Is Twistor Network Theory the Future of Loop Quantum Gravity?

[marcus]

Twistor Networks addresses several of the issues left open by EPRL and I suspect TN is destined to be the new "EPRL" phenomenon: the new Loop pace-setter.

So I suspect (having spent time looking at other potentially influential developments) that anyone who wants to follow Loop gravity research would be well advised to be reading
Speziale Wieland 1207.6348 and getting prepared to understand Speziale's ILQGS talk on 13 November.

So I'll list some titles and links in this thread. But also it would be very interesting if someone disagrees and thinks that some other reformulation of LQG that is currently being actively pursued has a better chance and could make a stronger showing at the upcoming Loops 2013 conference.

First of all here's the main paper.
http://arxiv.org/abs/1207.6348
The twistorial structure of loop-gravity transition amplitudes
Simone Speziale, Wolfgang Wieland
The spin foam formalism provides transition amplitudes for loop quantum gravity. Important aspects of the dynamics are understood, but many open questions are pressing on. In this paper we address some of them using a twistorial description, which brings new light on both classical and quantum aspects of the theory. At the classical level, we clarify the covariant properties of the discrete geometries involved, and the role of the simplicity constraints in leading to SU(2) Ashtekar-Barbero variables. We identify areas and Lorentzian dihedral angles in twistor space, and show that they form a canonical pair. The primary simplicity constraints are solved by simple twistors, parametrized by SU(2) spinors and the dihedral angles. We construct an SU(2) holonomy and prove it to correspond to the (lattice version of the) Ashtekar-Barbero connection. We argue that the role of secondary constraints is to provide a non trivial embedding of the cotangent bundle of SU(2) in the space of simple twistors. At the quantum level, a Schroedinger representation leads to a spinorial version of simple projected spin networks, where the argument of the wave functions is a spinor instead of a group element. We rewrite the Liouville measure on the cotangent bundle of SL(2,C) as an integral in twistor space. Using these tools, we show that the Engle-Pereira-Rovelli-Livine transition amplitudes can be derived from a path integral in twistor space. We construct a curvature tensor, show that it carries torsion off-shell, and that its Riemann part is of Petrov type D. Finally, we make contact between the semiclassical asymptotic behaviour of the model and our construction, clarifying the relation of the Regge geometries with the original phase space.
40 pages, 3 figures

 

[mitchell porter]

A basic question I know nothing about: How does EPRL look from the perspective of "naive perturbative quantum gravity"? I mean Feynman diagrams with a spin-2 particle, such as Feynman himself investigated in the 1960s. Can you get to EPRL by starting there and then generalizing it, or is EPRL such a different calculus that there's no such bridge?

Harder questions: how "loop twistor gravity" relates to the original twistor program as applied to gravity, and to the mainstream twistor revival.

The wellspring of the new twistor mainstream is the study of special supersymmetric theories where there are many fields in addition to gravity, but such theories have a "pure gravity" part (i.e. amplitudes for graviton-only processes) which still bear the mathematical imprint of the larger context. That part could be compared to the amplitudes of twistor EPRL.

The original twistor program for gravity should be more directly suited to comparison, but it's full of arcana like "twistor diagrams" and "sheaf cohomology".

 

[tom.stoer]

How does EPRL look from the perspective of "naive perturbative quantum gravity"? ... Can you get to EPRL by starting there and then generalizing it, ...

I don't think so. It's rather the other way round: there are approaches to derive results like graviton propagators and higher n-point functions using spin foams; I think this is work in progress.

how "loop twistor gravity" relates to the original twistor program as applied to gravity, ...

I see many "algebraic" similarities between the old twistor program and LQG in twistor language. But I don't knwo if they are "equivalent" in some sense. Penrose worked on the level of the (self-dual) gravitational field, not in the first oredr formalism. And it was especially the required restriction two self-duality which made his work a "dead end". I don't see whether the new approach is a way out b/c the twistors are applied to other variables, or whether the new approach will fail exactly due to the same reason, namely using time-gauge, trivial constraints and restrcution to self-duality i.e. SL(2,C) - SU(2).

