Remaining problems w LQG (and cosmology application)

[tom.stoer]

Yes, and every 3 or 4 years it magically becomes a different glass

The Hamiltonian glass remains the same, it remains half-empty and anomalous ...

In any case it is an exciting rapidly moving field.

I definately agree.

 

[marcus]

In any case it is an exciting rapidly moving field.

I definately agree.

I'm glad you agree! I heard recently that the next Loops conference will be at Erlangen, which I believe is where you live. This could be fun for you, to see these people in action, if there is enough room and you can attend some of the talks.

The field has essentially been "taken over" by a growing company of young people. Or so I think. It is essentially out of control and there is a lot of unanticipated unpredictable creativity. It could be fun to see what it is like in year or two. Actually I don't know WHEN the next Loops is planned to be. It has been held roughly every two years (biennial) so that would suggest 2015 in Erlangen.

BTW, Tom, I was not aware that there IS a "Hamiltonian glass."

Or maybe it is gathering dust on a high shelf. I think there is a "paradigm shift" occurring about how we think of general covariant QFT. Briefly, there is no fundamental time so there is no energy, and so the Dirac quantization is being abandoned in favor of Oeckl "boundary formulation".

To me as onlooker this makes a lot of sense because the right formulation should support BOTH general covariant QFT and general covariant SM (stat. mech.)
But for doing statistical mechanics and thermodynamics, I think the Dirac Hamiltonian constraint is USELESS. Please correct me if I'm wrong about this, Tom. This is how it looks to me. I think the Oeckl boundary formulation is really what one needs.

The boundary of a spacetime region---the interface thru which information passes---fundamentally DEFINES the region for me. There is no "initial" and "final" because there is no preferred time. So the Hilbertspace (and the amplitudes of vectors in it) must refer the boundary of the process inside. It's a kind of philosophical argument for why the Dirac Hamiltonian constraint is inappropriate for doing QFT and SM together in general covariant setting

 

[marcus]

In connection with Freidel's talk, I mentioned his August 2013 paper with Jon Ziprick titled "Spinning geometry=Twisted geometry". A new type of cellular decomposition is being defined, apparently very useful but hard for me to picture at first. I found it helpful to watch a more pedagogical/pictorial talk by Ziprick which takes the viewer though the construction, and describes the historical development from simpler cellular decompositions.
http://pirsa.org/displayFlash.php?id=13070057 Ziprick's is the first talk on the recording.

Those who are especially interested in the Hamiltonian approach to QG might be interested that this talk was given in the Canonical approach parallel session and at the end Ziprick explains that the new cellular decomposition provides an opening for the Canonical approach. I didn't understand how that is supposed to work, but it is one of his conclusions at the end.
One can get the slides PDF here:
http://pirsa.org/13070057
They go up to 26/98. Conclusions slide (25/98) says:
==quote==
Spinning geometries are isomorphic to twisted geometries, and represent the loop gravity phase space.

They are continuous, and have torsion and curvature supported on a closed network of helices.

The axes of the helices are defined by the holonomy data.

This is the most general cellular space with vanishing curvature and torsion outside of edges.

Spinning geometries provide a means to define continuous (A, e) fields from holonomy-flux data.

This opens a new door to dynamics, allowing us to draw from the general relativistic equations of motion.
==endquote==

 

[marcus]

Jon Engle is the "E" in EPRL and his research advances in the past couple of years have contributed substantially to putting LQG (and its application to cosmology) in good order. You may recall that his January paper ( http://arxiv.org/abs/1301.6210 ) showed how to embed the cosmology sector in the full theory. Essentially, we no longer have a separate "LQC" as distinct from LQG.
This month's paper by Engle (with his student Shirazi) is likely to turn out, I think, to be an important one. It takes a direction suggested by Thomas Thiemann (determining the measure factor in the Plebanski-Holtz path integral) and builds on work Engle did earlier with Thiemann and Muxin Han.
http://arxiv.org/abs/1308.2946
Purely geometric path integral for spin foams
Atousa Chaharsough 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

