# Introduction To Loop Quantum Gravity

by marlon
Tags: loop quantum gravity
P: 31
 Quote by selfAdjoint It's a procedure in complex variables, called analytic continuation.
So Wick rotation = analytic continuation?
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 Quote by Cinquero So Wick rotation = analytic continuation?
You have the integrals defined on the real axis, corresponding to Minkowski space, but they don't converge there, because they have factors like $$e^{ut}$$, which is unbounded as t goes to infinity. They are howeve analytic in the half plane above the real axis, and by continuation therefore on the imaginary axis, which correspond to $$i\tau = t$$, or euclidean four space. Then the integrals convege because the factors now read $$e^{ui\tau}$$ which is bounded for all tau. Then after you evaluate the integrals (they mostly reduce to a gaussian quadrature) you can rotate back.
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 Quote by selfAdjoint You have the integrals defined on the real axis, corresponding to Minkowski space, but they don't converge there, because they have factors like $$e^{ut}$$, which is unbounded as t goes to infinity. They are howeve analytic in the half plane above the real axis, and by continuation therefore on the imaginary axis, which correspond to $$i\tau = t$$, or euclidean four space. Then the integrals convege because the factors now read $$e^{ui\tau}$$ which is bounded for all tau. Then after you evaluate the integrals (they mostly reduce to a gaussian quadrature) you can rotate back.
You've got that exactly reversed. It's the imaginary exponentials that fail to converge and are converted by Wick rotation. Another way of saying the same thing is that a Wick rotation takes a QFT (which has a $$i\hbar$$ in the exponential) to a statistical mechanics (which has a $$-k/T$$ in the exponential). Your note is essentially saying that $$+k/T$$ diverges and this is true, but the solution in a Wick rotation is to rotate in the opposite direction. That way you end up with exponentials that converge. This all reminds me of the method of "steepest descent"[sp] that is used in Schroedinger's equation.

Stephen Hawking's latest paper uses "Euclidean Quantum Gravity":
http://arxiv.org/PS_cache/hep-th/pdf/0507/0507171.pdf

Does EQG have anything to do with LQG? My field is elementary particles, not gravitation. Sorry for the laziness. Hawking references a book I don't have immediate access to.

Carl
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 Quote by CarlB ... Stephen Hawking's latest paper uses "Euclidean Quantum Gravity": http://arxiv.org/PS_cache/hep-th/pdf/0507/0507171.pdf Does EQG have anything to do with LQG? My field is elementary particles, not gravitation. Sorry for the laziness. Hawking references a book I don't have immediate access to. Carl
"Euclidean QG" developed by hawking and friends in 1980s was a path integral AFAIK
and so it would be closer akin to Renate Loll Lorentzian path integral by CDT method ("causal dynamical triangulations") that we hear a lot about these days

Hawking never got Euclidean path integral to work, but he uses it to think with. It sounds a bit eccentric for him to call it the "only sane way to do nonperturbative QG"
the Lorentzian path integral people (Loll et al) have an equally nonperturbative approach that they are getting results with, including confirming a conjecture or two of hawking. No way is Loll's approach not sane. It is at least as sane as the Euclidean version.

I need to get you some online links. there is a 1998 survey of QG methods by rovelli which describes hawking Euclid. path integral. More recent online stuff do not discuss hawking's method very much because it is long obsolete except for him and one or two proteges. But I will get the link to the 1998 survey

Yes, here:
http://arxiv.org/abs/gr-qc/9803024
Strings, loops and others: a critical survey of the present approaches to quantum gravity
Carlo Rovelli
Plenary lecture on quantum gravity at the GR15 conference, Pune, India

