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March 30, 2005

BF

While we’re on a cosmological constant kick, I should mention a recent paper by Stephon Alexander. He claims to have found a mechanism by which the cosmological constant can be relaxed to a small value.

Which sounds pretty important. Anyone who’s been in the field for any length of time has tried and failed to find such a mechanism.

It’s well known that four-dimensional gravity has a CP-violating topological term,

(1)

S θ=θ348π 2trRR S_\theta = \frac{\theta}{348\pi^2} \int \tr R\wedge R which is very analogous to the θ\theta-term in QCD. If you have a Dirac fermion, chiral rotations are anomalous in a curved space background. A massless fermion makes the θ\theta-angle physically unobservable. If the fermion has a mass, then the linear combination θ¯=θ+arg(m) \overline{\theta}= \theta +\arg(m) is observable. In QCD, the apparent smallness of θ¯\overline{\theta} is a problem, and the Peccei-Quinn mechanism was invented to solve it. θ¯\overline{\theta} is replaced by a pseudoscalar field, the axion. Chiral symmetry breaking induces a potential for the axion which causes it to relax to zero.

Stephon points out that a very similar mechanism can be made to work in gravity, relaxing the gravitational θ\theta-angle.

“Who cares?” I hear you cry, “Gravity is so weak, gravitationally-induced CP-violating effects are unmeasurably small. Besides, I thought you were going to tell us about the cosmological constant.”

Stephon’s idea to connect the gravitational θ\theta-angle and the cosmological constant comes from a formulation of General Relativity due to Freidel and Starodubtsev.

Consider an oriented (and, for Stephon’s purposes, spin) four-manifold, XX. Let VXV\to X be a real, oriented rank-5 vector bundle with a fixed orthogonal structure. The structure group of VV is either SO(5)SO(5) or SO(1,4)SO(1,4), depending on whether we wish to think of XX as Euclidean or Minkowskian. As a topological restriction on VV, we will demand that there exists an embedding of the tangent bundle, TXVTX\hookrightarrow V.

The fields of our theory will consist of

  • BΩ 2( 2V)B\in \Omega^2(\wedge^2 V)
  • AA, an SO(5)SO(5) or SO(1,4)SO(1,4) connection on VV. FF is its curvature.
  • ΦΓ(V)\Phi\in\Gamma(V)
  • μ\mu a volume-form on XX.

The action is

(2)

S=B ABF ABα4lϵ ABCDEB ABB CDΦ Eβ2B ABB AB+μ(Φ AΦ Al 2) S= \int B^{A B}\wedge F_{A B} - \frac{\alpha}{4l} \epsilon_{A B C D E}B^{A B}\wedge B^{C D} \Phi^E - \frac{\beta}{2} B^{A B} \wedge B_{A B} +\mu\, (\Phi^A\Phi_A -l^2) α\alpha and β\beta are dimensionless constants and ll is a constant with units of length. The field μ\mu acts as a Lagrange multiplier, forcing a nonzero expectation value for Φ\Phi. We can use the SO(5)SO(5) gauge invariance to rotate Φ\Phi to point in some particular direction. The residual unbroken gauge group is SO(4)SO(4). Truth be told, Freidel and Starodubtsev start with this latter formulation in which Φ\Phi is just a constant SO(5)SO(5) vector. But, as we shall shortly see, it’s helpful to have a manifestly SO(5)SO(5)-invariant formalism.

Once you’ve imposed the Lagrange multiplier constraint, eDΦ:TXV e \equiv D \Phi : TX \to V plays the role of the vierbein. Since e AΦ A=0e^A\Phi_A=0, we have an orthogonal decomposition (when ee is nondegenerate), V=im(e)span(Φ)V= im(e)\oplus span(\Phi), and an orthogonal projector P AB=δ ABΦ AΦ B/l 2 P_{A B} = \delta_{A B} - \Phi_A \Phi_B /l^2 Denoting the projected indices with lowercase latin letters, and setting ω ab=A ab\omega^{a b} = A^{a b}, R ab(ω)=F ab(A)+1l 2e ae b R^{a b}(\omega) = F^{a b}(A) + \frac{1}{l^2} e^a\wedge e^b will be identified as the Riemann curvature. Integrating out the BB auxiliary field, (2) becomes

(3)

S= 12GR abe ce dϵ abcdΛ 06e ae be ce dϵ abcd 2γR abe ae b6(1γ 2)γΛ 0(De aDe aR abe ae b) 32Λ 0R abR cdϵ abcd+3γΛ 0R abR ab \array{ \arrayopts{\colalign{right left}} S =& -\frac{1}{2G}\int R^{a b}\wedge e^c \wedge e^d \epsilon_{a b c d} - \frac{\Lambda_0}{6} e^a\wedge e^b\wedge e^c\wedge e^d \epsilon_{a b c d}\\ & - \frac{2}{\gamma} R^{a b} \wedge e_a \wedge e_b - \frac{6(1-\gamma^2)}{\gamma\Lambda_0}(D e^a\wedge D e_a - R^{a b}\wedge e_a \wedge e_b) \\ & - \frac{3}{2\Lambda_0} R^{a b}\wedge R^{c d}\epsilon_{a b c d} + \frac{3\gamma}{\Lambda_0} R^{a b}\wedge R_{a b} } where Λ 0=3l 2,G=α 2β 2αl 2,γ=βα \Lambda_0 =\frac{3}{l^2},\quad G = \frac{\alpha^2-\beta^2}{\alpha}l^2,\quad \gamma =\frac{\beta}{\alpha}

The first line, you recognize as the Einstein-Hilbert term (in Palatini form) and a cosmological constant term. The second line contains terms which vanish for the Levi-Cevita connection: the second term is the Nieh-Yan class. The third line contains the Euler density and the aforementioned gravitational θ\theta-term.

Comparing (1) and (3) you find θ348π 2=3γGΛ 0 \frac{\theta}{348\pi^2} =\frac{3\gamma}{G\Lambda_0} So, says Stephon, relaxing θ\theta is tantamount to relaxing the cosmological constant.

