Abhay Ashtekar has just posted a surehanded insightful survey of the main approaches to QG, focusing on current Loop hamiltonian and spinfoam developments. The first 8 or 9 pages give historical perspective. The next section gives a pedagogical introduction which will serve well the needs of newcomers. The last third is a perceptive account of what problems are currently driving Loop research and what potential developments he sees on the horizon. This last was especially interesting. http://arxiv.org/abs/1201.4598 Introduction to Loop Quantum Gravity Abhay Ashtekar (Submitted on 22 Jan 2012) This article is based on the opening lecture at the third quantum geometry and quantum gravity school sponsored by the European Science Foundation and held at Zakopane, Poland in March 2011. The goal of the lecture was to present a broad perspective on loop quantum gravity for young researchers. The first part is addressed to beginning students and the second to young researchers who are already working in quantum gravity. 30 pages, 2 figures.
Among many good points Ashtekar makes is this one where he removes a possible cause of misunderstanding by clarifying the incremental progress which is the goal of quantum relativity---not full unification but possibly a key step in that direction. ==quote page 13== ...as is the case with classical general relativity, while requirements of background independence and general covariance do restrict the form of interactions between gravity and matter fields and among matter fields themselves, the theory would not have a built-in principle which determines these interactions. Put differently, such a theory would not be a satisfactory candidate for unification of all known forces. However, just as general relativity has had powerful implications in spite of this limitation in the classical domain, quantum general relativity should have qualitatively new predictions, pushing further the existing frontiers of physics. Indeed, unification does not appear to be an essential criterion for usefulness of a theory even in other interactions. QCD, for example, is a powerful theory even though it does not unify strong interactions with electro-weak ones. Furthermore, the fact that we do not yet have a viable candidate for the grand unified theory does not make QCD any less useful. ==endquote== I think it's clear that QG may turn out to be one of the steps along the road to a unified theory. But it is not itself a unification of forces. It aims to provide a quantum theory of geometry and matter without gettting into details about different matter species. Call it geometry-and-(generic)-matter if you like. Just as classic 1915 GR involves matter, so should the corresponding quantum theory.
I've always been unclear as to how the boundary state is chosen. It's interesting that he agrees a boundary state is the way to go, but that it's still not clear how to choose it (p25-26). It's also interesting that he's considers linking up with string theory (p26). Wouldn't that indicate that the canonical programme shouldn't work since it's meant to be a pure gravity theory?
Just to be clear about it. GR is not a pure gravity theory. The right hand of the equation is matter, the left hand is geometry. It is about the relationship between geometry and matter. quantum GR is not intended to be a pure gravity theory either. But in developing QG one can certainly work on limited cases with very simple matter, or a restricted amount of matter etc. One of the more interesting ideas for this was described on pages 19-20 (Domagala et al). Atyy I see no indication that he favors linking up with string, or believes that the theory needs it. What you refer to is a short passage on page 26 where he is speculating about future directions in research that MIGHT be explored. That is part of the job of a survey paper like this. He is laying out research possibilities to a broad audience of newcomers to the field and mentioning various things that might appeal to them. The paper is short---only 27 pages plus references. He mentions a lot of different ideas for research. At the end of that short paragraph on page 26 he says http://arxiv.org/abs/1201.4598: "string theory has two a priori elements: unexcited strings which carry no quantum numbers and a background space-time. Loop quantum gravity suggests that both could arise from the quantum state of geometry, peaked at Minkowski (or, de Sitter) space. The polymer-like quantum threads which must be woven to create the classical ground state geometries could be interpreted as unexcited strings. Excitations of these strings, in turn, may provide interesting matter couplings for loop quantum gravity."
That's something we can try to figure out. In the paragraph you are talking about he is essentially disussing Rovelli's work on the graviton propagator, or 2-point function. That work was done around 2007. I will get a link. IIRC the spinfoam (representing the dynamics) was caged inside a fixed spin-network which had labels that determined the distances. The spin network was the boundary state and allowed you to control the distance that the graviton was supposed to propagate. It was supposed to be attenuated by distance, according to inverse square.
GR is a pure gravity theory in the sense that the gravity degrees of freedom exist without matter, eg. the Schwarzshild solution. This was the initial point of view of canonical LQG. The contrasting viewpoint is unification, as tried by strings. So if loops and strings are related as Ashtekar speculates, then I don't think canonical LQG can work (or at least it's original philosophy wouldn't, maybe canonical LQG secretly contains matter).
