A purely geometric path integral for gravity

[marcus]

This paper seems to me especially interesting:

http://arxiv.org/abs/1308.2946
Purely geometric path integral for spin foams
Atousa Shirazi, Jonathan Engle
(Submitted on 13 Aug 2013)
Spin-foams are a proposal for defining the dynamics of loop quantum gravity via path integral. In order for a path integral to be at least formally equivalent to the corresponding canonical quantization, at each point in the space of histories it is important that the integrand have not only the correct phase -- a topic of recent focus in spin-foams -- but also the correct modulus, usually referred to as the measure factor. The correct measure factor descends from the Liouville measure on the reduced phase space, and its calculation is a task of canonical analysis.
The covariant formulation of gravity from which spin-foams are derived is the Plebanski-Holst formulation, in which the basic variables are a Lorentz connection and a Lorentz-algebra valued two-form, called the Plebanski two-form. However, in the final spin-foam sum, one sums over only spins and intertwiners, which label eigenstates of the Plebanski two-form alone. The spin-foam sum is therefore a discretized version of a Plebanski-Holst path integral in which only the Plebanski two-form appears, and in which the connection degrees of freedom have been integrated out. We call this a purely geometric Plebanski-Holst path integral.
In prior work in which one of the authors was involved, the measure factor for the Plebanski-Holst path integral with both connection and two-form variables was calculated. Before one discretizes this measure and incorporates it into a spin-foam sum, however, one must integrate out the connection in order to obtain the purely geometric version of the path integral. To calculate this purely geometric path integral is the principal task of the present paper, and it is done in two independent ways. Gauge-fixing and the background independence of the resulting path integral are discussed in the appendices.
21 pages

Shirazi gave a talk on it which is available as online Pirsa video. http://pirsa.org/13070086 (advance to minute 64:00) The talk is just 20 minutes and helps one get an intuitive appreciation of their result. The measure factor they get seems likely to be right because they calculated it from two different approaches and go the same answer.

Engle has been notably active in perfecting the EPRL (Engle-Pereira-Rovelli-Livine) path integral. Here's a bit of background information: https://www.physicsforums.com/showthread.php?t=706775

The usual form of loop gravity depends on two main variables, a connection ω and a tetrad often denoted by eⁱ which can be imagined as a moving frame, or as a Lorentz algebra valued one-form. That means e∧e is an algebra-valued two-form. (Technically called "Plebanski two-form." These authors work with a two-form X^IJ which is a constant multiple of e∧e (see equation 7 on page 4).

 

[marcus]

Atousa's slides start about 2/3 of the way down this slides PDF
http://pirsa.org/13070086.pdf
Hers are numbers 83 thru 91 of a set of 122, so you look for "83/122" at the lower righthand corner of the screen.
This set of 9 slides is a practical introduction to watching the 20 minute talk (http://pirsa.org/13070086) and the talk provides partial introduction/overview of the paper.

The relation of the two-form X^{I J} and e∧e, where e is the tetrad, appears around slide #5.

 

[marcus]

What this paper does is zip together Covariant (ie. spin foam) loop gravity with the Canonical QG version, making loop more cohesive.

Is covariant loop gravity now sufficiently coherent that it is ready for a regular TEXTBOOK presentation?

Let's review some of Engle's contributions:

1. Joining the main Loop theory to Loop cosmology:
Embedding loop quantum cosmology without piecewise linearity ( http://arxiv.org/abs/1301.6210 )
Other people have made important recent progress on this as well.
There is a growing field of specifically spinfoam cosmology. The restriction to the homogeneous isotropic case has been broken down. I would say the main theory and the cosmology application are now, for all practical purposes, joined, making the full theory potentially testable by observations.