"I review the present theoretical attempts to understand the quantum properties of spacetime. In particular, I illustrate the main achievements and the main difficulties in: string theory, loop quantum gravity, discrete quantum gravity (Regge calculus, dynamical triangulations and simplicial models), Euclidean quantum gravity, perturbative quantum gravity, quantum field theory on curved spacetime, noncommutative geometry, null surfaces, topological quantum field theories and spin foam models. I also briefly review several recent advances in understanding black hole entropy and attempt a critical discussion of our present understanding of quantum spacetime."
 PF Patron Sci Advisor P: 22,330 http://arxiv.org/abs/gr-qc/9803024 Strings, loops and others: a critical survey of the present approaches to quantum gravity Carlo Rovelli Section B. "Old hopes (becoming) approximate theories" ---quote Rovelli--- B. Old hopes -> approximate theories 1. Euclidean quantum gravity Euclidean quantum gravity is the approach based on a formal sum over Euclidean geometries [[my comment: HERE ROVELLI GIVES THE PATH INTEGRAL, BUT I CAN'T COPY IT EASILY, it is labelled equation (6)]] As far as I understand, Hawking and his close collaborators do not anymore view this approach as an attempt to directly define a fundamental theory. The integral is badly ill defined, and does not lead to any known viable perturbation expansion. However, the main ideas of this approach are still alive in several ways. First, Hawking’s picture of quantum gravity as a sum over spacetimes continues to provide a powerful intuitive reference point for most of the research related to quantum gravity. Indeed, many approaches can be sees as attempts to replace the ill defined and non-renormalizable formal integral (6) with a well defined expression. The dynamical triangulation approach (Section IVA) and the spin foam approach (Section VC2) are examples of attempts to realize Hawking’s intuition. Influence of Euclidean quantum gravity can also be found in the Atiyah axioms for TQFT (Section VC1). Second, this approach can be used as an approximate Second, this approach can be used as an approximate method for describing certain regimes of nonperturbative quantum spacetime physics, even if the fundamental dynamics is given by a more complete theory. In this spirit, Hawking and collaborators have continued the investigation of phenomena such as, for instance, pair creation of black holes in a background de Sitter spacetime. Hawking and Bousso, for example, have recently studied the evaporation and “anti-evaporation” of Schwarzschild-de Sitter black holes [61]... ---end quote--- Equation (6) here looks very much like Loll's dynamical triangulations path integral. but they start with exp(iS) where S is the Regge form of Einst action. Loll et al do a Wick rotation to get a euclidean version which gets used in the computer calculations. This equation (6) is still very much like what Loll CDT starts with, but instead of a metric [g] there is a TRIANGULATION T. so they are summing over all triangulations of a particular kind. Otherwise it looks formally the same. However there is a practical difference in that Loll et al can actually calculate. they do the sum (using montecarlo method) and get results. some of these results have born out hawking conjectures, so they cite him a lot. but his particular type of (euclidean) path integral i dont think any significant effort is being made to use it. to compare hawking EQG with current CDT path integral, have a look at the first 2 or 3 pages of these two papers http://arxiv.org/hep-th/0105267 http://arxiv.org/hep-th/0505154 you will see how close the CDT path integral is to Hawking's euclidean one.
 PF Patron Sci Advisor P: 22,330 in the past couple of pages of this thread we have been responding to questions from CarlB and cinquero and it may be time to regroup. I decided earlier that unless there is some reason not to do so we ought to make this thread serve as an introduction NOT ONLY to narrowly defined canonical LQG but to the main approaches to NONPERTURBATIVE QUANTUM GRAVITY. That includes canonical LQG but also spin foams, and other path integral approaches like CDT. selfAdjoint, at one point, proposed the term "Background Independent Quantum Gravity" for the general field. Renate Loll seems to favor "Nonperturbative QG". The organizers of the Loop 05 conference use the collective modifier "background independent/nonperturbative" And Lee Smolin has started to say "relational". But I think "nonperturbative" is going to win out as the mainest of mainstream term. As sideline observers we can't reform language, just have to go with the prevailing talk. I think one of the ambient ideas in the Loop 05 conference is that if you can forge a concept "NQG" and impress on people's minds the idea that there is research in "nonperturbative quantum gravity" then maybe a few more universities will establish professorships in NQG or faculty positions of some kind in NQG. It will be perceived as a lack not to have some research in nonperturbative QG being conducted in the physics department. It also means recommending each other's graduate students. if it is a field then there is more solidarity than if it is just a bunch of splinter group research lines. Hermann Nicolai definitely would like some professorships in German universities that are echo or counterpart to his lines of reseach at AEI, he has talked about that in Die Zeit interview. And AEI is hosting Loop 05. so it is time to assemble into a research field with an identifying label which is not String, and to get it recognized that a physics department has an embarrassing GAP if it doesnt have some research under way in Nonper Quavity.