Yes, that's be helpful! One thing I don't understand is how spacetime can have a boundary - wouldn't that require AdS space?
There are two main issues with LQG as of today: - incomplete understanding of quantization including dynamics (Hamiltonian, constraints, consistency, LQG and SF models) - coupling to matter and renormalization The first point is rather technical so I think it's clear why Ashtekar does not discuss these topics; the second is of major relevance due to the asymptotic safety program and the question of non-Gaussian fixed points when matter is coupled to gravity.
That's something we can try to figure out. In the paragraph you are talking about he is essentially disussing Rovelli's work on the graviton propagator, or 2-point function. That work was done around 2005-2008. I will get a link. IIRC the spinfoam (representing the dynamics) was caged inside a fixed spin-network which had labels that determined its proportions. The spin network was the boundary state and allowed you to control the distance that the graviton was supposed to propagate. It was supposed to be attenuated by distance, according to inverse square. The germ of the idea of using a fixed boundary state is in the 2005 paper. Beginning this far back may make it easier to understand because the earlier exposition spells it out in more detail. http://arxiv.org/abs/gr-qc/0508124 Look on page 3 The boundary state cage is just going to be the spin network bounding a 4 simplex! The spinfoam is just inside the 4 simplex itself. Everything is reduced to simplest form. It's going to get more complicated in the next paper but for now it's extremely rudimentary. At the top of page 3: "... The sums over permutations in the propagator give rises to a number of terms. Each of these can be interpreted as a spinfoam σ, by identifying closed sequences of contracted deltas as faces. Hence the amplitude.. can be written as a sum of amplitudes of spinfoams bounded by a given spinnetwork W.... an expression that is naturally interpreted (and can also be derived) as a sum over discretized 4-geometries bounded by a given discretized 3-geometry, namely as a definition of the Misner-Hawking sum-over-geometries formulation of quantum gravity, ..." I think that was the first graviton propagator paper---then there were a series 2006-2008 which eventually led to the replacement of the Barrett-Crane model by the EPRL. The next paper was 2006 http://arxiv.org/abs/gr-qc/0604044 see Figures 1, 2, 3...,6 on pages 26-30 By that time as you can see they are already using more complicated boundaries enclosing more complicated foams. But the germ of the idea was already in Rovelli's 2005 paper. The process did not stop until they had discovered there was trouble with the Barrett-Crane foam and replaced it (by around 2009)---then the dust kind of settled on that and there was the new formulation of LQG in 2010 and 2011. You could say that the conclusion of that arc of transition was the Zakopane Lectures http://arxiv.org/abs/1102.3660. When Ashtekar talks about "boundary state" he is acknowledging all that. The work on the graviton propagator was really critical. But now I think the field is ready for another unpredictable move. Ashtekar's paper should be a good one to study while trying to imagine what that could be.
The boundary is what divides the quantum experiment from the guy in the white coat. It defines a finite region of spacetime, whose geometry we are going to study. When Rovelli uses the "boundary formalism" he is not suggesting that the whole of the universe has a boundary. You can think of what the box encloses as approximately Minkowski space---not even as fancy as deSitter or anti-dS. The whole idea was to be able to derive an inverse square law. The boundary here is somewhat analogous to the box in which Schroedinger cat sits. It helps to define what the external experimenter can measure and observe. The boundary helps to distinguish between the quantum system being studied and the (classical?) world of the observer outside. Philosophically that's what it represents I think.
I understand it as a low energy approximation - maybe like what Giddings discusses in http://arxiv.org/abs/1105.2036. But if I recall from Rovelli's http://arxiv.org/abs/1102.3660, it seems that the whole spin foam framework requires this boundary to calculate anything - how can that be the case for cosmology - ie. outside a particle physics experiment? I suppose I should see how Vidotto approaches this.