For convenience I'll post the arxiv links for this paper's references [5 - 8]
[5] J. Engle, M. Han, and T. Thiemann, “Canonical path integral measures for Holst and Plebanski gravity. I. Reduced phase space derivation,” Class. Quant. Grav., vol. 27, p. 235024, 2009. http://arxiv.org/abs/0911.3433
[6] J. Engle, “The Plebanski sectors of the EPRL vertex,” Class. Quant. Grav., vol. 28, p. 225003, 2011. http://arxiv.org/abs/1107.0709 Corrigendum: Class. Quant. Grav. vol. 30, p. 049501, 2013. http://arxiv.org/abs/1301.2214
[7] J. Engle, “A proposed proper EPRL vertex amplitude,” Phys. Rev. D, vol. 87, p. 084048, 2013. http://arxiv.org/abs/1111.2865
[8] J. Engle, “A spin-foam vertex amplitude with the correct semiclassical limit,” Phys. Lett. B,
vol. 724, pp. 333–337, 2013. http://arxiv.org/abs/1201.2187

 

[marcus]

The 2009 Engle Han Thiemann paper (written while Engle was postdoc at Erlangen) was intended to be the first of a series, “Canonical path integral measures for Holst and Plebanski gravity. I."
However number II of the series never appeared.
This August 2013 paper of Engle Shirazi could be considered the sequel--the continuation of the 2009 work--shifting the focus to spin foams.
Here's the 2009 paper's abstract:
Canonical path integral measures for Holst and Plebanski gravity. I. Reduced Phase Space Derivation
Jonathan Engle, Muxin Han, Thomas Thiemann
(Submitted on 17 Nov 2009)
An important aspect in defining a path integral quantum theory is the determination of the correct measure. For interacting theories and theories with constraints, this is non-trivial, and is normally not the heuristic "Lebesgue measure" usually used. There have been many determinations of a measure for gravity in the literature, but none for the Palatini or Holst formulations of gravity. Furthermore, the relations between different resulting measures for different formulations of gravity are usually not discussed.
In this paper we use the reduced phase technique in order to derive the path-integral measure for the Palatini and Holst formulation of gravity, which is different from the Lebesgue measure up to local measure factors which depend on the spacetime volume element and spatial volume element.
From this path integral for the Holst formulation of GR we can also give a new derivation of the Plebanski path integral and discover a discrepancy with the result due to Buffenoir, Henneaux, Noui and Roche (BHNR) whose origin we resolve. This paper is the first in a series that aims at better understanding the relation between canonical LQG and the spin foam approach.
27 pages

==quote page 1 of EHT 2009==
Richard Feynman, in the course of his doctoral work, developed the path integral formulation of quantum mechanics as an alternative, space-time covariant description of quantum mechanics, which is nevertheless equivalent to the canonical approach [1]. It is thus not surprising that the path integral formulation has been of interest in the quantization of general relativity, a theory where space-time covariance plays a key role. However, once one departs from the regime of free, unconstrained systems, the equivalence of the path integral approach and canonical approach becomes more subtle than originally described by Feynman in [1]. In particular, in Feynman’s original argument, the integration measure for the configuration path integral is a formal Lebesgue measure; in the interacting case, however, in order to have equivalence with the canonical theory, one cannot use the naive Lebesgue measure in the path integral, but must use a measure derived from the Liouville measure on the phase space [2].
Such a measure has yet to be incorporated into spin-foam models, which can be thought of as a path-integral version of loop quantum gravity (LQG) [3, 4]. Loop quantum gravity is an attempt to make a mathematically rigorous quantization of general relativity that preserves background independence — for reviews, see [8, 6, 7] and for books see [9, 10]. Spin-foams intend to be a path integral formulation for loop quantum gravity, directly motivated from the ideas of Feynman appropriately adapted to reparametrization-invariant theories [4, 5]. Only the kinematical structure of LQG is used in motivating the spin-foam framework. The dynamics one tries to encode in the amplitude factors appearing in the path integral which is being replaced by a sum in a regularisation step which depends on a triangulation of the spacetime manifold. Eventually one has to take a weighted average over these (generalised) triangulations for which the proposal at present is to use methods from group field theory [3]. The current spin foam approach is independent from the dynamical theory of canonical LQG [11] because the dynamics of canonical LQG is rather complicated. It instead uses an apparently much simpler starting point: Namely, in the Plebanski formulation [14], GR can be considered as a constrained BF theory, and treating the so called simplicity constraints as a perturbation of BF theory, one can make use of the powerful toolbox that comes with topological QFT’s [12]. It is an unanswered question, however, and one of the most active research topics momentarily1, how canonical LQG and spin foams fit together. It is one the aims of this paper to make a contribution towards answering this question.
==endquote==