2. The semiclassical limit of the spinfoam path integral. This is summed up in Engle's chapter of the forthcoming Ashtekar Petkov Handbook of Spacetime
==quote http://arxiv.org/abs/1303.4636 ==
As shown in the recent work [22], the extra terms causing this problem are due precisely to the presence of the multiple sectors of solutions to the simplicity constraint presented in section 3.1, as well as the presence of different “orientations” as dynamically determined by the co-tetrad field e^I_μ. Once these sectors and orientations are properly handled [28, 29], one arrives at what is called the proper loop quantum gravity vertex amplitude. Its semiclassical limit includes only the single term consisting in the exponential of i times the classical action,
A⁺_v({λj_f,n_e,f }) ∼ λ⁻¹²C₁ e^iS_R
thereby solving the above problem and giving reason to believe that the resulting spin-foam model will yield a correct classical limit.
==endquote==
Here one sees the classical Regge action S_R = S_Regge recovered. v is a vertex of the spinfoam path integral and the data in curly brackets are edge e and face f information surrounding that vertex.

3. Now the third part of this is to firm up the relation between Covariant (spinfoam, path integral) with the Canonical treatment, and this seems to be what the "Purely geometric path integral" paper of Shirazi Engle is about.
===quote http://arxiv.org/abs/1308.2946 ==
In the path integral approach to constructing a quantum theory, the integrand of the path integral has two important parts: a phase part given by the exponential of i times the classical action, and a measure factor. The form of the phase part in terms of the classical action ensures that solutions to the classical equations of motion dominate the path integral in the classical limit so that one recovers classical physics in the appropriate regime. The measure factor, however, arises from careful canonical analysis, and is important for the path integral to be equivalent to the corresponding canonical quantum theory. In most theories, this means that it is important, in particular, in order for the path integral theory to have such elementary properties as yielding a unitary S-matrix that preserves probabilities. The importance of having the correct measure factor is thus quite high.

Spin-foams are a proposal for defining the dynamics of loop quantum gravity via path integral. In order for a path integral to be at least formally equivalent to the corresponding canonical quantization, at each point in the space of histories it is important that the integrand have not only the correct phase -- a topic of recent focus in spin-foams -- but also the correct modulus, usually referred to as the measure factor. The correct measure factor descends from the Liouville measure on the reduced phase space, and its calculation is a task of canonical analysis.

The covariant formulation of gravity from which spin-foams are derived is the Plebanski-Holst formulation, in which the basic variables are a Lorentz connection and a Lorentz-algebra valued two-form, called the Plebanski two-form. However, in the final spin-foam sum, one sums over only spins and intertwiners, which label eigenstates of the Plebanski two-form alone. The spin-foam sum is therefore a discretized version of a Plebanski-Holst path integral in which only the Plebanski two-form appears, and in which the connection degrees of freedom have been integrated out. We call this a purely geometric Plebanski-Holst path integral.

In prior work in which one of the authors was involved, the measure factor for the Plebanski-Holst path integral with both connection and two-form variables was calculated. Before one discretizes this measure and incorporates it into a spin-foam sum, however, one must integrate out the connection in order to obtain the purely geometric version of the path integral. To calculate this purely geometric path integral is the principal task of the present paper, and it is done in two independent ways. Gauge-fixing and the background independence of the resulting path integral are discussed in the appendices...
==endquote==

 

[marcus]

So to summarize the three points, Engle (and a number of others) have been making progress on three fronts that have to do with achieving greater coherence.

1. Joining the cosmology application to the main theory in a seamless way.

2. Connecting spinfoam path integral to Regge action, assuring proper semiclassical and likely classical limits. Note the attention to simplicity constraints.

3. Confirming commonality of covariant and canonical formulation of the theory.

=======================

The field is attracting researchers. But has covariant loop gravity become sufficiently coherent that it's ready for regular introductory textbook treatment? Has it solidified into a definite shape?

What is loop gravity, essentially? I think at a basic level it is what arises from taking BOTH quantum field theory and general relativity seriously, and striving to resolve the tension between the two.

In that sense it very likely does not NEED any further empirical data to guide its development. It can rely on the masses of data supporting two successful prior theories---which are in conflict and whose conflict needs to be reconciled. There are historical examples of this sort of thing.