 PF Patron Sci Advisor P: 22,330 Let's recap the introduction to the triangulations approach---Loll CDT. Here is a reading list from earlier in this thread http://www.physicsforums.com/showthr...294#post585294 Here's a short popularization by Loll, at her website, written for general audience http://www.phys.uu.nl/~loll/Web/research/research.html This PF thread has more stuff like that http://www.physicsforums.com/showthr...9&page=1&pp=15 ======================== To give an idea of where the field is at the moment, I am simply going to quote, in its entirety, the first paragraph of each of Loll's three most recent papers. These papers are dated May, June, July 2005. The first paragraph of a research paper often gives a bit of an overview or some perspective on the field. This is a fastmoving field and this will be one way to keep up with where things are at the moment. We have no more recent survey available. http://arxiv.org/hep-th/0505154 Reconstructing the Universe http://arxiv.org/gr-qc/0506035 Counting a black hole http://arxiv.org/hep-th/0507012 Taming the cosmological constant...topology change Very encouraging progress has been made recently in constructing spacetime dynamically from a nonperturbative gravitational path integral, by studying the continuum limit of causal dynamical triangulations [1, 2, 3, 4]. The quantum geometries generated in this way exhibit semiclassical properties at sufficiently large scales: they are four-dimensional [5, 6] and the large-scale dynamics of their spatial volume is described by an effective cosmological minisuperspace action [7]. Their short-distance behaviour is highly nonclassical, including a smooth dynamical reduction of the spectral dimension from four to two [8] and evidence of fractality [6]. Despite recent progress [1, 2], little is known about the ultimate configuration space of quantum gravity on which its nonperturbative dynamics takes place. This makes it difficult to decide which (auxiliary) configuration space to choose as starting point for a quantization. In the context of a path integral quantization of gravity, the relevant question is which class of geometries one should be integrating over in the first place. Setting aside the formidable difficulties in “doing the integral”, there is a subtle balance between including too many geometries – such that the integral will simply fail to exist (nonperturbatively) in any meaningful way, even after renormalization – and including too few geometries, with the danger of not capturing a physically relevant part of the configuration space. Nonperturbative quantum gravity can be defined as the quest for uncovering the true dynamical degrees of freedom of spacetime geometry at the very shortest scales. Because of the enormous quantum fluctuations predicted by the uncertainty relations, geometry near the Planck scale will be extremely rugged and nonclassical. Although different approaches to quantizing gravity do not agree on the precise nature of these fundamental excitations, or on how they can be determined, most of the popular formulations agree that they are neither the smooth metrics... (or equivalent classical field variables) of general relativity nor straightforward quantum analogues thereof. In such scenarios, one expects the metric to re-emerge as an appropriate description of spacetime geometry only at larger scales. I'll try to interpret some---as time permits. but hopefully this is already fairly clear and doesn't need much explication
 PF Patron Sci Advisor P: 22,330 I had better keep a list of links to the prediction polls that folks at PF have so that when the time comes to look we can easily find the thread with the predictions Background independence talks at Strings 06 http://www.physicsforums.com/showthread.php?t=85207 (when the programme of talks is posted, check to see who was right) August-September hits on Smolin latest http://www.physicsforums.com/showthread.php?t=83578 (in late September 2005, start checking http://citebase.eprints.org/cgi-bin/...hep-th/0507235 to see if they are counting and registering downloads of "The case for background independence") String Forecast Poll http://www.physicsforums.com/showthread.php?t=81739 (around March 2006 check SLAC/Stanford for the 2005 HEP Topcites. This year around March 2005 they brought out the 2004 Topcites as usual. But they have not yet done the full job with Michael Peskin's review, which is worrisome. the list to check is whatever is analogous to this http://www.slac.stanford.edu/library...4/annual.shtml ) Will Loll etc. achieve sum over topologies in 4D? http://www.physicsforums.com/showthread.php?t=81626 (this prediction poll has no definite declared cut-off date, which was an oversight. we will have to use reasonableness and see whether, in a reasonable time, Loll et al manage to extend the results on topology change to higher dimensions)
 PF Patron Sci Advisor P: 22,330 Hi Cinquero, I responded to your post #82 by editing the previous post. Hope you saw the note. At the moment just need a place to stash the links to the audio of a two-part Bojowald talk given last Friday and concluded today at Penn State. He is talking about the LQG model Black Hole. the audio of the first part is here http://www.phys.psu.edu/events/index...-20.2005-12-25 Loop Quantum Cosmology of the Kantowski-Sachs Model Gravity Theory Seminar by Martin Bojowald from Albert Einstein Institute (Germany) Friday at 11:00 AM in 318 Osmond (9/16/2005) and the second part (which was today) is here http://www.phys.psu.edu/events/index...pe_ids=0&span= Spherically Symmetric Quantum Geometry Gravity Theory Seminar by Martin Bojowald from Albert Einstein Institute Friday at 11:00 AM in 318 Osmond (9/23/2005) at the same page there was also this audio http://www.phys.psu.edu/events/index...pe_ids=0;span= Generalizing Quantum Mechanics for Quantum Gravity IGPG Seminar by James Hartle from University of California, Santa Barbara Monday at 3:00 PM in 318 Osmond (9/19/2005) and this audio as well http://www.phys.psu.edu/events/index...-20.2005-12-25 Quantum Nature of the Big-Bang: Numerical Issues Gravity Theory Seminar by Thomas Pawlowski & Parampreet Singh Friday at 11:00 AM in 318 Osmond (9/9/2005) Ashtekar has announced that he has a paper, written with Thomas Pawlowski & Parampreet Singh, to appear about this topic: LQG picture of the big bang. Several of these seminar talks relate to the Ashtekar Bojowald collaboration about LQG of big bang and black hole, see for example their recent paper http://www.arxiv.org/gr-qc/0509075 Quantum geometry and the Schwarzschild singularity
 P: 31 Actually, mys request for hyperlinks was in regard to: "Bibliography of Publications related to Classical Self-dual variables and Loop Quantum Gravity" :)
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 Quote by Cinquero Actually, mys request for hyperlinks was in regard to: "Bibliography of Publications related to Classical Self-dual variables and Loop Quantum Gravity" :)
Ah!, I see what you mean. Corichi writes the URLs out for online sources, so one could paste them in and get to them, but in the PDF version these URLs do not automatically function as hyperlinks, as they might if he had provided an HTML version. I understand you may be joking, but it wouldn't be a bad idea for Corichi to make an up-to-date selective HTML bibliography of online quantum gravity sources.

