Progress on LQG dynamics (the Master Constraint program)

[marcus]

Five new papers by Thomas Thiemann and Bianca Dittrich

http://arxiv.org/abs/gr-qc/0411138
Testing the Master Constraint Programme for Loop Quantum Gravity I. General Framework
42 pages


"Recently the Master Constraint Programme for Loop Quantum Gravity (LQG) was proposed as a classically equivalent way to impose the infinite number of Wheeler-DeWitt constraint equations in terms of a single Master Equation. While the proposal has some promising abstract features, it was until now barely tested in known models. In this series of five papers we fill this gap, thereby adding confidence to the proposal. We consider a wide range of models with increasingly more complicated constraint algebras, beginning with a finite dimensional, Abelian algebra of constraint operators which are linear in the momenta and ending with an infinite dimensional, non-Abelian algebra of constraint operators which closes with structure functions only and which are not even polynomial in the momenta. In all these models we apply the Master Constraint Programme successfully, however, the full flexibility of the method must be exploited in order to complete our task. This shows that the Master Constraint Programme has a wide range of applicability but that there are many, physically interesting subtleties that must be taken care of in doing so. In this first paper we prepare the analysis of our test models by outlining the general framework of the Master Constraint Programme. The models themselves will be studied in the remaining four papers. As a side result we develop the Direct Integral Decomposition (DID) for solving quantum constraints as an alternative to Refined Algebraic Quantization (RAQ)."

http://arxiv.org/abs/gr-qc/0411139
Testing the Master Constraint Programme for Loop Quantum Gravity II. Finite Dimensional Systems
23 pages

"This is the second paper in our series of five in which we test the Master Constraint Programme for solving the Hamiltonian constraint in Loop Quantum Gravity. In this work we begin with the simplest examples: Finite dimensional models with a finite number of first or second class constraints, Abelian or non-Abelian, with or without structure functions."

http://arxiv.org/abs/gr-qc/0411140
Testing the Master Constraint Programme for Loop Quantum Gravity III. SL(2,R) Models
33 pages

"This is the third paper in our series of five in which we test the Master Constraint Programme for solving the Hamiltonian constraint in Loop Quantum Gravity. In this work we analyze models which, despite the fact that the phase space is finite dimensional, are much more complicated than in the second paper: These are systems with an SL(2,R) gauge symmetry and the complications arise because non-compact semisimple Lie groups are not amenable (have no finite translation invariant measure). This leads to severe obstacles in the refined algebraic quantization programme (group averaging) and we see a trace of that in the fact that the spectrum of the Master Constraint does not contain the point zero. However, the minimum of the spectrum is of order hbar^2 which can be interpreted as a normal ordering constant arising from first class constraints (while second class systems lead to hbar normal ordering constants). The physical Hilbert space can then be be obtained after subtracting this normal ordering correction."

http://arxiv.org/abs/gr-qc/0411141
Testing the Master Constraint Programme for Loop Quantum Gravity IV. Free Field Theories
23 pages

"This is the fourth paper in our series of five in which we test the Master Constraint Programme for solving the Hamiltonian constraint in Loop Quantum Gravity. We now move on to free field theories with constraints, namely Maxwell theory and linearized gravity. Since the Master constraint involves squares of constraint operator valued distributions, one has to be very careful in doing that and we will see that the full flexibility of the Master Constraint Programme must be exploited in order to arrive at sensible results."

http://arxiv.org/abs/gr-qc/0411142
Testing the Master Constraint Programme for Loop Quantum Gravity V. Interacting Field Theories
20 pages

"This is the final fifth paper in our series of five in which we test the Master Constraint Programme for solving the Hamiltonian constraint in Loop Quantum Gravity. Here we consider interacting quantum field theories, specifically we consider the non-Abelian Gauss constraints of Einstein-Yang-Mills theory and 2+1 gravity. Interestingly, while Yang-Mills theory in 4D is not yet rigorously defined as an ordinary (Wightman) quantum field theory on Minkowski space, in background independent quantum field theories such as Loop Quantum Gravity (LQG) this might become possible by working in a new, background independent representation."

 

[marcus]

Here is Thomas Thiemann's homepage at the Albert Einstein Institute (Golm)
http://www.aei.mpg.de/cgi-bin/interface/people.cgi?key=thiemann
It has a photo.