==quote pages 1,2 of Shirazi Engle 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 path integral approach to quantum gravity in which one does not sum over classical gravitational histories, but rather quantum histories arising from canonical quantization. Specifically, in spin-foams, one sums over histories of canonical states of loop quantum gravity. These histories possesses a natural 4-dimensional space-time covariant interpretation, whence each can be thought of as a quantum space-time. This approach allows one to retain the understanding gained from loop quantum gravity, such as the discreteness of area and volume spectra, while at the same time formulating the dynamics in a way that makes space-time symmetries more manifest [1].

The starting point for the derivation of spin-foams is the Plebanski-Holst formulation of gravity [2–6], in which the basic variables are a connection ω and what is called the Plebanski two-form, Σ. However, in the final spin-foam sum, the connection ω is usually not present, and one sums over only spins and intertwiners, which determine certain eigenstates of Σ alone. Because of this, the continuum path integral most directly related to the spin-foam sum is a Plebanski-Holst path integral in which only Σ appears, and in which the connection has been integrated out. We call such a path integral purely geometric because Σ directly determines the geometry of space-time.

Because of the quantum mechanical nature of the histories used in spin-foams, ensuring that the summand has the required phase part and measure factor is not completely trivial. Only within the last couple years has the correct phase part been achieved [7, 8]. Regarding the measure factor, a first step has been carried out in the work [5], where the correct measure factor is calculated for the Plebanski-Holst path integral with both ω and Σ present. However, until now, the measure factor for the path integral with Σ alone, necessary for spin-foams, had not yet been calculated. To carry out this calculation is the main purpose of the present paper. In order to be sure about all numerical factors, we do this in two different ways: (1.) by starting from the path integral in [5] and then integrating out the connection degrees of freedom, and (2.) by starting from the ADM path integral and introducing the necessary variables from there. We find that these two ways of calculating the measure factor exactly match, as must be the case, as the canonical measure factor ultimately descends from the Liouville measure on the reduced phase space, which is independent of the formulation of gravity used [5].

The path integral derived in this paper is ready to be discretized and translated into a spin-foam model, a task which will be carried out in forthcoming work. Furthermore, when this is accomplished, we would like to emphasize that, because both primary and secondary simplicity constraints are already incorporated in the continuum path integral [5,9], they will be automatically incorporated in the resulting spin-foam model as well.
==endquote==

 

[marcus]

The conclusion paragraphs of the Shirazi Engle paper give what I think is currently the most understandable overview of the present status of the (LQG) spin foam program. People often ask about this, so I'll quote.

==Shirazi Engle http://arxiv.org/abs/1308.2946 page 13==
Spin-foams are a path integral approach to quantum gravity based on the Plebanski-Holst formulation of general relativity. The basic variables of the Plebanski-Holst formulation are a Lorentz connection and the Plebanski two-form, the pull-backs of which to any Cauchy surface are conjugate to each other. The Plebanski two-form by itself completely determines the space-time geometry. In spin-foams, one sums over histories of spins and intertwiners which label eigenstates of the Plebanski two-form. Because of this, the spin-foam sum may be understood as a discretization of a Plebanski-Holst path integral in which the connection degrees of freedom have been integrated out — that is, it is a discretization of what we have called a purely geometric Plebanski-Holst path integral.

In order to ensure that a path integral quantization be equivalent to canonical quantization, it is important that the correct canonical path integral measure be used. The path integral measure for Plebanski-Holst, with both connection and Plebanski two-form variables present, was calculated in the earlier work [5]. 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==

This seems to be a systematic top-down approach with a reasonable chance of resulting in consistent path integral QG. Many of the details have been attended to, and there is a clear next step.

Among the details one can see Engle and others attending to: relation of cosmology sector to full theory ("LQC" seems to mean several quite different things to different people so one has to be cautious using the the word), and relation of covariant to canonical LQG, and simplicity constraints (mentioned earlier), and rigorous definition of the measure factor in the path integral.