“Bollocks!” you say, “There’s no symmetry relating those terms in the action. Why should their coefficients be related in this way?”

And, indeed, you’re right. There are 6 terms in (3), whereas there were only 3 independent coupling constants in (2). Of course we got relations. But that’s just because Freidel and Starodubtsev failed to write down the most general action consistent with the symmetries.

Consider adding to the action,

(4)

S= c 12l 3ϵ ABCDEΦ ADΦ BDΦ CF DE +c 22l 5ϵ ABCDEΦ ADΦ BDΦ CDΦ DDΦ E +c 3l 2DΦ ADΦ BF AB \array{ \arrayopts{\colalign{right left}} S' =&\int \frac{c_1}{2l^3} \epsilon_{A B C D E} \Phi^A D\Phi^B \wedge D\Phi^C \wedge F^{D E}\\ & + \frac{c_2}{2 l^5} \epsilon_{A B C D E} \Phi^A D\Phi^B\wedge D\Phi^C\wedge D\Phi^D\wedge D\Phi^E\\ & + \frac{c_3}{l^2} D\Phi^A\wedge D\Phi^B \wedge F_{A B} }

These contribute terms of exactly the same form as existing first three terms in (3). With this addition, all six terms in (3) have independently variable coefficients. In particular, the first two terms in (4) contribute a shift in the cosmological constant, Λ=Λ 0+Λ 1\Lambda=\Lambda_0+\Lambda_1. Relaxing the θ\theta angle has, as far as I can tell, nothing to do with relaxing the cosmological constant.

It was a very clever paper. Everything Stephon did was OK, except for starting with the action of Freidel and Starodubtsev.

Update (3/31/2005):

In response to Urs, let me elaborate a bit on why Freidel and Starodubtsev recast GR in this fashion. In the α0\alpha\to 0 limit, the action SS becomes topological1. It is invariant under a much larger set of gauge transformations, namely AA+βΨ,BB+(DΨ+ΨΨ) A\to A+ \beta\Psi,\quad B\to B +(D\Psi + \Psi\wedge\Psi) for arbitrary ΨΩ 1( 2V)\Psi\in \Omega^1(\wedge^2 V), which (for nonzero β\beta) allows you to gauge away all the local degrees of freedom. They then hope to introduce nonzero α\alpha as a small perturbation. That’s essentially treating 1/GΛ1/G\Lambda as a small parameter, and doing perturbation theory in it. Never mind that, in the real world, this “small parameter” is 10 12010^{120}.

Update (4/2/2005):

Since many people may be bored and fail to read through all the comments below, I think I should bump this up “above the fold.” In response to Lee Smolin, I explained why I find Freidel and Starodubtsev’s work interesting.

Despite the fact that F&S failed to write down the full list of terms in the action compatible with the SO(5) symmetry, the full list is nonetheless finite. All of the fields (after gauge-fixing SO(5) to SO(4)) have positive form-degree, and there are only a finite number of SO(4)-invariant 4-forms that one can write down using them.

Thus, if a sensible perturbation theory can be constructed for this theory (e.g., preserving gauge invariance of the quantum effective action), then this theory is guaranteed to be renormalizable, as there can only be a finite number of primitive divergences to cancel and once you’ve remembered to write down all the terms consistent with the symmetries there are counterterms available to cancel all of them.

That’s very, very interesting.

Update (4/3/2005):

I couldn’t help myself. If we’re gonna have a discussion about this, it’s best to have the relevant formulæ at hand. So (with a slight change in notation in (4)) here’s the effect of adding it to the action.
  1. The coefficients of the topological terms in (3) are exactly the same functions of α,β,l\alpha,\beta,l as they were before. The coefficient of the Nieh-Yan class is still 1βl 2\frac{1}{\beta l^2}, the coefficient of the Euler class is still 32π 2αα 2β 2\frac{32\pi^2\alpha}{\alpha^2-\beta^2} and, most importantly for Stephon, θ348π 2=βα 2β 2\frac{\theta}{348\pi^2}=\frac{\beta}{\alpha^2-\beta^2} as before.
  2. The coefficients of the non-topological terms are, however, shifted.
(5)

G= l 2α 2β 2α11c 1(α 2β 2)/α Λ= 3l 21+2(c 2c 1)(α 2β 2)/α1c 1(α 2β 2)/α γ= βα1c 1(α 2β 2)/α1+c 3βα(α 2β 2)/α \array{ \arrayopts{\colalign{right left}} G =& l^2 \frac{\alpha^2-\beta^2}{\alpha} \frac{1}{1-c_1(\alpha^2-\beta^2)/\alpha}\\ \Lambda =& \frac{3}{l^2} \frac{1+2(c_2-c_1)(\alpha^2-\beta^2)/\alpha}{1-c_1(\alpha^2-\beta^2)/\alpha}\\ \gamma =& \frac{\beta}{\alpha} \frac{1-c_1(\alpha^2-\beta^2)/\alpha}{1+c_3\frac{\beta}{\alpha}(\alpha^2-\beta^2)/\alpha} } In particular, the previous relation between θ\theta and γ/GΛ\gamma/G\Lambda no longer holds.

These words were for the Minkowskian (SO(1,4)SO(1,4)) theory. In the Euclidean (SO(5)SO(5)) theory, we should take βiβ\beta\to i\beta and c 3ic 3c_3\to i c_3 in the above formulæ (as I explain below).


1 Their actual calculations mostly take place at β=0\beta=0, where, instead, GΛ0G\Lambda\to 0 as α0\alpha\to 0. That theory has the same lousy UV behaviour as plain-old GR. They’d really like to work at nonzero β\beta, which is what I’ve described.

Posted by distler at March 30, 2005 8:46 PM

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16 Comments & 3 Trackbacks

Re: BF

I had looked into Freidel & Starodubtsev’s paper a while ago after having seen people getting excited about the rumor which said that

Topological M-theory contains the topological sector of loop quantum gravity.

Apparently Cumrum Vafa explictly said so.

But really what happened is that BF theory shows up in topological M-theory and that BF theory has always been a preferred toy example for some constructions in LQG.