Bianca is also one of the graduate students at AEI-Golm
but her homepage has no photo.

She is also parttime at Perimeter and so you can see her homepage there:

Just go here
http://www.perimeterinstitute.ca/people/index.php
and select Graduate Students from the menu
and click on her name.

there is also the Audio and Slides you can download from Bianca's talk
at PennState about the Master Constraint program. She shows remarkable patience with some unnamed excited person in the audience who keeps talking as if it was his Seminar Talk and not hers. I like the fact that she never seems threatened or nervous in spite of a difficult lecture situation.

Naturally this seminar talk has a lot of the same material as these 5 papers that they just posted---but it has audio and slides, so you can see the drawn diagrams and get some extra queues. So it is a possibly useful supplement. I will get the PennState link

Yes, go here
http://phys.psu.edu/events/index.html
Select "this semester" in other words Spring 2004 semester.
Scroll down to 9 February, it is Thomas Thiemann talk about Master Constraint
http://phys.psu.edu/events/index.html?event_id=847;event_type_ids=0;span=2003-12-26.2004-05-31

Then also scroll down to 13 February, it is Bianca also about master constraint, the testing in various situations.
http://phys.psu.edu/events/index.html?event_id=850;event_type_ids=0;span=2003-12-26.2004-05-31

both are good talks and give extra insight that you might not get in the papers, so it is complementary.

 

[selfAdjoint]

I read enough of the first paper to see some interesting things..

1. Thiemann accepts anomalies. His Master Constraint program is not intended to eliminate them, but to push off the encounter with them until you actually start doing dynamics with it.

2. Refined quantization/group averaging has been replaced (I didn't yet read far enough to see just how).

3. There's still a long row to hoe; all these papers do is demonstrate how useful the Phoenix program will be, if it works. They still have to show it works (sort of the inverse of string theory, where it works magnificently but they still have to show it can be useful).

 

[selfAdjoint]

Master Constraint Program I: What is the Master Constraint?

I am going to post here what I have come to understand of the Master Constraint Program, why it exists, what it does, and the hope for what it can achieve. I am going to build this post be successive edits, working back and forth between various papers and gettting quotes. I will cite the link for each quote as I post it. I will cite the series of papers by Dittrich and Thiemann on Testing the Master Constraint Progem as Testing I, II, and so forth.

First of all, the Master Constraint is a replacement for The Hamiltonian Constraint.

What is the Hamiltonian Constraint?
I haven't been able to find a good cite for the definition, so here goes.
Back in the day, Wheeler and deWitt worked on a program to quantize GR that they called geometridynamics. As part of it, they developed an expression for the Hamiltonian of GR. Lo and behold this expression, valid at every point of spacetime, was (1) identically equal to zero, and (2) a multiple of the square root of the determinant of the metric tensor at the point, hence non-polynomial.

The first property provides a condition that is always true wherever and whenever you are in spacetime; this expression, evaluated at your therethen, is zero. This is the Hamiltonian constraint, and every theory that proposes to quantize gravity must deal with it. The second prperty means none of the methods traditionally used to cope with a constraint will work.

Why does the Hamiltonian constraint need to be replaced?
The Hamiltonian Constraint is not invariant under diffeomorphism. So you cannot solve the Hamiltonian constraint as it stands in a diffeomorphic theory.

In short, progress on the solution of the Hamiltonian constraint
has been slow because of a technical reason: the Hamiltonian constraints themselves are not
spatially diffeomorphism invariant. This means that one cannot first solve the spatial diffeomorphism
constraints and then the Hamiltonian constraints because the latter do not preserve
the space of solutions to the spatial diffeomorphism constraint [10]. On the other hand, the
space of solutions to the spatial diffeomorphism constraint [10] is relatively easy to construct
starting from the spatially diffeomorphism invariant representations on which LQG is based [11]
which are therefore very natural to use and, moreover, essentially unique. Therefore one would
really like to keep these structures. The Master Constraint Programme removes that technical
obstacle by replacing the Hamiltonian constraints by a single Master Constraint which is a
spatially diffeomorphism invariant integral of squares of the individual Hamiltonian constraints
which encodes all the necessary information about the constraint surface and the associated
invariants.