For Engle's treatment of LQC as of January see http://arxiv.org/abs/1301.6210
Embedding loop quantum cosmology without piecewise linearity
but the approach has improved significantly since then, e.g. go to minute 83:00 of http://pirsa.org/displayFlash.php?id=13070039 for a 20 minute talk. This is how he embeds the cosmology sector into full LQC as of July.
Buffering will take roughly 8-10 minutes--that's when you advance thru the the first 80 minutes of the recording, with sound muted while you work on something else.
Slides PDF is available at http://pirsa.org/13070039
The July 2013 Diffeomorphism invariance and dynamical symmetry reduction. The student collaborator's name is Mathew Hogan. he also gives a talk. I'll look for the link to the recording.
http://pirsa.org/13070087 (51:00)

 

[marcus]

Atousa C. Shirazi gave a short talk at Loops 2013 on the material in her paper with Engle.
http://pirsa.org/13070086 (talk starts around minute 64:00)
The full list of recorded conference talks
http://pirsa.org/C13029
Alphabetical list by speaker:
https://www.physicsforums.com/showthread.php?p=4468610#post4468610
Atousa somehow got listed in the C's because of her middle name. Sorry :-(

 

[marcus]

I think it is misleading to depict Loop gravity (with cosmo sector) as a "glass half empty" that has been sitting around half empty--IOW a static picture. That is an impression someone might have who had not been paying attention.

Paying attention would mean reading an up-to-date review article on spinfoam like
http://arxiv.org/abs/arXiv:1303.4636 (chapter of Springer Handbook of Spacetime, in press)

and I think to get an idea of where Loop cosmology is at present one would need to watch a couple of 20 minute Pirsa talks, namely
http://pirsa.org/13070039 starting at minute 83:00
http://pirsa.org/13070087 starting at minute 51:00

The corresponding articles are in preparation and I expect them to be posted reasonably soon. The cosmology work is new enough that for now we just have the two conference talks.

The point is that there has been this long-outstanding problem of how do you put Loop cosmology into the full theory. An homogenous isotropic sector of the full theory ("symmetry reduction") has to be defined and it has to be done in a diffeomorphism invariant way.
IOW quantize first, then reduce, and do it without breaking general covariance.
The title of the relevant Pirsa talk is Diffeomorphism invariance and dynamic symmetry reduction.

 

[sheaf]

I think it is misleading to depict Loop gravity (with cosmo sector) as a "glass half empty" that has been sitting around half empty--IOW a static picture. That is an impression someone might have who had not been paying attention.

But LQG must be a static field - there's no Hamiltonian so it can't have evolved!

 

[marcus]

But LQG must be a static field - there's no Hamiltonian so it can't have evolved!

A witty in-joke *chuckle chuckle* obviously as a field of research LQG is evolving rapidly and, I think, actually approaching its goal of a path integral formulation (with zero Hamiltonian).

Not everyone reading may realize, so I'll mention it, in the case of a general covariant theory the Hamiltonian must be identically zero and so is essentially useless for representing time-evolution.
To get a useful grip on dynamics you need to take the path integral approach. How to formulate a quantum spacetime geometry path integral?

The Springer press is bringing out a Handbook of Spacetime, which should give some idea of what the currently prevailing wisdom about that is. I think in terms of research communities working specifically on that (quantum space-time geometry path integral) we are seeing the most activity, largest numbers of young researchers entering the field, fastest growth in the area of spinfoam path integral.

As I said, covariance dictates that the Hamiltonian must be identically zero on the physical states. But that does not mean it isn't interesting! As soon as one has a spinfoam version of dynamics one should check to see what the corresponding Hamiltonian operator is and make sure it vanishes. Wieland posted a paper along those lines recently. Even though the Hamiltonian is not the primary dynamical method, people still like to check consistency.

If you pay attention to the field you will of course realize there is remarkable progress in this area as well. For example see Wieland's January paper:
http://arxiv.org/abs/1301.5859
Hamiltonian spinfoam gravity
This paper presents a Hamiltonian formulation of spinfoam-gravity, which leads to a straight-forward canonical quantisation. To begin with, we derive a continuum action adapted to the simplicial decomposition. The equations of motion admit a Hamiltonian formulation, allowing us to perform the constraint analysis...Next, we canonically quantise. Transition amplitudes match the EPRL (Engle--Pereira--Rovelli--Livine) model, the only difference being the additional torsional constraint affecting the vertex amplitude.
28 pages, 2 figures

 

[MTd2]

Boundary formalism, PI, SF, ... doesn't help per se.