In particular Smolin, Freidel and others are expressing hopes that by reformulating GR as BF plus additional terms it might be possible to quantize GR by ‘perturbing’ about the topological BF theory.

Of course that doesn’t make BF theory the ‘topologivcal sector of LQG’.

Posted by: Urs Schreiber on March 31, 2005 2:11 AM | Permalink | Reply to this

BF=BS

The “BF” nature of this formulation is a complete fake. BB is an auxiliary field which can be integrated out exactly.

Sometimes, it is useful to keep the auxiliary fields in the action: when you have a symmetry algebra that closes off-shell with the auxiliary fields present, but only on-shell after integrating them out.

That’s true in “honest” BF theory, but it’s not true here. Just integrate BB out, and formulate the theory in terms of AA, Φ\Phi and μ\mu. You have lost nothing.

Posted by: Jacques Distler on March 31, 2005 2:25 AM | Permalink | PGP Sig | Reply to this

Re: BF

I would like to reply to some comments made in this blog on our work and the work of stephon. It is said :So, says Stephon, relaxing θ is tantamount to relaxing the cosmological constant.

The argument made by stephon is more subtle than that it seems to me. The mechanism is that if massive fermions get a vev then this affect the effective theta term but the main effect in stephon argument is that also it create a potential for the pseudo scalar. The effective value of this potential at the minima is negative and that relaxes the cosmological constant. From my understanding this argument is robust under change of the initial choice of the theta parameter. May be stephon should comment on that himself.

It is said :Truth be told, Freidel and Starodubtsev start with this latter formulation in which Φ is just a constant SO(5) vector

It is true that our starting point is that but if you take the pain to read further the paper we arrive after at an other formulation which is SO(5) invariant and do not need any fixed vector. It gives back classical GR equation of motion via a dynamical symmetry breaking. Essentially in this formulation the vector is a composite operator determined entirely by the B field.

It is also said: That’s essentially treating 1/GΛ as a small parameter, and doing perturbation theory in it. Never mind that, in the real world, this “small parameter” is 10 12010^{120}.

That would be pretty stupid I agree. Hopefully what we are doing is to treat GΛ as a small parameter (when the immirzi parameter is 0) and this is indeed a pretty small parameter which is 10 12010^{-120} :) For non zero immirzi parameter α\alpha which is our perturbation parameter is equal to GΛ/(1γ 2)G\Lambda/(1-\gamma^2) and this is indeed pretty small as long as the immirzi parameter is not tuned to be one.

Concerning the fact that you could in principle choose freely the coupling constant of the topological terms freely I agree in principle if one take the point of view of an independent dynamical vector field. First, in the version where the vector field is a composite field one cannot freely write every additional terms. Also, In our work we are interested only in the dynamical aspect of gravity and we dont really consider the topological terms anyway or give physical prediction from them.

The main emphasis is put on the immirzi parameter which is not a theta term and still induces gravitational cp violation. Wether this term leads to a relevant and measurable effect should be investigated. In our context it clearly has a strong effect on the quantum dynamics (by acting as a regulator) more should come on this issue.

It is also said: The “BF” nature of this formulation is a complete fake. B is an auxiliary field which can be integrated out exactly.

This statement is in principle not wrong, it just completely misses the point of the paper which is that once you use the B field you can set up a perturbation theory without having to split the metric into a background and a fluctuation and more importantly the coupling constant is dimensionless and naively very very small. This perturbation theory is in nature very different from the usual one and we invite people to look more closely to it. We haven’t proven yet that this totally new perturbation theory is free of any problem even if there is encouraging evidence in the case where the immirzi parameter is not 0.

Posted by: Laurent Freidel on April 1, 2005 3:10 PM | Permalink | Reply to this

Re: BF

Dear Laurent,

Thanks for your comments. I took the liberty of turning on the itexToMML filter for your comment (makes the formulæ more readable).

The argument made by stephon is more subtle than that it seems to me. The mechanism is that if massive fermions get a vev then this affect the effective theta term but the main effect in stephon argument is that also it create a potential for the pseudo scalar.

You are describing the Peccei-Quinn mechanism (for relaxing the θ\theta-angle) which is, believe me, well-understood. The point, however, is that there is no a-priori relation between the θ\theta-angle and the cosmological constant.

There is a relation in your theory. But that is an accident of your having missed some terms that should have been included in your Lagrangian. Once those terms are included, each of the six couplings in your Lagrangian is independent. There are no relations between them.

It is true that our starting point is that but if you take the pain to read further the paper we arrive after at an other formulation which is SO(5) invariant …

You didn’t write down the “missing” terms in that formulation, either.

For non zero immirzi parameter α\alpha which is our perturbation parameter is equal to GΛ/(1γ 2)G\Lambda/(1-\gamma^2) and this is indeed pretty small.

I was talking about taking α0\alpha\to 0 at fixed β\beta. You say you want to take α0\alpha\to 0 at fixed γ\gamma, which means taking β0\beta\to 0 as well (so that β/α=γ\beta/\alpha=\gamma is fixed).

I’m fine with that, if that’s what you want to do, but it’s not the limit in which the theory becomes topological. That was my only point: the real world is very far from the topological theory.

But the bigger point is that the coefficients of the Einstein-Hilbert, Cosmological and Immirzi terms in the action are not related in the way you think they are.

The three terms I wrote down are independent, additional contributions to the Einstein-Hilbert, Cosmological and Immirzi terms. Therefore, just as Stephon made a mistake in assuming that

(1)θ348π 2=3γGΛ \frac{\theta}{348\pi^2} = \frac{3\gamma}{G\Lambda}

you are making a mistake in assuming that

(2)α=GΛ3(1γ 2) \alpha = \frac{G\Lambda}{3(1-\gamma^2)}

That is true only when c 1=c 3=Λ 1=0c_1=c_3=\Lambda_1=0. There’s no reason those couplings should be taken to vanish. In any case, they will surely be induced under radiative corrections.

Sorry to make your perturbation theory more complicated. But, you would surely discover these terms later, even if I didn’t tell you about them now.