The original twistor program for gravity should be more directly suited to comparison, but it's full of arcana like "twistor diagrams" and "sheaf cohomology".

Penrose invented both twistors and spin networks. Did he construct any relation between them?

 

[marcus]

Hi Tom, hi Mitchell, I first want to post the links for the main paper's references [1, 2, 3, 4] which it gives as the history of TN. Twistor loop gravity has a very short recent history starting in 2010 with Freidel Speziale 1006.0199.

http://arxiv.org/abs/1006.0199
From twistors to twisted geometries
Laurent Freidel, Simone Speziale

http://arxiv.org/abs/1107.5002
Twistorial phase space for complex Ashtekar variables
Wolfgang M. Wieland

http://arxiv.org/abs/1107.5274
Holomorphic Lorentzian Simplicity Constraints
Maité Dupuis, Laurent Freidel, Etera R. Livine, Simone Speziale
(Submitted on 26 Jul 2011, last revised 20 Feb 2012)
We develop an Hamiltonian representation of the sl(2,C) algebra on a phase space consisting of N copies of twistors, or bi-spinors. We identify a complete set of global invariants, and show that they generate a closed algebra including gl(N,C) as a subalgebra. Then, we define the linear and quadratic simplicity constraints which reduce the spinor variables to (framed) 3d spacelike polyhedra embedded in Minkowski spacetime. Finally, we introduce a new version of the simplicity constraints which (i) are holomorphic and (ii) Poisson-commute with each other, and show their equivalence to the linear and quadratic constraints.
20 pages, explicit counting of the holomorphic constraints added.

http://arxiv.org/abs/1108.0369
Twistor Networks and Covariant Twisted Geometries
Etera R. Livine, Simone Speziale, Johannes Tambornino
(Submitted on 1 Aug 2011, last revised 13 Feb 2012)
We study the symplectic reduction of the phase space of two twistors to the cotangent bundle of the Lorentz group. We provide expressions for the Lorentz generators and group elements in terms of the spinors defining the twistors. We use this to define twistor networks as a graph carrying the phase space of two twistors on each edge. We also introduce simple twistor networks, which provide a classical version of the simple projected spin networks living on the boundary Hilbert space of EPRL/FK spin foam models. Finally, we give an expression for the Haar measure in terms of spinors.
18 pages.

After this abbreviated history of Twistor Networks, I want to review the Speziale Wieland paper's abstract. This is where, among other interesting results, the EPRL dynamics (the transition amplitudes) are recovered.

http://arxiv.org/abs/1207.6348
The twistorial structure of loop-gravity transition amplitudes
Simone Speziale, Wolfgang Wieland
The spin foam formalism provides transition amplitudes for loop quantum gravity. Important aspects of the dynamics are understood, but many open questions are pressing on. In this paper we address some of them using a twistorial description, which brings new light on both classical and quantum aspects of the theory. At the classical level, we clarify the covariant properties of the discrete geometries involved, and the role of the simplicity constraints in leading to SU(2) Ashtekar-Barbero variables. We identify areas and Lorentzian dihedral angles in twistor space, and show that they form a canonical pair. The primary simplicity constraints are solved by simple twistors, parametrized by SU(2) spinors and the dihedral angles. We construct an SU(2) holonomy and prove it to correspond to the (lattice version of the) Ashtekar-Barbero connection. We argue that the role of secondary constraints is to provide a non trivial embedding of the cotangent bundle of SU(2) in the space of simple twistors. At the quantum level, a Schroedinger representation leads to a spinorial version of simple projected spin networks, where the argument of the wave functions is a spinor instead of a group element. We rewrite the Liouville measure on the cotangent bundle of SL(2,C) as an integral in twistor space. Using these tools, we show that the Engle-Pereira-Rovelli-Livine transition amplitudes can be derived from a path integral in twistor space. We construct a curvature tensor, show that it carries torsion off-shell, and that its Riemann part is of Petrov type D. Finally, we make contact between the semiclassical asymptotic behaviour of the model and our construction, clarifying the relation of the Regge geometries with the original phase space.
40 pages, 3 figures