Either you have to define a consistent, anomaly-free constraint algebra incl. Hamiltonian constraint,
or you have to define a consistent, anomaly-free path integral including measure, effective action, ...

I don't see that either of these approaches has succeeded in providing such a consistent definition. For all constructions it is unclear whether this is the case.

Quoting you for the sake of http://arxiv.org/abs/1310.7786

"Group field theory as the 2nd quantization of Loop Quantum Gravity

Daniele Oriti"

To what degree does this paper solves this problems?

 

[tom.stoer]

it's the topmost paper on my desk I'll try to read asap

 

[marcus]

BTW I recently saw the abstract summary of Wolfgang Wieland's thesis which I expect may be posted later this year. There are four main results, which I think are all out there on arxiv already in individual papers. Wieland already has a number of research papers and my impression is the thesis kind of packages their results in a coherent way, together with some explanatory detail that makes them more accessible. Should be useful (to me and likely others). Here's how I'd paraphrase the four main results, or groups of results.

1. The canonical analysis is completed with the original SELF-DUAL Ashtekar variables (yet without dropping the B-I parameter) and the equations retain the nice polynomial form. The "reality condition" required by the original complex Ashtekar variables becomes the requirement that a spatial spin connection be torsionless.

2. Freidel-Speziale's result linearizing the Loop gravity phase space using TWISTORS is generalized from SU(2) to SL(2,C). We're talking about the phase space of selfdual holonomy-flux variables, as per #1.

3. Spinfoam dynamics is studied in terms of these twistorial variables. One gets a "spinfoam Hamiltonian", that is a Hamiltonian based on a discrete combinatorial structure. I could be mistaken but I think this is a development away from the differential manifold (continuum) basis one usually sees in the canonical approach.

4. Quantization. Since the action is a polynomial in the spinors, canonical quantisation is rather straightforward. Transition amplitudes reproduce the EPRL spinfoam model.

Anyone wanting more information on particular can look through the arxiv listing of recent work.

An interesting characteristic of this research is that it all takes place in a sort of middle ground in between the canonical Hamiltonian LQG approach on the one hand and the covariant path integral spinfoam approach on other hand.

 

[marcus]

Another major paper that ties up loose ends in Loop gravity using the twistor formulation that has taken center stage during the past two years.

http://arxiv.org/abs/1311.3279
Null twisted geometries
Simone Speziale, Mingyi Zhang
(Submitted on 13 Nov 2013)
We define and investigate a quantisation of null hypersurfaces in the context of loop quantum gravity on a fixed graph. The main tool we use is the parametrisation of the theory in terms of twistors, which has already proved useful in discussing the interpretation of spin networks as the quantization of twisted geometries. The classical formalism can be extended in a natural way to null hypersurfaces, with the Euclidean polyhedra replaced by null polyhedra with space-like faces, and SU(2) by the little group ISO(2). The main difference is that the simplicity constraints present in the formalims are all first class, and the symplectic reduction selects only the helicity subgroup of the little group. As a consequence, information on the shapes of the polyhedra is lost, and the result is a much simpler, abelian geometric picture. It can be described by an Euclidean singular structure on the 2-dimensional space-like surface defined by a foliation of space-time by null hypersurfaces. This geometric structure is naturally decomposed into a conformal metric and scale factors, forming locally conjugate pairs. Proper action-angle variables on the gauge-invariant phase space are described by the eigenvectors of the Laplacian of the dual graph. We also identify the variables of the phase space amenable to characterize the extrinsic geometry of the foliation. Finally, we quantise the phase space and its algebra using Dirac's algorithm, obtaining a notion of spin networks for null hypersurfaces. Such spin networks are labelled by SO(2) quantum numbers, and are embedded non-trivially in the unitary, infinite-dimensional irreducible representations of the Lorentz group.
22 pages, 3 figures

One way to think about what's going on here is this: to arrive at a manifestly covariant formulation the spinfoam transition amplitudes have to refer to the entire boundary not merely to "initial" and "final" spacelike components.

Perhaps it's significant that among the first works referenced is some research by various from among Alexandrov, Conrady, Hnybida, Kadar. And also referenced is some joint work in progress by Alexandrov and Speziale.