Constraint Algebra & Gupta-Bleuler in LQG & SF Models | W. Wieland

[tom.stoer]

There's a new paper dealing with constraint algebra and Gupta-Bleuler quantization in LQG and SF models.

http://arxiv.org/abs/1012.1738
Complex Ashtekar variables and reality conditions for Holst's action
Authors: Wolfgang Wieland
(Submitted on 8 Dec 2010)
Abstract: From the Holst action in terms of complex valued Ashtekar variables additional reality conditions mimicking the linear simplicity constraints of spin foam gravity are found. In quantum theory with the results of You and Rovelli we are able to implement these constraints weakly, that is in the sense of Gupta and Bleuler. The resulting kinematical Hilbert space matches the original one of loop quantum gravity, that is for real valued Ashtekar connection. Our result perfectly fit with recent developments of Rovelli and Speziale concerning Lorentz covariance within spin-form gravity.

The idea is to relate Ahtekar's variables plus reality conditions of canonical LQG to simplicity constraints of SF models.

Whoever is interested in these topics and has some experience regarding constraint quantization a la Dirac: let's discuss if this can be correct; I still think that they 'cheat' when trying to avoid second class constraints and Dirac quantization.

• (24f) means that the algebra does not close; the following lines of reasoning do not seem to be convincing
• I do not see how they prove consistency of (30) with (24)
• (40) is the main result which shows the equivalence between reality conditions and simplicity constraints; but (41) shows that C is not compatible with other constraints
• Then unfortunately they stop calculating secondary and tertiary constraints which is required to complete the analysis
• In (47) and (48) they explain what they want to do - and it simply looks wrong; they basically neglect that the two sets of constraints do not commute and are therefore not consistent;
• afaik Gupta-Bleuler fails in QCD b/c the method is not able to deal with ghosts; here they have a similar situation but they apply Gupta-Bleuler w/o worrying about ghosts; why?
• in section 5.2 they admit that they did not treat T which is second class correctly and that for H there is still no conclusive and consistent expression available

So my conclusion is that the paper does not fix the problems of simplicity constraints in the SF formulation, but that it translates the errors made there to the canonical framework. This is a remarkable result not b/c the problems are resolved but b/c they become explicit in the canonical framework.

 

[marcus]

Tom, as I am sure you know, Wolfgang is a PhD student at Marseille. I think it makes sense to look at his earlier (2010) work in the light of his later work---his 2011 papers. I will get the abstracts.
http://arxiv.org/find/grp_physics/1/au:+Wieland_W/0/1/0/all/0/1

AFTER the one you mentioned, he has three more recent papers of which the two latest are solo:

1. http://arxiv.org/abs/1107.5002
Twistorial phase space for complex Ashtekar variables
Wolfgang M. Wieland
We generalise the SU(2) spinor framework of twisted geometries developed by Dupuis, Freidel, Livine, Speziale and Tambornino to the Lorentzian case, that is the group SL(2,C). We show that the phase space for complex valued Ashtekar variables on a spinnetwork graph can be decomposed in terms of twistorial variables. To every link there are two twistors---one to each boundary point---attached. The formalism provides a new derivation of the solution space of the simplicity constraints of loop quantum gravity. Key properties of the EPRL spinfoam model are perfectly recovered.
18 pages, to appear in: Class. Quantum Grav

2. http://arxiv.org/abs/1105.2330
Complex Ashtekar variables, the Kodama state and spinfoam gravity
Wolfgang Wieland
Starting from a Hamiltonian description of four dimensional general relativity in presence of a cosmological constant we perform the program of canonical quantisation. This is done using complex Ashtekar variables while keeping the Barbero--Immirzi parameter real. Introducing the SL(2,C) Kodama state formally solving all first class constraints we propose a spinfoam vertex amplitude. We construct SL(2,C) boundary spinnetwork functions coloured by finite dimensional representations of the group, and derive the skein relations needed to calculate the amplitude. The space of boundary states is shown to carry a representation of the holonomy flux algebra and can naturally be equipped with a non-degenerate inner product. It fails to be positive definite, but cylindrical consistency is perfectly satisfied.
30 pages, 3 figures

3. http://arxiv.org/abs/1012.4719
Spinfoam fermions
Eugenio Bianchi, Muxin Han, Elena Magliaro, Claudio Perini, Carlo Rovelli, Wolfgang Wieland
We describe a minimal coupling of fermions and Yang Mills fields to the loop quantum gravity dynamics. The coupling takes a very simple form.
8 pages, no figures

Just as a side comment, in your post you use the pronoun "they" to refer to Wolfgang Wieland and his work that he is presenting in the paper. It doesn't sound right to call a single individual "they", although there is a convention in English of an author referring to himself as "we".

I don't know if Wolfgang's later work has any relevance to the paper you mentioned about complex Ashtekar variables. I would be interested to know your view on this. Especially about the last one.

 

[tom.stoer]

Does any paper address my questions?

 

[marcus]

Tom, you asked for comment on Wieland's 2010 paper: I think what stands out is that he is clearly going off in own direction, with complex Ashtekar variables.