 

[tom.stoer]

I studied the paper a second or third time. My summary is that they are rewriting LQG in terms of twistors, but that they are using - at the most relevant steps - the same ideas and procedures as most standard LQG approaches; these are
- gauge fixing of temporal gauge
- discretization on graphs
- reality condition = simplicity constraints
- solution for first class constraint, master constraint for second class constraint

So my conclusion is that their approach is to a large extent identical to the standard one. Therefore its a valuable consistency check to re-derive EPRL, but it's not a new or independent approach.

Hope I don't miss something essential ...

 

[marcus]

So my conclusion is that their approach is to a large extent identical to the standard one. ...

I'm glad to hear this! So when they resolve issues in their new treatment it probably shows that those issues are resolvable also in the earlier one. It just was harder to see how to do it, then, in the older formulation.

==quote http://arxiv.org/abs/1207.6348 page 29==
...This is achieved assigning a pair of twistors with equal norms to each link of the graph. In this way, we embed the non-linear holonomy-flux algebra in a much simpler algebra of canonical Darboux form. The first advantage of doing so shows up in dealing with the simplicity constraints. In the usual path to the quantum theory, one solves the (primary and secondary) simplicity constraints at the continuum level, and then smears the resulting SU(2) variables. Here we have shown that swapping reduction and smearing is also possible. One smears the covariant SL(2,C) variables, and the SU(2) variables are recovered solving the discretized simplicity constraints...
==endquote==

Also on the question of secondary constraints they seem hopeful their twistor formulation will make it easier to address a long-standing problem:

==page 30==
It remains to formulate an explicit discretization of the secondary constraints, and study the gauge sections they identify. This has been an important open question in the field for many years. The twistorial formalism offers a way to address it, and we hope to come back to this in future research. For the moment, we verified our treatment of the secondary constraints using the simple case of a flat 4-simplex, which is also the one relevant for the EPRL spin foam model. Unlike the case of primary constraints, the solution to the secondary constraints involves a non-local graph structure, and can not be found on each link separately.

Twistors lead to significant insights also in the quantum theory...
==endquote==

 

[tom.stoer]

marcus, I don't really share this optimistiv view b /c of the following: they introduce no new concepts to treat the constraints (up to now). Everything they did so far was standard, just using new variables. So if they succeed it's mostly due to technical simplification in the new new variables. But if the whole approach (discretization on graphs - reality condition = simplicity constraints - solution for first class constraint, master constraint for second class constraint) is a dead end, then it's a dead end in all these deeply related approaches.

My feeling is that something is rotten in the state of loop quantum gravity (discretization on graphs - reality condition = simplicity constraints - solution for first class constraint, master constraint for second class constraint)

 

[Naty1]

Here is what Penrose has to say, page 952-955, THE ROAD TO REALITY...2004

His perspective might be interesting, even if now out of date, as he has been working on related issues since the 1950's...

As a general observation, Penrose likes the discretization inherent in the newer approach...

32.7 status of loop quantum gravity

... both the original Ashtekar variables and later descriptions in terms of loop variables strike me as powerful and highly original developments...the loop states do appear to address at least some of the profound problems raised by general covariance...set against this is the somewhat disturbing fact that the theory...adopts a connection ...which does not appear geometrically correct...there is still the fundamental difficulty that the full Einstein Hamiltonian has yet to be unambiguously encompassed in the loop-variable framework even thought constraint equations are handled by the use of spin-networks...a formulation based on a three space description [in terms of S] rather than being a more global spacetime one...[this] brings a whole "Pandora's box' of problems...The difficulty has to do with the issue of how time evolution according to the Einstein equation is to be properly expressed in a a generally covariant 4 space formalism...the problem of time in quantum gravity...My own perspective..is that the issue is unlikely to be resolved without the problem of the state vector reduction R being satisfactorily addressed, and that this, in turn, will require a drastic revision of general principles...

