperature.

If the theory doesn’t undergo a phase transition, I don’t see why it makes sense to talk about it having different “phases.”

Posted by: Jacques Distler on December 29, 2009 10:35 AM | Permalink | PGP Sig | Reply to this

Re: GraviGUT

Sure. Yes temperature is a relevant parameter, but as you ask, one would like to have some coupling constant or mass scale (then becoming temperature dependent) that could describe the triggering of the phase transition.

While there are parameters in such theories (but they are dimensionless, seen from above), to my knowledge a possible phase transition has still to be proven/disproven, so at this stage one can just conjecture that this happens.

And all this even before enlarging the gauge group. That’s a long way.

best, f

Posted by: FNesti on December 30, 2009 3:56 AM | Permalink | Reply to this

Re: GraviGUT

Readers of this blog will be pleased to know that the paper
has been published in Phys. Rev. D81 025010

Posted by: Roberto Percacci on March 11, 2010 10:46 PM | Permalink | Reply to this