If you want to encourage him to do this you could email him. Be sure to mention PF. He--or else a good friend of his--has often visited us, I believe, and supplied helpful information.
 PF Patron Sci Advisor P: 22,330 This is an update of post #56 which was about a book edited by Abhay Ashtekar scheduled to be published this year by World Scientific. here is the publisher's webpage http://www.worldscibooks.com/physics/5876.html A Hundred Years of Relativity. Several chapters of this book are already online as preprints: Martin Bojowald http://arxiv.org/abs/gr-qc/0505057 Elements of Loop Quantum Cosmology Larry Ford http://arxiv.org/abs/gr-qc/0504096 Rodolfo Gambini and Jorge Pullin http://arxiv.org/abs/gr-qc/0505023 Discrete space-time Hermann Nicolai http://www.arxiv.org/abs/gr-qc/0506031 Gravitational Billiards, Dualities and Hidden Symmetries Thanu Padmanabhan http://arxiv.org/abs/gr-qc/0503107 Understanding Our Universe: Current Status and Open Issues Alan Rendall http://arxiv.org/abs/gr-qc/0503112 Clifford Will http://arxiv.org/abs/gr-qc/0504086 Was Einstein Right? Testing Relativity at the Centenary ======== other stuff: Ashtekar there are several useful surveys, such as http://arxiv.org/abs/gr-qc/0410054 Gravity and the Quantum http://arxiv.org/abs/gr-qc/0404018 Background Independent Quantum Gravity: A Status Report
 PF Patron Sci Advisor P: 22,330 Agons, moments of truth. for some reason I keep thinking back to the times Smolin spoke up at the Toronto string panel discussion, and each time immediately afterwards he was put down by you know who. http://www.physicsforums.com/showthread.php?t=84585 http://www.fields.utoronto.ca/progra...005/panel.html and I remember Atiyah at Santa Barbara interrupted and almost derailed as he tried to get across his "old man's crazy thoughts". http://www.physicsforums.com/showthread.php?t=96806 http://online.kitp.ucsb.edu/online/strings05/atiyah/ and as if to compensate there is Gerard 't Hooft's response after listening to some strange and quite possibly wrong ideas from Atiyah---That sounds like physics!
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There may be hints of slow shift in research interest from string to non-string approaches to quantum gravity. The latter include Causal Dynamical Triangulations (CDT), spinfoams, Loop Quantum Gravity (LQG) and others on the Loops '05 conference programme.
this shift in research activity, if it exists, is hard to verify and measure statistically. here is one indicator---by itself not conclusive but something to watch along with the rest.
Last month at selfAdjoint's suggestion I included "heterotic, superstring" in the list of keywords and did a search using the Harvard ADS abstract service engine.
http://www.physicsforums.com/showthr...185#post789185
 I'll be glad to try. The main thing is just to have a fixed set of keywords you can apply year by year, to get the trend in papers with those keywords. In accordance with your suggestion, I checked how many papers were published each year (October thru September) with the words "brane" or "M-theory" or "AdS/CFT" or "superstring" or "heterotic" in the abstract. 2001 1202 2002 1097 2003 970 2004 959
For continuity I tried the same check today. This is now the papers November thru October, year by year, with any of the same keywords in the abstract.
2001   1220
2002   1083
2003    972
2004    938
Part of this could certainly have nothing to do with a concurrent increase in QG research output in the non-string lines of investigation. It is very iffy and difficult to link the two trends! But at a level of anecdotal evidence one does encounter cases of people who have switched over.