 

[marcus]

The development of the theory is remarkably cogent. One can argue fairly convincingly that each step along the way is logically necessary or at least well motivated by analogies with what has been successful in other quantum field theory contexts.

There's one outstanding issue as I see it which is the sign of the tetrad. The theory has to be based on the tetrad, not on a metric, because of fermion matter. The sign of the tetrad determines which way fermion phase rotates. It is possible to have an "anti"-spacetime in which the tetrad orientation everywhere is reversed and all the phases rotate oppositely.

In Jon's approach the opposite sign tetrad is SUPPRESSED by hand, so to speak. That makes everything turn out nice and realistic. But maybe there is another way to handle this, and something to learn from the mathematical possibility of this kind of reversal of the orientation of geometry.
I'll set that aside for the moment, too difficult to be worrying about it.

The main thing is, aside from that, to get a sense of the logic of how the theory is developed.
Basically it stems from the REGGE version of GR, modified to be Einstein-Cartan instead of pure Einstein, and the realization that distance MAKES NO SENSE below around Planck scale. You have to give up on Riemann geometry below a certain scale. Have to go, but will try to sketch the ideas more later.

 

[marcus]

Lengths less than Planck being meaningless makes us quit using the classical manifold as a model of realworld geometry. So we have to find a non-manifold implementation of GR, which can accept a minimal length. Classical geometry/gravity as we know it is GR, and the only non-manifold implementation I'm aware of is Regge.
In Regge context space is made of tetrahedra, and spacetime is made of pentachora (foursimplices=pentachora).

We're used to a classical tetrahedron's shape being determined by 6 numbers, or more exactly by the lengths of the six segments that frame it. But by analogy with the way angular momentum works at small scale, a fuzzy tet shape is given b only 5 numbers (a volume and four areas). That amount of information is all Heisenberg uncertainty let's you know about the tet.

So whatever theory we come up with for quantum gravity/geometry will not be based on a manifold but geometry made of fuzzy tets and the like. I'll try to get back to this later. The challenge will be to marshall fuzzy tets etc into some kind of choreographic order that resembles GR at large scale.

You might think this line of reasoning would lead to something like Loll et al causal dynamical triangulations (and it might) but my hunch is that CDT has no intrinsic minimal length scale and that its tetrahedra are classical, not fuzzy.

 

[marcus]

What I have to motivate or explain is why one does not stay with a Regge-like triangulation geometry made of fuzzy tets, fuzzy pents and so on, but instead we go over to the DUAL of a fuzzy triangulation.
Each pent is made to correspond to a vertex. (imagined as the centroid of that foursimplex)
Each tet corresponds to an edge (the edge radiating from the center spearing through that tetrahedral wall of the pent).
The triangle side where two tets meet corresponds to a face spanning the two edges dual to those tets... Sorry have to go. Back later...

OK I'm back. The dual Δ* represents the essential information about the original triangulation Δ.
Each vertex v stands for a chunk of space-time, a five-walled fuzzy pent.
Each edge radiating out from v represents a wall, a fuzzy tet.
And if a pair of edges stand for two CONTIGUOUS tets then those two edges are connected by a FACE, to show that their corresponding tets touch. The face is dual to a fuzzy triangle in the original Δ

So the dual Δ* transcribes all the essential information, and though I'm not emphasizing this, it is more general: you might have a vertex from which more than 5 edges issue, corresponding classically to a polyhedron in 4d with more walls than a 4-simplex. This kind of structure can describe spacetime geometry built not only of fuzzy simplices but also of fuzzy polyhedra more general than simplices. The structure is called a cell complex.

What I'm arguing is, given that we can't expect spacetime to be a 4d manifold, the simplest most reasonable thing to try is triangulation by fuzzy simplices, and a cell complex is the most convenient way to describe such a triangulation---moreover GR (the version able to handle fermions) can be implemented.

Anything else, seen from here, looks like a detour.