Posted by: Jacques Distler on April 1, 2005 5:44 PM | Permalink | PGP Sig | Reply to this

BF

(sent via email and posted by JD)

Dear Jacques,

Thanks, as always, for your interest in this work. I have the impression that you are, however, missing the key point about the mechanism proposed by Stephon Alexander in his last paper. Unless I am mistaken, his mechanism is not invalidated by the addition of terms such as you suggest. To see why, look at the key equation (38), which is, with clarifying labels

(1)Λ eff(Δ,ϕ)=Λ bare total+mΔcos(ϕ) \Lambda_{eff} (\Delta , \phi') = \Lambda_{\text{bare total}} + m \Delta \cos(\phi' )

The point is that Λ bare total\Lambda_{\text{bare total}} need not be the same as Λ\Lambda introduced in equation 12 through the parameters α\alpha and β\beta of the modified BF action. It may contains all contributions from matter, quantum corrections and any additional terms induced by quantum corrections such as the ones you suggest. The point of Alexander’s mechanism, as I understand it, is that, whatever is in Λ bare total\Lambda_{\text{bare total}}, the above equation gives a mechanism for diminishing it, because both Δ\Delta and ϕ\phi' are expectation values of matter field operators in the ground state. Hence, if the ground state minimizes vacuum energy density, and hence Λ eff\Lambda_{eff}, it will do it first by putting ϕ\phi' to the minimum of the cosine, after which it will adjust Δ\Delta so that

(2)Λ eff(Δ,ϕ)=Λ bare totalmΔ \Lambda_eff (\Delta , \phi' ) = \Lambda_{\text{bare total}} - m \Delta

is as small as allowed by boundary conditions and other constraints. Whether this adjusts Λ eff\Lambda_eff to zero or some non-zero value is not shown in this paper. But what is shown is that this mechanism will generally reduce Λ eff\Lambda_{eff} to a value less than Λ bare total\Lambda_{\text{bare total}}.

If I can make a few other comments:

  • Please correct the comment about perturbation theory in 1/GΛ1/G\Lambda. Freidel and Starodubtsev discuss not this, but a perturbation theory in GΛG\Lambda. As Laurent says, the whole point of their formulation is to permit a perturbation theory around the topological BF theory. In a forthcoming paper of theirs this is studied in some detail, and I suspect you and your readers will find the results interesting.
  • In case you are interested, there are results on a strong coupling perturbation theory in 1/GΛ1/G \Lambda, see papers on this by Gambini and Pullin and myself and Rovelli. But this is not what they do.
  • While it is not relevant for this issue, I might emphasize that the underlying issue is whether the intuition we have from ordinary perturbation theory around fixed backgrounds applies to diffeomorphism invariant theories. We claim that the standard intuitions about which theories are finite is invalid for diffeomorphism invariant and background independent theories because uv finiteness is enforced once one reduced to diffeomorphism invariant states and histories. This is the essential point, and it is established by rigorous results, both at the Hamiltonian level (see Thiemann’s forthcoming book, or gr-qc/0110034, gr-qc/0210094) and at the path integral level through uv finiteness theorems in spin foam models (gr-qc/0011058, gr-qc/0104057). You should really study these results, and either refute them or take them into account.
  • Re the comments on topological M theory and LQG, see hep-th/0503140 where LQG methods are applied to quantize Hitchin’s action in d=7d=7. LQG cannot be a sector of anything, because it is not a theory, it is a method for quantizing diffeomorphism invariant gauge theories, but these results show that it can be applied usefully to the theory Hitchin and others studied.
  • The idea of breaking SO(5) to SO(4) dynamically, as you describe in your eq 2 is an old one, one version is in hep-th/0311163.
  • Finally, recently I sent you references you requested on rigorous results in LQG including the uniqueness theorems. I’d be curious if you had a chance to study them and had any comments.

Thanks, Lee

Posted by: Lee Smolin on April 2, 2005 9:34 AM | Permalink | Reply to this

Re: BF

Hence, if the ground state minimizes vacuum energy density, and hence Λ eff\Lambda_{eff}, it will do it first by putting ϕ\phi' to the minimum of the cosine, after which it will adjust Δ\Delta so that

(1)Λ eff(Δ,ϕ)=Λ bare totalmΔ \Lambda_eff (\Delta , \phi' ) = \Lambda_{\text{bare total}} - m \Delta

is as small as allowed by boundary conditions and other constraints.

Umh, no. Δ\Delta is the fermion condensate. Stephon doesn’t specify what strong dynamics drives the fermions to condense (it sure as heck ain’t gravity), but for the purposes of his mechanism (as, indeed, for the conventional Peccei-Quinn mechanism), Δ\Delta is some externally-determined constant, not a field which can relax.

Please correct the comment about perturbation theory in 1/GΛ1/G\Lambda. Freidel and Starodubtsev discuss not this, but a perturbation theory in GΛG\Lambda. As Laurent says, the whole point of their formulation is to permit a perturbation theory around the topological BF theory.

Perturbation about BF theory is, as I said above, perturbation theory about α=0\alpha=0, with β0\beta\neq0 fixed. That is perturbation theory in 1/GΛ1/G\Lambda. In their paper, Freidel and Starodubtsev consider mostly perturbation theory in α\alpha about α=β=0\alpha=\beta=0 which is, indeed, perturbation theory in GΛG\Lambda.

Laurent, in his comment, indicates that what they really intend to study is perturbation theory about α=β=0\alpha=\beta=0 with γ=β/α\gamma=\beta/\alpha fixed. That is, “nth order” in perturbation theory involves terms proportional to α pβ np\alpha^p\beta^{n-p}.

That was far from clear from their paper.

While it is not relevant for this issue, I might emphasize that the underlying issue is whether the intuition we have from ordinary perturbation theory around fixed backgrounds applies to diffeomorphism invariant theories. We claim that the standard intuitions about which theories are finite…

I don’t even pretend to understand that claim.

I will, however grant the following (which is why I find this work quite interesting). Despite the fact that F&S failed to write down the full list of terms in the action compatible with the SO(5) symmetry, the full list is nonetheless finite. All of the fields (after gauge-fixing SO(5) to SO(4)) have positive form-degree, and there are only a finite number of SO(4)-invariant 4-forms that one can write down using them.