Here's a recent paper by Etera Livine and a co-author in Dittrich group at AEI, Meché Martin-Benito. http://arxiv.org/abs/1111.2867 Classical Setting and Effective Dynamics for Spinfoam Cosmology The development of spinfoam approach to cosmology is just beginning. Still at rudimentary toymodel stage. This paper is probably the most recent window on these beginnings. It reminds me that the boundary can be disconnected. It can consist of an initial state and a final state. Offhand I don't see how this can deal with anything but a spatially finite universe like hypersphere S^{3} or 3-torus T^{3}. One would pick some arbitrary interval of time like from one minute before bounce to one minute after bounce. And fix some initial and final quantum states of geometry----initial and final spin network states. Then the boundary consists of two disconnected components. And the bulk is spinfoam histories that bridge between initial and final. That picture is more aligned with the "transition amplitude" language. I'll get a page reference. You can see from the Table of Contents that it is mostly about HAMILTONIAN approach but the last section, section IV, gets into spinfoam cosmology: IV. Spinfoam Dynamics 23 A. The Spinfoam Cosmology Setting 23 B. Spinfoam Amplitude and Dynamics for BF Spinfoam 26 C. Asymptotic Behavior and FRW Equation 28 D. Recovering the Hamiltonian Constraint 29 E. How to Depart from Flat Cosmology? 31 F. Cosmological Dynamics with Holomorphic Simplicity Constraints 32 Here is an excerpt from page 25: ==quote== 4. The Group Field Theory Point of View and the Issue of Renormalization Here, we have taken the point of view of fixing both the boundary graph Γ on which our spin networks live and the bulk spinfoam 2-complex ∆. Our goal is to compute the corresponding spinfoam amplitudes describing the evolution and dynamics of the spin networks for this fixed choice of bulk structure and interpret as a mini-superspace model (for cosmology). An alternative would be to fix the structure of the boundary but sum over all “admissible” bulks. In order to do this, we need to define the list of admissible 2-complexes and to fix their relative weights in the sum. This is done automatically by the group field theory formalism which provides us with a non-perturbative definition of the sum over spinfoam histories for fixed boundaries (see e.g. [32, 33]). ==endquote== Incidental BTW http://www.iem.csic.es/departamentos/qft/CV/CV_Martin-Benito.html I'm just guessing Meché as a nickname for Mercedes. A friend in Bogota Colombia goes by Mechás but I think Meché is more common.
It was a 2007 paper by Rovelli and Alesci. I'll look it up. http://arxiv.org/abs/0708.0883 The complete LQG propagator: I. Difficulties with the Barrett-Crane vertex
That reminds me! I'd like to find a good way for a new person to find out about the Loop Gravity UNSETTLED HAMILTONIAN SITUATION. As far as I know (AFAIK) there is no Hamiltonian at present, only several proposals. They have not been fully worked out. In at least one case a proposal has been worked out in 3D but not in 4D. In at least one other case only an idea has been presented. I would like to get other people's ideas. Some may think that LQG has a definite hamiltonian (they may disagree with me.) First, I can give some indication of the unsettled situation by linking to some technical papers but this is definitely NOT A GOOD INTRODUCTION because exploratory proposals are the complete opposite from textbook-expository style introductions. So these are just things to have heard of and realize how much in flux the situation is. Just to have heard of, not even to know anything definite about. Laurent Freidel is certainly someone to watch and he and Valentin Bonzom have one: http://arxiv.org/abs/1101.3524 The Hamiltonian constraint in 3d Riemannian loop quantum gravity "...This fills the gap between the canonical quantization and the symmetries of the Ponzano-Regge state-sum model for 3d gravity." Carlo Rovelli and Alesci have one: http://arxiv.org/abs/1005.0817 A regularization of the hamiltonian constraint compatible with the spinfoam dynamics "...The resulting constraint can generate the 1-4 Pachner moves and is therefore more compatible with the dynamics defined by the spinfoam formalism. We calculate its matrix elements and observe the appearence of the 15j Wigner symbol in these." Etera Livine and Valentin Bonzom have one: http://arxiv.org/abs/1110.3272 A new Hamiltonian for the Topological BF phase with spinor networks "...We introduce a new scalar Hamiltonian, based on recent works in quantum gravity and topological models, which is different from the plaquette operator..." It's really important that the Hamilton be graph-changing, and e.g. be capable of a 1-to-4 Pachner move. Space can expand by giving birth to new vertices. I don't understand how this deficiency persisted so long. It's a good sign that the 15j Wigner symbol shows up (basic to spinfoam dynamics). Also I just noticed that Valentin Bonzom, a young postdoc researcher, shows up in two of the three cases.
Thanks a lot. And I have another question: is there any good and easy-to-read reference on Hamilton Constraint? I checked some Thiemann's paper, like Quantum Spin Dynamics series, which is very hard to follow? I wish I could hear from your recommendation. Thanks.