(http://www/arxiv.org/gr-qc/11140 )

Why is the Hamiltonian Constraint not diffeomorphism invariant?
When you build the constraint algebra out of commutation relations, this algebra has functions where a Lie Algebra has the structure constants of its group. It's still a good algebra, but it isn't Lie, and its theory is all but non existent. In particular its representations are unknown. To prove the diffeomorphism invariance of the Hamiltonian constraint you would have to use tose representations. This is essentially the off-shell closure problem noted by Nicolai, Peeters, and Zamaklar in hep-th/0501114.

As we have
stressed, it is this constraint which reflects the main problems of quantum gravity, not only in
the loop approach but also in geometrodynamics. However, in the latter approach, it is at least
possible to carry out some elementary checks at the dynamical level, and to demonstrate by
explicit computation its failure to produce a theory of quantum gravity with all the requisite
properties. For the time being, such an examen crucis does not appear possible for LQG due to
the mathematical complexity of the Hamiltonian constraint, despite many other proposed tests
of its viability in the literature.

(http://www.arxiv.org/hep-th/0501114 )


WHat is the master constraint?
Assume the raw Hamiltonian function is C(x), at each point x of a spacelike hypersurface, part of a preestablished foliation of spacetime, then Thiemann's master constraint is

$$M =\frac{1}{2}\int_{\sigma} d^3x \frac{C(x)^2}{\sqrt{det(g)(x)}}$$

Where sigma is that hypersurface. You see that he squares the Hamiltonian expression, divides out that determinant, and then integrates over the hypersurface, obtaining one expression to be zero instead of an infinity of them, removing the non-polynomial awkwardness, and, by the square, greatly easing later calculations. Of course the question arises: is this justified?, along with: does it work? The Dittrich and Thiemann Testing papers are to answer these questions.

Now squaring a constraint as displayed in (1.7) looks like a rather
drastic step to do in view of the following fact: A weak Dirac Observable O is determined by the
infinite number of relations {C(x),O}M=0 = 0, x ∈ σ. However, the condition {M,O}M=0 = 0 is
obviously trivially satisfied for any O so that the Master Constraint seems to fail detecting weak
Dirac observables. However, this is not the case, it is easy to see that the single Master Relation
{{M,O},O}M=0 = 0 (1.8)
is completely equivalent to the infinite number of relations {C(x),O}M=0 = 0, x ∈ σ. Therefore
the Master Constraint encodes sufficient information in order to perform the constraint analysis.

(http://www.arxiv.org/gr-qc/0411138 )
(More to come).

 

[marcus]

I want to be able to follow along (especially with you providing commentary!) so I have printed out two of the Thiemann/Dittrich papers

http://arxiv.org/abs/gr-qc/0411138
Testing I. General Framework

http://arxiv.org/abs/gr-qc/0411140
Testing III. SL(2,R) Models

the second is "just in case" because you referred to it in your post.
these papers look like they will be hard going for me
but I am encouraged by Nicolai's paper and by your interest to
think that they may be worth trying to understand better.

 

[marcus]

We should try to understand the first paper in this series

http://arxiv.org/abs/gr-qc/0411138
Testing I. General Framework

it will be hard.

Let us also remember that resolving the trouble with the dynamics is the big agenda item and that there are currently 3 horses in the race. One is Thiemann, with this Master Constraint.

One is Gambini, where the hamiltonian constraint is actually replaced by an operator that advances you step by step in a discretized evolution.
LQG formulation can be applied at each stage, according to his most recent paper.

And one is Ambjorn and Renate Loll---a kind of Feynmanesque path integral story about the evolution of the shape of the world.