Thus, if a sensible perturbation theory can be constructed for this theory (e.g., preserving gauge invariance of the quantum effective action), then this theory is guaranteed to be renormalizable, as there can only be a finite number of primitive divergences to cancel and once you’ve remembered to write down all the terms consistent with the symmetries there are counterterms available to cancel all of them.

That’s very, very interesting.

I sent you references you requested on rigorous results in LQG including the uniqueness theorems. I’d be curious if you had a chance to study them and had any comments.

When one states and tries to prove a rigorous theorem, one often has to specify all sorts of technical assumptions. Sometimes these technical assumptions are necessary for the proof, but one expects that the result is true more broadly. Sometimes the result is just plain false if you relax any of the assumptions.

In the case of the “uniqueness theorem” for LQG quantization, the one and only case where we know that a canonical quantization of gravity exists is in 2+1 dimensions. And there, it violates the assumptions of this “theorem”, and there exist multiple inequivalent quantizations, none of which look like the LQG quantization.

I have no idea whether there exists a canonical quantization of gravity in higher dimensions. (Personally, I’m kinda sceptical, but I’ll keep an open mind.) But, given the experience from 2+1 dimensions, I have no reason to expect that if one exists, it satisfies the assumptions of this theorem.

P.S.: Sorry you had trouble posting your comment. If you post something that’s actually TeXable (i.e., all the equations are enclosed in either $…$ or \[…\]) and choose (say) the “itex to MathML with parbreaks” filter, it should “just work.”

Posted by: Jacques Distler on April 2, 2005 10:37 AM | Permalink | PGP Sig | Reply to this

Re: BF

Hi Jacques and Laurent
Thanks for reading my paper! I’ve spent 3 whole years thinking about non-perturbative, background independent ways to deal with the cosmological constant and this is the best I can do so far. Over this time I’ve tried to poke every possible hole I could think of. So its good to get help from very smart people like you.

I will not give my technical response now. But I want to make two points that are important to me.

1) My mechanism DOES NOT rely on minimizing the bare Theta parameter whatsoever. The vev of a pseudoscalar fixes the value of the bare cosmological constant due to minimizing the effective theta parameter (which is what I called \phi’). The bare cosmological constant (if for the moment you allow me to use the scaling I have with the gravitational theta parameter). The bare cosmological constant is subtracted from the energy gap of a fermion bilinear. This is very subtle and I urge everyone to reread my paper very caferully. I am not relaxing theta.

2) My mechanism does not rely on the construction on BF theory. I used this for calculational convenience.
There are two other formulations of generaly relativity which exploit the relationship between the gravitational theta parameter and the bare cosmological constant. My point in this paper was to show that de-Sitter space has large diffeomorphisms which contribute to the bare cosmological constant. A fact discussed by Witt and Friedmann. So by taking into account the theta sector of 4-d GR we could normailze the cosmological constant from a fermionic sector which necessarily has a negative contribution to the CC (here I quote Marvin Weinstein and BJ Bjorken).

More later.
And sincere thanks again for the useful critique.,


Stephon

Posted by: Stephon Alexander on April 2, 2005 12:52 PM | Permalink | Reply to this

Theta Theta

Thanks for commenting Stephon, and thanks for taking this post in the helpful spirit it was intended.

I have many technical questions about your paper, but I want to understand the “big picture” first.

Say I came to you and said,

Stephon, I’ve found a solution to the Cosmological Constant Problem! I introduce a scalar field, φ\varphi, with a potential, V(φ)V(\varphi), such that, at the minimum, V(φ min)<0\V(\varphi_{\text{min}})\lt 0. Then its effect is to partially cancel out the “bare” cosmological constant:

(1)Λ eff=Λ+V(φ min)\Lambda_{\text{eff}}=\Lambda + \V(\varphi_{\text{min}})

You would, justifiably, say to me,

Jacques, old man, you’re beginning to show your age. You have solved nothing. Unless you can explain why the depth of the potential V(φ)V(\varphi) should be correlated with the value of Λ\Lambda, you haven’t explained why they should cancel to such exquisitely high precision.

The interesting thing about the Peccei-Quinn mechanism is that the location of the minimum of the axion potential is correlated with the “bare” value of θ\theta that you are trying to cancel. Thus we have a natural mechanism for relaxing θ\theta to zero.

What I think you are trying to do is exploit the purported relation between the gravitational θ\theta-angle and the cosmological constant to provide the requisite correlation between the depth of the axion potential and the cosmological constant to be canceled.

But, what I hope to have convinced you is that there is no a-priori relation between the “bare” gravitational θ\theta-angle and the “bare” cosmological constant. In the theory under discussion, they are independent couplings. They only appeared to be correlated because F&S forgot to write down a few terms in their action.

There are two other formulations of generaly relativity which exploit the relationship between the gravitational theta parameter and the bare cosmological constant.

If you’ve read a paper which claims that, then I suggest you find a nice café and sit down with a pad of paper to construct the terms which the author of the paper missed.

I assure you that they are there.

Posted by: Jacques Distler on April 2, 2005 4:53 PM | Permalink | PGP Sig | Reply to this

Re: BF

Dear Jacques

I am not sure how to handle equations in this interface so feel free to improve the presentation.

[To a first approximation, just choose one of the TeX-enabled filters (e.g., “itex to MathML with parbreaks”) and type something TeXable. The equations (and only the equations) will be converted to MathML. If you want other effects, like *asterisks* for emphasis or

> the greater-than sign for

blockquoted material

choose the “Markdown with itex to MathML” filter.
–JD]

I guess we both agree that the theory we are talking about contains 6 parameters three (G,Λ,γ)(G,\Lambda,\gamma) are bulk parameters and three are theta angles and any reformulation of this system which agree on the bulk terms differs only by topological terms. We also agree that this 6 parameters are independent.