By coincidence I just started responding to that question a few minutes ago in the preceding post! Your question reminded me! Here is what I had written so far: ==quote post #15== ...I'd like to find a good way for a new person to find out about the Loop Gravity UNSETTLED HAMILTONIAN SITUATION. As far as I know (AFAIK) there is no Hamiltonian at present, only several proposals. They have not been fully worked out. In at least one case a proposal has been worked out in 3D but not in 4D. In at least one other case only an idea has been presented. I would like to get other people's ideas. Some may think that LQG has a definite hamiltonian (they may disagree with me.) First, I can give some indication of the unsettled situation by linking to some technical papers but this is definitely NOT A GOOD INTRODUCTION because exploratory proposals are the complete opposite from textbook-expository style introductions. So these are just things to have heard of and realize how much in flux the situation is. Just to have heard of, not even to know anything definite about. Laurent Freidel is certainly someone to watch and he and Valentin Bonzom have one: http://arxiv.org/abs/1101.3524 The Hamiltonian constraint in 3d Riemannian loop quantum gravity "...This fills the gap between the canonical quantization and the symmetries of the Ponzano-Regge state-sum model for 3d gravity." Carlo Rovelli and Alesci have one: http://arxiv.org/abs/1005.0817 A regularization of the hamiltonian constraint compatible with the spinfoam dynamics "...The resulting constraint can generate the 1-4 Pachner moves and is therefore more compatible with the dynamics defined by the spinfoam formalism. We calculate its matrix elements and observe the appearence of the 15j Wigner symbol in these." Etera Livine and Valentin Bonzom have one: http://arxiv.org/abs/1110.3272 A new Hamiltonian for the Topological BF phase with spinor networks "...We introduce a new scalar Hamiltonian, based on recent works in quantum gravity and topological models, which is different from the plaquette operator..." It's really important that the Hamilton be graph-changing, and e.g. be capable of a 1-to-4 Pachner move. Space can expand by giving birth to new vertices. I don't understand how this deficiency persisted so long. It's a good sign that the 15j Wigner symbol shows up (basic to spinfoam dynamics). Also I just noticed that Valentin Bonzom, a young postdoc researcher, shows up in two of the three cases. ==endquote== In addition to those three, there is also another Hamilton proposal from Etera Livine, Daniele Oriti, and James Ryan http://arxiv.org/abs/1104.5509 Effective Hamiltonian Constraint from Group Field Theory "...Our strategy is to expand group field theories around non-trivial classical solutions and to interpret the induced quadratic kinematical term as defining a Hamiltonian constraint on the group field and thus on spin network wave functions..."
I see that in this list of 4 papers proposing Hamiltonians for LQG Livine and Bonzom both appear twice. So they are people to watch as we look for the establishment of a satisfactory Loop Hamiltonian, and also likewise are Freidel, Rovelli, Alesci, Oriti, and Ryan. It's pretty exciting. Starting around 2009 or 2010 Loop research began a period of rapid development. Much of what people are dealing with is of fairly recent origin. To respond to your question, which was specifically about INTRODUCTORY material. I would say this 1. One way into the subject is through Loop cosmology. That is a radically simplified version of LQG. It has a definite Hamiltonian. It says stuff about the beginning of expansion. The universe is much simpler than the general theory because it looks like on average constant curvature and there is a "universe time" that cosmologists use. The main authority in the application to cosmology is Abhay Ashtekar so you can just browse his papers on arxiv until you find something suitable. He has one called "Introduction to LQG through cosmology." He has a recent pedagogical review of straight LQG which is the topic of this thread. 2. Since the Hamiltonian approach to LQG is still unsettled and not yet ripe for an introductory presentation IMHO, another way to get into the subject is to learn the spinfoam approach. For example http://arxiv.org/abs/1102.3660. If that is not suitable, there are more introductory treatments, I could try to help dig up some. 3. A straightforward approach that might provide an introduction to the OLD (Thiemann) version of the Loop Hamiltonian? This would work if you are near a college or university and can use the library. If they don't have this textbook, suggest they get a copy! The section on the Hamiltonian constraint is pages 117-123. http://www.amazon.com/First-Course-Loop-Quantum-Gravity/dp/0199590753 A First Course in Loop Quantum Gravity Rodolfo Gambini, Jorge Pullin Oxford University Press. I haven't looked at the Gambini Pullin textbook myself so I can't reliably recommend. But as a first course text for advanced undergrads it shouldn't be too dense. You could browse a library/bookstore copy without buying, to be sure. I'll keep thinking about this, Karmerlo, and may have something more in a day or two. Also others perhaps with a completely different point of view, may have suggestions!
Thanks! I'll read it. Another introduction to the Thiemann Hamiltonian is http://arxiv.org/abs/1007.0402 Introductory lectures to loop quantum gravity Pietro Doná, Simone Speziale