Since the effort in non-string QG is still small compared with string, this shift (if it is occurring) could be viewed simply as diversification. One could take it NOT AS A SIGN THAT ONE THING IS RIGHT AND ANOTHER WRONG but that for whatever reason people are branching out in more directions, and trying non-string ones.

I will try to get some figures on non-perturbative QG research output trends.
 PF Patron Sci Advisor P: 22,330 non-string QG research seems to have increased during the same period. For a rough indication of this I use the keyword search engine at arxiv.org, to find the number of preprints submitted each year with certain terms in the abstract. It gets some papers it shouldn't (that just happen to have the right keywords) and it misses some. Here the results: 2001 98 2002 121 2003 140 2004 184 In case anyone is interested here are links to these arxiv.org searches, and to some others just to have them handy. Year 2001: http://arXiv.org/find/grp_physics/1/...0/1/0/2001/0/1 Year 2002: http://arXiv.org/find/grp_physics/1/...0/1/0/2002/0/1 Year 2003: http://arXiv.org/find/grp_physics/1/...0/1/0/2003/0/1 Year 2004: http://arXiv.org/find/grp_physics/1/...0/1/0/2004/0/1 Last twelve months: http://arXiv.org/find/grp_physics/1/...0/1/0/past/0/1 Year to date, 2005: http://arXiv.org/find/grp_physics/1/...0/1/0/2005/0/1 BTW here I have been looking only at the 2001-2004 period. Can we say anything about current trends? Well it may take a while for the 2005 numbers to stabilize and it may be too soon to say anything much. But I think nonperturbative QG research is experiencing LOLL-SHOCK and is in a temporary lull where people are considering re-directing their efforts more in line with Causal Dynamical Triangulations (because of recent seemingly important results). John Baez has been very frank about this. Last month (October) he presented an overview of Spinfoam where he pointed out CDT results and asked could Spinfoam be modified (introducing an analogous causality structure?) to be more like CDT and could it get similar results, and then maybe it would surpass CDT because of inherent advantage in some other department. this was the talk he gave at Loops '05 and posted at his website. There is a lot of new stuff to digest right now, besides CDT there is Thiemann single constraint program, another approach which is not standard LQG and which look attractive to some LQG people, and there is Freidel's result that highlights DSR as a possibly necessary feature of QG. For whatever reason there has been almost no QG appearing on arxiv.org since the 10-14 October conference. Loll-shock is my best guess but their could be other reasons.
 PF Patron Sci Advisor P: 22,330 Here is a great talk by Sundance Bilson-Thompson given 16 November 2005 at Perimeter Institute http://streamer.perimeterinstitute.c...uldResize=true it is split screen, slides and video, he occasionally goes to the blackboard to explain stuff and the camera gets that too. he is talking about his preon model (simple basis for a sketchy approximation of the standard model) which he and Lee Smolin are currently trying to connect with the spin networks of LQG. Please let me know if this link does not work for you. It worked for me when I tried it.

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