When we wrote down the action (which was introduced before by Lee and Artem we just add the beta term), we cared only about dynamics and we never stated that the theta angle are fixed by the bulk terms no more than anybody writing down the Einstein action or Yang-mills think this imply that θ\theta is zero, for the purpose of perturbative dynamics this is just boundary term and you ignore them. What I learn from stephen work is that we should may be care more about them.

To come back to the point this means that your action is interesting and i will definitely give more thought to it, but up to boundary terms field redefinition and renormalisation of the bulk parameters it can be recasted in the form we work with and that’s all that matter for perturbative dynamical question. Of course this suggests that the bulk and boundary parameters will run and that’s why I called the scaling α=GΛ\alpha = G\Lambda naive and this is something we worry about. (With the understanding that defining an invariant analog of renormalisation group in a background independant context is far from trivial.)

That was my only point: the real world is very far from the topological theory.

Nobody has ever claimed the contrary.

First, the world we are discussing in this work is a world made only of pure gravity and no matter. In this world when we take the limit G0G\to 0, that is when we ignore the dynamical nature of gravity we are left with only one solution which is de Sitter space (this is the solution of BF). When GG is not zero the theory is not topological of course.

We can extend what we are doing to the more interesting case of gravity with matter (something should appear on that) in this case when we take the G=0G =0 limit, only the gravity sector becomes topological what we are left with is usual field theory on a fixed de Sitter background. This is the world of particle physicist where gravity is totally frozen.

This amounts to show that we can write the usual matter Feynman graph in a unusual way as expectation value of some operators in BF theory (since it is not yet published you have the right to remain skeptical even if there is a published proof of that for 3D scalar field theory).

The main point is that the world of particle physics which is the world where gravity possess absolutely no dynamics, where backreaction is ignored, blackhole do not form etc… is identical to a world where gravity is treated as a topological field theory (hence is not dynamical). In some sense this is not really surprising or deep even if it took us some amount of work to arrive there.

[An example you might be familiar with which is vaguely related comes from open string with B field: write it in first order by adding an auxiliary BB field and take the limit α0\alpha'\to 0, you get a topological string model describing open string non commutativity, cf seiberg-witten]

What it gives us is a new angle to adress the problem of the world where gravity is also dynamical and quantum. What’s different here is the fact that we can adress this question in a background independent manner and it is important for people who think that background independence is a key to any theory of gravity. It does’nt mean that this will be obviously successful of course even if the usual arguments which rely on a background choice don’t obviously apply. If it doesn’t work we (or I) will have learn something new. We will see.

Since you complained that we go too much away from the theta discussion i will refrain from replying to some disturbing comments you made on 2+1 Gravity and LQG, let me just say that 2+1 gravity is really well studied and understood in the context of LQG and spin foam, {\it believe me}. One example of the latest thing one can do in this context is compute the effective action of a scalar field obtained after integration of quantum gravity fluctuation and prove the appearance of effective non commutativity and lorentz deformation. I can provide you a large list of reference pointing to this very interesting subject of mathematical physics

Let me move on a remark on the relaxation mechanism of stephon.

So we all agree that the relaxation describe by stephon is independent of theta and only on the fermions mass and condensate properties you worry rightly that this relaxation might not solve the cosmological constant, something that stephon did not claim to do.

Since your original criticism was mostly on the particular value of theta stephon choose as a starting point (which relates btw two physical constants not yet constrained by experiments and usually assumed to be zero). The question i have is suppose for the sake of the argument that there is a non perturbative reason that fixes θ\theta to be related to the cosmological constant and γ\gamma in the way originally stated. Why Would you expect in this case that the relaxation mechanism might be much more efficient?

Posted by: laurent freidel on April 4, 2005 9:59 AM | Permalink | Reply to this

Effective action

To come back to the point this means that your action is interesting and i will definitely give more thought to it, but up to boundary terms field redefinition and renormalisation of the bulk parameters it can be recasted in the form we work with and that’s all that matter for perturbative dynamical question.

Well, if you want to be careful about things, you should note that, in the Euclidean theory (i.e., SO(5)SO(5), as opposed to SO(1,4)SO(1,4)), the CP-odd terms you have noted have imaginary coefficients. In all the formulæ above, you should take βiβ\beta\to i\beta and c 3ic 3c_3\to i c_3 (and, for Stephon, θiθ\theta \to i\theta). In particular, the Immirzi parameter of the Euclidean theory is pure-imaginary (i.e., you should take γiγ\gamma\to i\gamma as well).

When you do that, the ubiquitous combination, κ(α 2β 2)/α\kappa\equiv(\alpha^2-\beta^2)/\alpha becomes κ=(α 2+β 2)/α\kappa=(\alpha^2+\beta^2)/\alpha and there’s nothing special happening at β=α\beta=\alpha. There is, however, something special happening at c 1=1/κc_1=1/\kappa, though that’s outside of the range of validity of perturbation theory.

This amounts to show that we can write the usual matter Feynman graph in a unusual way as expectation value of some operators in BF theory (since it is not yet published you have the right to remain skeptical even if there is a published proof of that for 3D scalar field theory).

Huh? The most immediate effect of introducing matter is to renormalize the gravitational theory. Matter loops induce corrections to the operators you have in your theory, and induce new local operators that you don’t have in your theory.

If all the matter is massive (unlike in the real world), then the effect of integrating out the matter can be summarized in a local effective Lagrangian for the gravity sector, where the higher-dimension operators have coefficients given by inverse powers of the mass. This effective Lagrangian is a good vehicle for computing physics on length scales longer than the Compton wavelength of the matter. At shorter distance scales, it breaks down, and must be replaced by the full quantum theory, including the matter.

The “Cosmological Constant Problem” is that, in this effective theory, the value of G effΛ effG_{\text{eff}}\Lambda_{\text{eff}} is at least m 4/M pl 4m^4/M_{pl}^4, which is typically much, much larger than 10 12010^{-120}.

Moreover, since, in order to write down the matter Lagrangian, one had to introduce the inverse vierbein, one gets all possible couplings in the gravitational effective Lagrangian, not just the ones one can build with fields of positive form degree.

But, frankly, I’d be happy if you could come up with a perturbatively-renormalizable theory of pure gravity in de Sitter space.