And the question I'm asking about Engle's version of the story, which looks like it is taking shape in a fairly comprehensive consistent way, is whether it is coherent enough at this point to be presentable in a graduate-level textbook helping to define the field.

 

[tom.stoer]

marcus, thanks for the summary.

Of course I wanted to start reading this paper immediately after its appearence on arxiv, but I have to admit that I did not hab time to do that. So let me ask a couple of questions to you instead: if Engle has the correct phase and measure, ...
1) what is the next step to prove absence of anomalies?
2) what is the next step to reconstruct a Hamiltonian
3) does his formulation address the "off-shell" closure of constraints? I guess no b/c he integrated them out

A question which I wanted to ask a couple of times: starting with triangulation, PL manifolds or something like that means that dual graphs are valid only if they are indeed dual to a triangulation. Taking graphs seriously we find that the set of general graphs is larger than the set of graphs having a dual triangulation. So the questions are
4) were exactly are the graphs restricted to the ones dual to a triangulation?
5) what is the condition to eliminate graphs no respecting this duality
(Unfortunately all "modern" treatments, especially SF models, are formulated with triangulations only and cannot address these questions; they have to be answered in a formulation using graphs where you can explicitly formulate a condition which "kills" the non-dual graphs)

 

[marcus]

marcus, thanks for the summary.

Of course I wanted to start reading this paper immediately after its appearence on arxiv, but I have to admit that I did not hab time to do that. So let me ask a couple of questions to you instead: if Engle has the correct phase and measure, ...
1) what is the next step to prove absence of anomalies?
2) what is the next step to reconstruct a Hamiltonian
3) does his formulation address the "off-shell" closure of constraints? I guess no b/c he integrated them out

A question which I wanted to ask a couple of times: starting with triangulation, PL manifolds or something like that means that dual graphs are valid only if they are indeed dual to a triangulation. Taking graphs seriously we find that the set of general graphs is larger than the set of graphs having a dual triangulation. So the questions are
4) were exactly are the graphs restricted to the ones dual to a triangulation?
5) what is the condition to eliminate graphs no respecting this duality
(Unfortunately all "modern" treatments, especially SF models, are formulated with triangulations only and cannot address these questions; they have to be answered in a formulation using graphs where you can explicitly formulate a condition which "kills" the non-dual graphs)

These are great questions! and they really should be asked of Jonathan Engle.
Why don't you write him?
jonathan.engle@fau.edu
Maybe also send a copy to his student Atousa Shirazi achahars@fau.edu as well.

 

[marcus]

Hi Tom, hopefully you've had time to take another look at the paper and can comment a bit. I see Etera Livine's October paper on spin network coarse-graining (along with Engle's work) as the two most consequential (specifically loop) developments appearing so far this year. I can't choose one over the other--torn between them, so to speak.

I started a thread about Livine's coarse-graining paper. https://www.physicsforums.com/showthread.php?t=717348 It would be good to hear any thoughts you might have on it.

About your NEXT STEP questions, you may have noticed that Engle Shirazi actually TELL US at the end what they think the next step has to be. They say their next paper will be about that. They have found the measure factor (they are pretty sure they are right about that) but they have not yet DISCRETIZED it to live on a spinfoam, so that it will be calculable purely from the spinfoam data.
==quote conclusions Engle Shirazi==
In the present work, we have calculated the pure geometric form of this path integral, whose discretization will yield the necessary measure factor for spin-foams. We have calculated the measure for this path integral in two independent ways (1.) by integrating out the connection from the path integral derived in [5], and (2.) by ensuring consistency with the canonical ADM path integral. Both methods lead to the same final measure factor, providing a check on the detailed powers of the space-time and spatial volume elements present. The next step is to discretize this measure on a spin-foam cell complex, expressing it directly in terms of spins and intertwiners. This will involve non-trivial choices which will in part be fixed by considerations of gauge-invariance. This will be discussed in a later, complementary paper.
==endquote==
Who knows if this will succeed? Hopefully it will and they can take the next step. I think this must be done before they can proceed to address the other questions you raised.