In order to get any new understanding we have to focus our attention on just ONE HORSE and so let it be Thiemann

If he boggles us, we can still fall back and try to understand one of the other two (which are more elementary math-wise, with less powerful and abstract tools----Thiemann is the one using the most heavy machinery)

 

[marcus]

one question to consider is What was wrong that Thiemann is offering to fix?
it seems like a good idea to check out the history a little so I will go back and have a look at thiemann's 2003 paper for a moment

http://arxiv.org/gr-qc/0305080

"The Hamiltonian constraint remains the major unsolved problem in Loop Quantum Gravity (LQG). Seven years ago a mathematically consistent candidate Hamiltonian constraint has been proposed but there are still several unsettled questions which concern the algebra of commutators among smeared Hamiltonian constraints which must be faced in order to make progress. In this paper we propose a solution to this set of problems based on the so-called Master Constraint which combines the smeared Hamiltonian constraints for all smearing functions into a single constraint. If certain mathematical conditions, which still have to be proved, hold, then not only the problems with the commutator algebra could disappear, also chances are good that one can control the solution space and the (quantum) Dirac observables of LQG. Even a decision on whether the theory has the correct classical limit and a connection with the path integral (or spin foam) formulation could be in reach.
..."

In the introduction he speaks of three 1997 papers [15,16,17] which shot down his 1996 Hamiltonian and caused several years hiatus in work on LQG dynamics

"...Despite this success, immediately after the appearanceof [2, 3, 4, 5, 6, 7, 8, 9] three papers [15,16,17] were published which criticized the proposal by doubting the correctness of theclassical limit of the Hamiltonian constraint operator. ... While the arguments put forward are inconclusive (e.g. the direct translation of the techniques used in the full theory work extremely well in Loop Quantum Cosmology [18]) these three papers raised a serious issue and presumably discouraged almost all researchers in the field to work on an improvement of these questions. In fact, except for two papers [19] there has been no publication on possible modifications..."

[15] L. Smolin, “The classical Limit and the Form of the Hamiltonian Constraint in NonPerturbative Quantum General Relativity”, gr-qc/9609034

[16] D. Marolf, J. Lewandowski, “Loop Constraints : A Habitat and their Algebra”, Int.J.Mod.Phys.D7:299-330,1998, [gr-qc/9710016]

[17] R. Gambini, J. Lewandowski, D. Marolf, J. Pullin, “On the Consistency of the Constraint Algebra in Spin Network Gravity”, Int.J.Mod.Phys.D7:97-109,1998, [gr-qc/9710018]

I believe that these people (Lewandowski, Marolf, Smolin, Gambini, Pullin) are credited with having pointed out the trouble----the non-closure of the constraint algebra and so forth. Thiemann details the difficulties that were pointed out:

"The origin of the potential problems pointed out in [15, 16, 17] can all be traced back to simple facts about the constraint algebra:

1. The (smeared) Hamiltonian constraint is not a spatially diffeomorphism invariant function.

2. The algebra of (smeared) Hamiltonian constraints does not close, it is proportional to a spatial diffeomorphism constraint.

3. The coefficient of proportionality is not a constant, it is a non-trivial function on phase space whence the constraint algebra is open in the BRST sense.

These phrases are summarized in the well-known formulas (Dirac or hpersurface deformation algebra)..."

well I should post this, even though I have not yet come to any conclusions.
these 1997 events are the background for the Master Constraint effort and the series of papers by Thiemann/Dittrich which perform test-case studies of the master constraint to see how it well it does.

 

[selfAdjoint]

You are absolutely right. The 2003 paper was his return to the constraint problem. He is still referrring to his efforts on it by the old name Phoenix program, under which he had soared and been shot down. It took great courage to do this. But of course the Phoenix is reborn from its own ashes.
In this paper Thiemann announced his aims for the new program.

In the paper I have called Testing I (gr-qc/0411138) he and B. Dittrich build the tool they will use to carry out those aims, and in the remaining four of the outstanding papers (Testing II-V; gr-qc/0411139 -0411142) they test it on increasingly complex cases, and in the forthcoming paper they claim to have achieved the taming of the Hamiltonian constraint and the construction of consistent Dirac observables in full 3+1 dimensional quantum gravity.

I will look into the 2003 paper too, so we can have understanding in stereo.

 

[marcus]

I have been reading "Testing I."
I have not printed out the 2003 paper "Phoenix" yet (i thought I could get away without doing that) but will do so now and have a closer look.
I am again astonished that so much of a graduate "functional analysis" course comes into play---all these names of theorems now mocking me and my decrepit memory like birdcalls from the past: RieszMarkov, Lusin, RadonNikodym, see how you wasted your youth? they call, you never suspected any of this would ever involve the real world (I never did, it was all just abstract mathematics)

 

[marcus]

funny thing, in "Phoenix" section 6, he refers to the MC programme (or some consequence of it) as "this new form of path integral approach"
and mentions something I never heard of before called the
"Osterwalder-Schrader reconstruction theorem" which connects
path integral and Hamiltonian formulations.
so that it seems there is a possible hookup of this MC programme with path-integral approaches like Dynamical Triangulations

I would find that awfully interesting if it actually happened.