As far as I know, nobody else does this. Not since early 1990s I think. The original variables were complex and one had to introduce "reality conditions" at the end to get physical results,which should be real numbers. Then the innovation of Barbero and Immirzi was to introduce the barbero-immirzi parameter which had the effect of making the equations more complicated but however REAL.

When I look at Wolfgang's last 3 solo papers I see something very surprising. Namely that all on his own he is resurrecting the old complex valued geometric variables and exploring what he can do with them---given our greater knowledge that has accumulated over the past 20 years when they were eclipsed by the Immirzi parameter.

 

[tom.stoer]

The goal is to demonstrate that (linearized) simplicity constraints in SF models and reality conditions in canonical LQG with complex Ashtekar variables are closely related; what I understand is that there is something wrong with the quantization in approach A, that one can translate the problem to approach B and show that it again goes wrong. That deepens our knowledge regarding the problem - but unfortunately seems to miss a solution.

 

[marcus]

Does any paper address my questions?

Well you raised your questions in reference to Wieland's QG approach, citing equations from the December 2010 paper. The relation of this approach to LQG is not known. I was just now looking at the May 2011 and July 2011 papers to respond to your question.

I see in the May 2011 paper many questions about the relation of this new approach to LQG. Even the most basic issues, like the discreteness (or not) of the area spectrum, have not been studied!

My guess is that very little is known about this new approach. One does not even know how close or far apart it is from LQG.

And even if one could answer the questions you asked about Wieland QG in your post#1, it would not necessarily say anything about LQG.

 

[tom.stoer]

... it would not necessarily say anything about LQG.

marcus, it does!

there are several approaches (canonical LQG with real variables, complex variables, covariant LQG, now twistor variables, spin foams in several variants); they all agree on the kinematical Hilbert space; and they all agree in the sense that they are incomplete regarding reality conditions ~ simplicity constraints, dynamics, and regarding establishing relations between them;

SFs have (partially) been inventied to bypass problems in the canonical approach;

but there ain't no such thing as a free lunch

 

[marcus]

...all agree in the sense that they are incomplete regarding reality conditions ~ simplicity constraints, dynamics, and regarding establishing relations between them...

I'd be interested if you could point me to a paper that discusses the equivalence of simplicity constraints (of vintage 2009 EPRL spinfoams) with the older reality conditions of the 1990s.
There may be some logical correspondence between them and I'm the only one who is not aware of it!

You take a rather sweeping view, at the moment. I think that well, every approach being actively worked on is incomplete. They must be incomplete, else why are people working on them?

But as I see it this does not represent agreement. The different approaches are, I think, incomplete in quite different ways and to different extents.

 

[tom.stoer]

I'd be interested if you could point me to a paper that discusses the equivalence of simplicity constraints (of vintage 2009 EPRL spinfoams) with the older reality conditions of the 1990s.

Wieland's paper discusses this relation; and Alexandrov mentions it, too.

You take a rather sweeping view, at the moment. I think that well, every approach being actively worked on is incomplete. They must be incomplete, else why are people working on them?

In his Zakopane lectures http://arxiv.org/abs/1102.3660 Rovelli claims that

"LQG utilizes the Ashtekar's formulation of GR and its variants, and can be derived in different ways. The three major ones are: canonical quantization of GR, covariant quantization of GR on a lattice, and a formal quantization of geometrical shapes. Surprisingly, these very different techniques and philosophies converge towards the same formalism."

It becomes clear that the convergence is restricted to kinematics and that the dynamics is rather incomplete in all approaches.

The different approaches are, I think, incomplete in quite different ways and to different extents.

I do not want to copy everything from recent discussions we had on the Alexandrov papers; please refer to them.

https://www.physicsforums.com/showthread.php?t=544728
https://www.physicsforums.com/showthread.php?t=545596
https://www.physicsforums.com/showthread.php?t=539793

Nevertheless a short summary of my perception may be interesting:

• The canonical formulation lacks a precise and unique definition, quantization and regularization of the Hamiltonian constraint, the construction is a bit baroque (Rovelli in his "twenty five years" paper http://arxiv.org/abs/1012.4707); this is essentially where the canonical formulation gets stuck
• Spin foam models have been invented to avoid these problems; but strictly speaking a [Lagrangian] PI is constructed from H and its matrix elements which correspond to vertex operatords [plus integrating out momenta]; b/c H is far from being well-defined in LQG, SFs cannot be derived from H but have to be "defined" guessing the correct vertex operators
• These two approaches agree on the kinematical setup, but there are ambiguities in the SF approach which can be traced back to the missing of the well-defined starting point - which would be H
• There are several variants within each approach (canonical one: real vs. complex Ashtekar variables, covariant LQG a la Alexandrov) which converge regarding the kinematical framework but which deviate regarding the full implementation of constraint, the dynamics - and which show different but somehow related quantization ambiguities
• It is interesting that reality conditions for complex Ashtekar variables correspond to simplicity constraints in SF models; both are second class (in the Dirac sense) and strictly speaking must be solved before quantization b/c implementing them on the kinematical Hilbert space changes the symplectic structure, may alter the propagating degrees of freedom, may introduce anomalies, affects the measure in the PI, ...
• Due to the structure of the second class constraints their effects are typically invisible in the semiclassical limit (a determinant in the measure like Fadeev-Popov always comes with factors of hbar; anomalies in operator algebras are usually suppressed by an additional factor of hbar); so missing them may result in inconsistencies visible in the quantum regime only
• Quantizing GR seems to be nothing else but consistent implementation of constraints; once this has been achieved the theory has a precise definition; unfortunately all approaches invent some tricks, deviate from standard quantization rules, treat different constraints differently (Gauss, diff., Hamiltonian, simplicity),
• Especially for diff. inv. and the Hamiltonian constraint it is by no means clear whether these symmetries are implemented correctly or whether they are broken; unfortunately here the above mentioned problems are masked by the discretization approach which seems to be both a solution of the diffeomorphism invariance and an approximation to be overcome in some continuuum or scaling limit; there are recent attempts to study this discretization in the classical phase space formalism w/o quantization to separate effects from discretization and quantzation
• The problem is that neither formalism provides an "off-shell closure" of constraints; to be clear on that: correctly implementing constraints and reducing the Hilbert space to its physical subsector is welcome - but after having done that there is no way to check the consistency i.e. absence of anomalies b/c there are simply no constraints, no symmetry operations; so in the physical Hilbert space or PI indications regarding inconsistencies are rather indirect
• Last but not least there is the mystery regarding the cosmological constant: at first glance it seems to be an ingredient of the theory just like any other couplig constant, but then it plays a major role in constructing the kinematics, i.e. in defining the quantum deformation of SU(2) and the truncated SU_q(2) spin networks; how can it be that the quantum deformation is based on a pure algebraic setup using the cosmological constant as a constant (!) when there are indications from the asymptotic safety program that Lambda is "running" w.r.t. renormalization group energy scaling? please note that up to now a RG approach to LQG is missing

To summarize: there is growing evidence that the incompleteness of the different approaches has a common origin which can be traced back to the lack of understanding in the quantization procedures itself.

 

[suprised]

...there is growing evidence that the incompleteness of the different approaches has a common origin...

Indeed, it may be that trying to quantize an effective theory will never work, ie., without introducing the extra degrees of freedom that are needed for unitarity.

 

[tom.stoer]

Hey suprised,

thanks for participating

Indeed, it may be that trying to quantize an effective theory will never work, ie., without introducing the extra degrees of freedom that are needed for unitarity.

This seems to be a premature conclusion.

There are problems intrinsic to the LQG program - and only these problems are addressed here. There is no evidence from the LQG program itself that the theory is an effective one, that certain degrees of freedom are missing, that it may violate unitarity etc. There are indeed indications that gravity can be quantized w/o adding further degrees of freedom (asympotic safety). All the problems I am mentioning here cannot be traced back to a physical origin but only to a lack of understanding of the correct quantization procedure. They would remain as mathematical problems even if the LQG community withdraws LQG as a theory of nature.

I know what you mean and where your arguments are coming from, but my feeling is (currently - it changes from time to time) that most problems in LQG and in string theory are intrinsic, techical problems created by the approaches, not questions asked by nature, and that therefore both approaches are not yet in a stage where they can participate or learn much from each other. Even background independence which is (my impression) one of the most severe issues in string theory is not a severe issue b/c of LQG, but b/c string theory itself tells us that some pieces are still missing. And why should LQG learn from string theory that we need additional degrees of freedom when a) string theory is by no means complete, b) string theory is not able to tell us what the unique fundamental degrees of freedom are and c) nobody has ever observed stringy effects?

If you want to convince me that "2+2=5" is wrong, then please use algebra, not string theory ;-)

What I want to say is the following: LQG can fail due to several different reasons
1) it's intrinsically inconsistent - in order to understand that we need math, not strings
2) it's consistent but gravity in the quantum regime works differently in nature - then nature should tell us
3) quantum gravity and unification are deeply connected - then LQG will fail b/c it misses exactly this point
4) quantum gravity and unification are losely connected; this is interesting b/c there is the possibility that QG may cure UV problems (asymptotic freedom and / or safety for 'gravity + matter couplings', especially 'gravity + standard model') - then 'LQG + standard' model is nothing else but another step towards the UV w/o adressing unification (just like the SM itself).
5) ...

Only in rare cases (3) the failure of LQG could be traced back to strings; string theory as of today is not (yet) a sound basis to kill LQG; and LQG is by no means a good starting point to kill strings.

So please tell me what you think about the intrinsic problems of LQG ;-)

 

[suprised]

Well, let's not open again Pandora's box, things have been said often enough. I simply want to convey the opinion of most of my colleageus, and this is based on the insight how certain problems are solved within string theory (unitary, black holes), and it is hard to imagine how that can possibly ever work without the "right" degrees of freedom. But we are open to surprises.

 

[tom.stoer]

regarding Pandora: do you know which item was most deeply hidden in her box?

 

[suprised]

A black hole?

 

[tom.stoer]

no, hope!

http://en.wikipedia.org/wiki/Pandora's_box

 

[suprised]

lol.. that fits well to quantum gravity... but the secret truth is: "der Weg ist das Ziel"