[Right now, I don’t even see that. It looks to me that the null-space of the quadratic form in the quadratic part of your action is “too large”. The quadratic form is degenerate, even on the subspace transverse to the gauge orbits. But, probably, I have misunderstood something.]

Since you complained that we go too much away from the theta discussion i will refrain from replying to some disturbing comments you made on 2+1 Gravity and LQG

The comment was in reference to an email conversation with Lee. He pointed me to a theorem (due to Sahlmann, Thiemann and others) to the effect that there is a unique quantization of gravitational theories (invariant under spatial diffeomorphisms) for dimensions d>2d\gt 2.

In 2+1 dimensions, it is known that there are multiple inequivalent quantizations of “pure gravity.”

The question i have is suppose for the sake of the argument that there is a non perturbative reason that fixes θ\theta to be related to the cosmological constant and γ\gamma in the way originally stated. Why Would you expect in this case that the relaxation mechanism might be much more efficient?

That’s not my expectation, it’s Stephon’s.

Put differently, if there is no relation between θ\theta and Λ\Lambda, why do you (or anyone) believe that the depth of the axion potential should be related to the “bare” value of Λ\Lambda that we wish to cancel?

Posted by: Jacques Distler on April 4, 2005 3:42 PM | Permalink | PGP Sig | Reply to this

Re: Effective action

Physics blogging* is pretty addictive, i feel like starting up smoking again, you should may be among the itex signs some warning signs analogous to the one they put on cigarettes pack :-) *Even if it doesn’t replace a good chat where the misunderstanding can be cleared up much more easily.

just choose one of the TeX-enable

obviously i havent learn how to do that yet i will try. when and how do i itex? should i download the program?

Huh? The most immediate effect of introducing matter is to renormalize the gravitational theory. Matter loops induce corrections to the operators you have in your theory, and induce new local operators that you don’t have in your theory.

You are right but that’s not what i was refering to (I was not very clear i admit). What we are doing (ref hep-th/0502106 ) is a problem where you start with a scalar field in 3d coupled to 3d gravity and you integrate out the metric field not the scalar field.

There are many reasons for doing that one of them being why not:) a more serious one of them being that quantum gravity is suppose to tell us on which background the field propagate and its only after integrating out gravity that you know.

If you do that (which we where able to do) you get an effective action for the scalar field which, you can prove in this case, is the action of a field propagating on a very special non commutative three d geometry.

Moreover, since, in order to write down the matter Lagrangian, one had to introduce the inverse vierbein, one gets all possible couplings in the gravitational effective Lagrangian, not just the ones one can build with fields of positive form degree.

You are right that this is a very big problem. The beauty of the reformulation we have is that it allows us to have a formulation in which you can couple matter without inverting the vierbien and only invoke forms of positive degree. (since this is one of the thing i am suppose to work on instead of being here you will have to wait for more detail).

In the cited paper you can get some idea of how it works in 3 d. In 3 d this problem is also very serious since it spoils the beauty and integrability properties of the form formulation of 3d gravity. The main idea is first to couple particle to gravity (this doesn’t need inverse form) and by extension to all feynman graph. We know how to compute the averaging over gravity for each Feynman graph of the matter field. And from there we can reconstruct the effective action of the matter field (gravity has disappeared). It amounts to do second quantisation after the coupling to gravity rather than before which lead as you noticed to a dead end.

(i.e., you should take γ→iγ as well).

I guess you are thinking of some Wick rotation of Lorentzian gravity. It this field it is important to distinguish between Wick rotated gravity (I should admit that I always miserably failed to understand how you can give any meaning to that) and euclidean quantum gravity (if you have a better denomination feel free) where you quantised the euclidean gravity system (you want to solve the euclidean hamiltonian constraints). Thats the second choice we are presenting in order to avoid dealing at first with non compact gauge group.

You might be happy to learn that even if we succeed to do the euclidean gravity case we will have learned a lot but we will still have to worry about the Lorentzian case. This is the same in 3d, you can quantize both euclidian and lorentzian gravity and they are not related by any analytic continuation. For instance the hilbert space of 3d euclidean and lorentzian gravity with positive cosmological constant are very different the first one is finite dimensional the second one infinite dimensional. In obvious contradiction with a huge amount of sloppy litterature on the subject of de sitter space.

Posted by: laurent freidel on April 4, 2005 7:17 PM | Permalink | Reply to this

Re: Effective action

obviously i havent learn how to do that yet i will try. when and how do i itex?

should i download the program?

Nothing to download. No programs to install.

Just select one of the “itex with …” filters from the Text Filter: popup menu in the Comment-Entry window (there’s a description of each filter, with links to further documentation, below the menu).

You can include equations in TeX, and they will be automatically converted to MathML. Just click PREVIEW and you will see them magically appear.

What we are doing (ref hep-th/0502106 ) is a problem where you start with a scalar field in 3d coupled to 3d gravity and you integrate out the metric field not the scalar field.

Remember my caveat about massive fields? When you integrate out a massive field, the resulting effective action is local. If you try integrating out a massless field (like the metric in 3+1 dimensions, or even a massless scalar), the resulting effective action for the remaining fields is horribly nonlocal.

I’ll take a look at your paper about the 2+1 d case, where, since there are no propagating degrees of freedom in the metric, you seem to be able to get away with integrating it out.

(i.e., you should take γ→iγ as well).

I guess you are thinking of some Wick rotation of Lorentzian gravity.

Nope. I’m saying that, in Euclidean field theory, the CP-even terms in the action are real, whereas the CP-odd terms are pure-imaginary. In your formulation, α\alpha is real, but β\beta is imaginary. This leads to real coefficient for the Einstein-Hilbert, Cosmological Constant and Euler Density terms in the action, but imaginary coefficients for the Immirzi, Nieh-Yan and Pontryagin terms.

The ‘Wick-rotated’ Minkowskian theory is an SO(1,4)SO(1,4) gauge theory on a Euclidean-signature 4-manifold. A completely different animal.