Anyone have an opinion about whether these issues are addressed in the newer papers?? Seems like Stoer's concerns suggest not. Time evolution is SO fundamental.

 

[marcus]

...
My feeling is that something is rotten in the state of loop quantum gravity (discretization on graphs - reality condition = simplicity constraints - solution for first class constraint, master constraint for second class constraint)

Well people's feelings are interesting. A hunch can be valuable, though not to be confused with fact.

But you don't present evidence that the approach IS a dead end---you just say IF it is...

I think the Spez-Wiel paper is exactly the kind of thing one should see if the approach is NOT a dead end because it discovers some new light. Solving the simplicity constraint by restricting to a class of twistor they call *simple* twistors.
Finding that area and dihedral angle form a *canonical pair*
and also that area and dihedral correspond two two parts of a *simple* twistor---which can be seen as consisting of a spinor and a real number.

The Spez-Wiel paper has a whole bunch of fresh mathematical insight, and this is the kind of thing that, to me, indicates the approach is working and going ahead.

However I cannot ARGUE with what is, in you, a deeply rooted feeling of wrongness. What I see are not reasons on which to base an argument, just signs of growing enlightenment and mathematical clarification.

I expect we will both be listening to Speziale's seminar talk on 13 November.

 

[mitchell porter]

Regarding comments #2 and #3: "Gravity, Twistors and the MHV Formalism" uses twistors to describe scattering of gravitons off an anti-self-dual background.

On page 2, the authors write: "The MHV formulation is essentially chiral. For gravity, this chirality suggests deep links to Plebanski's chiral action, to Ashtekar variables and to twistor theory. It is the purpose of this article to elucidate these connections further and to go some way towards a non-linear formulation that helps illuminate the underlying nonperturbative structure."

It would be very interesting if someone familiar with LQG could take a look at that paper and see if they understand any of it. This is one of the comparisons I mentioned in comment #2. It looks as if LQG twistor networks, on the one hand, and gravitational MHV diagrams, on the other hand, are both approaches to quantum gravity based on twistor combinatorics. This paper even starts with a Plebanski action (page 5) - though it's chiral, and I think LQG characteristically uses a non-chiral version?

 

[tom.stoer]

But you don't present evidence that the approach IS a dead end---you just say IF it is...

I don't say that it's a dead end, but I strongly believe that some essential ingredient is missing and that this does not come from the twistor approach; the reason is that the approach is different, but not at the essential weak points of the program; there it is nearly identical, or at least uses the same kind of reasoning.

 

[marcus]

there it is nearly identical, or at least uses the same kind of reasoning.

That's interesting. What page(s) in the Speziale Wieland paper are you talking about?

Where in the paper is the approach DIFFERENT? Where in the paper is it "nearly identical, or at least..."?

If you give me some specific pages to look at, I may be able to get a concrete idea of what you think "essential weak points" are.

 

[tom.stoer]

Where in the paper is the approach DIFFERENT?

It uses different variables

Where in the paper is it "nearly identical, or at least..."?

In the way the constraints are 'solved'; and in the discretization

 

[marcus]

In a couple of days (Tuesday 13 November) Simone is giving his ILQGS talk. There is a lot going on in LQG. A lot of different, potentially important research lines are being pursued. To counteract my tendency to get absorbed in one and overlook the others, I have a list of a half-dozen different active research fronts that I try to recall and mentally review now and then. Twistor LQG is an important one.