 

[selfAdjoint]

I looked up the Osterwalder Schrader theorem in my copy of Haag's Local Quantum Physics. Translating a little from his notation, the theorem concerns a quantum theory defined in Euclidean 4-space, and what it has to satisfy in order to define a valid theory in Minkowski spacetime. The great example of this of course is when you use Wick rotation to tame path integrals and you wind up with a well-defined theory in Euclidean space. Then you want to "rotate back", and the O-S theorem tells when you are justified in doing that.

The Euclidean theory has to be "nice" in the sense that it satisfies the Wightman axioms and so is defined by a set of Wightman functions. Then there is a condition called reflection positivity that the Wightman functions have to satisfy in order to be able to get back to Minkowsi spacetime. If you like I can work this up and post it, but I don't really think it supports your conjecture.

BTW where are you with Thiemann's "Phoenix Project" paper now? I am slogging through section 2, where he compares his original formulation of the Hamiltonian constraint problem with the new results from his Master Constraint initiative. This isn't the most exciting part, but I want to at least get some background. And of course as he points out, a lot of the original program is carried over unchanged.

 

[marcus]

BTW where are you with Thiemann's "Phoenix Project" paper now? I am slogging through section 2, ...

I spent a while yesterday reading "Phoenix", in sections one and two.
It is hard. I can't say that I am making progress because I slip and slide.
I keep wanting someone to come in and explain it to me---give me an overview.

A lot of time, when I'm reading or trying to read Thiemann, I am
kicking myself: why didnt i read this Phoenix paper when it came out in 2003, and other times I have moments of admiration for his courage. he really took a long shot. Back in 2003 one would have said (I would have said, if I had taken the time to consider it) that it was bound to fail.

and I worry: maybe it does fail. isn't there some catch?
so I dither some, and read the same thing over. If there is something wrong I know i am not good enough to catch it. If there is not something wrong then apparently it's substantial work, with broad consequences.
I guess I am gradually learning something about the MC approach tho, so thanks for getting me to pay some attention finally!

BTW your post #4 on this thread is very helpful as part of an overview.
Wd be grateful any time you care to enlarge on it.

 

[selfAdjoint]

Boy do I ever know what you mean! I've been to the end of section 2 a couple of times, but each time I doubt my conclusions and turn back.

Right now I am afraid I am thrashing around with 2.1 "Cotriad regularization", and this phrase: "since $$e^j_a$$ is not a polynomial in the elementary phase space coordinates $$(A^j_a,E^j_a)$$ consisting of an SU(2) connection A and a canonically conjugate Ad_SU(2)-covariant vector density E of weight one..."

I know what a vector density of weight one is - at least I know the change of coordinates definition, and that you integrate a vector density to get a vector. And I suppose the phrase "canonically conjugate" is clear to me in this phase space context. But what does that Ad stuff mean? I have always had a problem with the second of the Ashtekar variables, and this is just another siege of it. Somehow a clear picture of what is going on fails to emerge.

 

[marcus]

... what does that Ad stuff mean? I have always had a problem with the second of the Ashtekar variables, and this is just another siege of it...

I presume Ad refers to the adjoint representation of the Lie algebra su(2) which also IIRC defines some map on the group SU(2). I don't claim to understand this right off but for starters here is mathworld about adjoint representation
http://mathworld.wolfram.com/AdjointRepresentation.html
(I don't think any of this is news to you).
Personally i hope that his saying that about E (the second of the Asht. variables) being valued in the "Ad-covariant" vectors is just a side comment and of no particular importance.
It looks like that section 2.1 is a review (of material in his papers [2,3,4,...]) so whatever IS essential in section 2.1 may be something we have to look back in earlier papers for. arghh and all that. but supper is on the table and we should be happy for lentils and polish sausage and not let this Ad spoil a nice evening. I will be back soon.