For instance the hilbert space of 3d euclidean and lorentzian gravity with positive cosmological constant are very different the first one is finite dimensional the second one infinite dimensional. In obvious contradiction with a huge amount of sloppy litterature on the subject of de sitter space.

I have similar sentiments about the sloppy literature on the subject.

Posted by: Jacques Distler on April 4, 2005 10:41 PM | Permalink | PGP Sig | Reply to this

Re: Effective action

I’m saying that, in Euclidean field theory, the CP-even terms in the action are real

I am not sure what general rule you are following to fix the reality conditions, the one I follow is some kind of reflexion positivity which states that the euclidean action i should put in the exponent should be invariant under a combination of inversion on space time (T) and complex conjugation. This rules gives me a i theta for yang-mills since the yang-mills action is invariant under spacetime inversion but the cp term is not. It also gives a pure imaginary weight in the exponent for chern-simons theory as it should. Now for the formulation we are discussing everything is expressed in terms of forms so they have the same behavior under spacetime inversion. The CP difference comes if you consider inversion in the lorentz space which does not for me determine the reality condition (but may be you have a different way to look at that). The problem with taking beta imaginary can be easily seen if you look at pure BF (α=0\alpha =0 β\beta not zero) in this case the theory is a constrained system and if i want to make sense of it the constraint should always comes with an i in the exponential to produce the projection on the kernel of the constraint (one of the main problem i have with euclidean gravity). So if i take your prescription (BF with real coefficient and BB with imaginary one) the pure BF theory cannot even be defined at the quantum level and i have to stop even before taking the first step.

Concerning 2+1 the unicity result Lee was describing is a uniqueness result refering to the {\it kinematical} hilbert space. It gives a framework in which we should think of quantisation in diff invariant theory whose phase space is a pair of electric and magnetic fields, and it is very robust. It tells me that i can always write down such hilbert space in the spin network form but it doesn’t tell me that there is no ordering ambiguity in writing down an hamiltonian.

Within this framework one should now solve the dynamics, that is impose the constraints and construct the corresponding physical scalar product, this is what spin foam models are about and there is no uniqueness theorem for that. In three d we know one solution wich is non perturbative, which work for all genus which can be consistently extended to all toology changing amplitude and which gives unitary representation of the mapping class group. At this level there is now uniqueness theorem for the physical scalar product. The problem you are refering to is dynamical and as far as i know proven only for the torus. It is not clear to me that this inequivalent quantisation all satisfy the consistency requirement i was mentionning before namely all genus and good behavior under the mapping class group. It is an interesting open question.

[Right now, I don’t even see that. It looks to me that the null-space of the quadratic form in the quadratic part of your action is “too large”. The quadratic form is degenerate, even on the subspace transverse to the gauge orbits. But, probably, I have misunderstood something.]

May be, may be not (it depends on what you mean exactly) but there is a point where you are right, the free theory have a huge kernel. In our perturbation theory we take as a “free” (or “quadratic”) theory the BF theory with beta term and alpha as a perturbation (beta should be small in the physical regime but lets not worry about that now since the beta term is topological we can treat it non perturbatively) At zero order we have just BF and we should divide by the volume of the BF gauge group so the amplitude at zero order is normalised to be one. Now as you mention the “free” theory as a huge kernel and it will show up in the first order perturbation term in alpha which breaks locally BF symmetry, this means that at this order a local fraction of the gauge group becomes dynamical (the gauge group is still the one of BF except at one point) and we have to compute its contribution, this computation can be reduced to a finite dimensional integral by using the gauge group symmetry where it apply. We are left with the integration over one copy of the gauge transformation acting at a point divided by its volume. For non zero beta we can argue that the volume is finite (there is a spin foam model of BF where this is true and more arguments for that in the paper). The first analysis we have, shows (and thats where the scary technical part is and the result is not final yet) that this contribution start by 1/β1/\beta plus an expansion in beta. So effectively one of the small coupling constant is 1/γ1/\gamma (γ=0\gamma =0 is a priori not in the applicability range of what we are doing).

I know that this is a very unusual perturbation theory since we break the symmetry of the topological free theory and that almost everybody expect us to fail there. It is just terra incognita we are in, I can’t rely on any books or reference where this is tried and may be you know of some unsuccessfull* attempt to do the same type of perturbation away from the seiberg-witten limit, which is not a bad analogy *(which shows that it can’t be done, the successfull ones i would know about it).

concerning 3d, the effective action for the scalar field we obtain is non local since it is living on a non commutative space time. (It contains a cutoff in momentum space given by the planck mass).

Posted by: laurent freidel on April 5, 2005 11:40 AM | Permalink | Reply to this

Re: Effective action

So if i take your prescription (BF with real coefficient and BB with imaginary one) the pure BF theory cannot even be defined at the quantum level and i have to stop even before taking the first step.

In the pure BF theory (β0\beta\neq0, α=0\alpha=0), the Euclidean action, after integrating out BB is iθ8π 2TrFF\frac{i\theta}{8\pi^2}\int Tr F\wedge F.

The “ii” is important. That’s what makes θ\theta an angle.

Posted by: Jacques Distler on April 5, 2005 1:43 PM | Permalink | PGP Sig | Reply to this

Re: Effective action

In the pure BF theory (β≠0, α=0), the Euclidean action, after integrating out B is iθ8π 2∫TrF∧F.

Ok we agree on that, some miscommunication. So in the BF formulation, you propose that one should take in the exponent i[BF+βBB+iαBBϵ]i[BF +\beta BB +i\alpha BB\epsilon] instead of i[BF+βBB+αBBϵ]i[BF +\beta BB +\alpha BB\epsilon]. I would be happy to understand your reasons or where you disagree with the one i presented.

Posted by: laurent Freidel on April 5, 2005 2:16 PM | Permalink | Reply to this

“i”s

I don’t particularly care about the action with the auxiliary field present (since one has a choice as to whether BB is even or odd under CP, the answer is ambiguous).

In the action with the auxiliary field integrated out, however, it is unambiguous where the "ii"s should go. They go where I said.

Posted by: Jacques Distler on April 5, 2005 4:59 PM | Permalink | PGP Sig | Reply to this
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