twistorLQG (Speziale's ILQGS talk and 1207.6348)
tensorialGFT (Carrozza's ILQGS talk and 1207.6734)
holonomySF (Hellmann's ILQGS talk and 1208.3388)
dust (Wise's ILQGS talk and 1210.0019)
hybrid LQC
An Extension of the Quantum Theory of Cosmological Perturbations to the Planck Era (1211.1354)
The pre-inflationary dynamics of loop quantum cosmology: Confronting quantum gravity with observations (in prep)
GR Thermo
General relativistic statistical mechanics (1209.0065)
Horizon entanglement entropy and universality of the graviton coupling (Bianchi's ILQGS talk and 1211.0522)
Interpretation of the triad orientations in loop quantum cosmology (1210.0418)

I think the last topic, general relativistic thermodynamics (and also statistical mechanics) has a lot of potential development ahead, and I recently added the Kiefer and Schell paper http://arxiv.org/abs/1210.0418 as an indication of where that is going. Kiefer Schell have the purity/mixedness of quantum states run on a continuum from zero to one. I'm impressed by that. A state is a trace-class operator ρ on the hilbert space, a generalized "density matrix". Pure states are those for which tr(ρ²) = 1 and they can gradually decohere and the purity index can gradually come down from 1 to zero. Kiefer Schell have a quantum state of geometry do this as it interacts with the matter in the environment.

 

[marcus]

Simone S.'s talk was yesterday and the slides PDF is posted, so we can read that and see what is new since the Speziale Wieland paper.
EDIT: The audio is now posted online.
PDF: http://relativity.phys.lsu.edu/ilqgs/speziale111312.pdf
AUDIO: http://relativity.phys.lsu.edu/ilqgs/speziale111312.wav

More information about the ILQGS:
http://relativity.phys.lsu.edu/ilqgs/

The outline on slide#2 has two parts.
Classical Theory (slides 2-19)
‚ Description of the covariant phase space in terms of twistors algebra of the area matching and simplicity constraints
‚ Smearing the connection-tetrad algebra to the holonomy-flux algebra before or after solving the simplicity constraints is equivalent: T ̊SU(2) with AB holonomy
‚ Notion of simple twistors solving the simplicity constraints

Quantum theory (slides 20-24)
‚ Hilbert space represented via homogeneous functions on spinor space (instead of cylindrical functions)
‚ Dynamics as integrals in twistor space
‚ Embedding of the Regge data of the EPRL asymptotics in the initial phase space

Talk is based on S. paper with Wieland, which we have, and a new one with Miklos Langvik, in prep.

Simplicity constraints: slide 10
Solution of simplicity constraints: slide 12
Geometric interpretation: slide 14
Discussion of "simple twistor" = type used to solve simplicity constraints: slide 17

A discrete Levi-Civita connection can be constructed WITHOUT the shape-matching restriction which limits twisted geometries to the Regge case. (reference to HRVW paper 1211.2166)
This tends to validate working with twisted geometries. But there is still the open problem
"Is there a consistent classical dynamics for twisted geometries?": slide 19

At the end of the section on the quantum theory, at slide 24, this open problem is re-emphasized and the following remark is added:
"‚ A priori this is not necessary: correct semiclassical limit may also emerge from coarse graining graphs, and not graph by graph But we need to find this out!"

The audio is posted, links given above.

 

[mitchell porter]

Phil Gibbs and commenters discuss whether twistors can unify loop and string.

 

[xristy]

Phil Gibbs and commenters discuss whether twistors can unify loop and string.

And I note that sometime ago you had mentioned:

Regarding comments #2 and #3: "Gravity, Twistors and the MHV Formalism" uses twistors to describe scattering of gravitons off an anti-self-dual background...It would be very interesting if someone familiar with LQG could take a look at that paper and see if they understand any of it.

that passed without comment and now at http://pirsa.org/C13029/1 it seems the organizers have thought it relevant to schedule back-to-back talks:

Livine: Spinor and Twistor Networks in Loop Gravity

and

Skinner: Twistor Strings for N=8 Supergravity

The latter has a nearly subliminal mention of the SBT at the start. So it appears that there is some sentiment that there may connections.