Introduction To Loop Quantum Gravity

marcus

the Einstein action measures roughly speaking how much some spacetime is off from being a well behaved classical solution of the classical Einstein equation of General Relativity. So if you minimize the einstein action it you get the classical equation back.
so it measures how much the "path" is screwing up and getting distracted from its studies and cutting classes and taking dope and all what it isnt supposed to be doing---how "busy" it is with messing up---that "busy-ness" is the action. believe me you want to cut down on it.

a spacetime is just a path from the beginning of the world to the end, a path in "geometry space" if you can picture the space of all geometries.

the quantum idea is the universe doesnt just follow one path, it is a fuzzy mixture, well that is a rather distracting idea so lets not get into that.

the extremely beautiful thing is that with simplex geometries, with geometries built of triangles, YOU CAN IMPLEMENT THE ACTION FUNCTION JUST BY COUNTING DIFFERENT KINDS OF TRIANGLES

so even a computer, merely able to count up things in its memory, can do it

so we get back to our story where Loll and friends are running a computer model of spacetime, and the model is doing "sweeps" consisting of a million or so "Monte Carlo moves" which are localized elementary rearrangements of the simplex building blocks

each time they roll the dice and pick a monty move at random, they calculated some "badness" or "action" numbers to see whether to ACCEPT OR REJECT the proposed move!

this is how the localized microscopic "DYNAMICAL PRINCIPLE" that Loll talks about enters into the picture

it is this action principle operating at a microscopic Planckian or even maybe sub-Planckian level that the overall spacetime grows from. However it looks, whether it has 4 dimensions or 3 dimensions or some fraction etc, whatever its geometry, it grows out of many many local applications of the action principle at micro-scale.

I will try to find a quote.

marcus

Yeah, this is going to seem very dry and overdetailed but it shows how the quantum spacetime dynamics, the path integral action principle, was implemented:

<<3. Numerical implementation

We have investigated the infinite-volume limit of the ensemble of causal triangulated four-dimensional geometries with the help of Monte Carlo simulations at finite four-volumes N₄ = N_(4,1) + N_(3,2) of up to 362,000 four-simplices. A simplicial geometry is stored in the computer as a set of lists, where the lists consist of dynamic sequences of labels for simplices of dimension n from zero to four, together with their position and orientation with respect to the time direction. Additional list data include information about nearest neighbours, i.e. how the triangulation “hangs together”, and other discrete data (for example, how many four-simplices meet at a given edge) which help improve the acceptance rate of Monte Carlo moves. The simulation is set up to generate a random walk in the ensemble of causal geometries of a fixed time extension t. The local updating algorithm consists of a set of moves that change the geometry of the simplicial manifold locally, without altering its topological properties. These can be understood as a Lorentzian variant of (a simplified version of) the so-called Alexander moves [21, 22, 23], in the sense that they are compatible with the discrete time slicing of our causal geometries. For example, the subdivision of a four-simplex into five four-simplices by placing a new vertex at its centre is not allowed, because vertices can only be located at integer times tau . Details of the local moves can be found in [8]. As usual, each suggested local change of triangulation is accepted or rejected according to certain probabilities depending on the change in the action and the local geometry. (Note that a move will always be rejected if the resulting triangulation violates the simplicial manifold property.) The moves are called in random order, with probabilities chosen in such a way as to ensure that the numbers of actually performed moves of each type are approximately equal. We attained a rather high average acceptance rate of about 12.5%, which was made possible by keeping ...>>

By the way Alexander wrote his book in 1930. that is how far these "Monty Carlo moves" go back. they are just modified Alexander moves. Pachner is also cited. So this his how they "shuffle the deck".

Now I can say what part the Wick rotation plays. The Wick rotation changes the complex weights into real weights which can be dealt with as PROBABILITIES in this process of choosing the next random move, in "shuffling the deck" or randomizing spacetime geometry by Monte Carlo moves.

the probabilities enter each time you do a local rearrangement of some building blocks, you check whether that local microscopic rearrangement would be favored or disfavored by the Einstein equation. you do that by comparing badness. and it is still random----there is still always a chance that you can do a move that increases the badness, that happens lots in fact---but the probabilities are weighted against it (the House of general relativity wins over the long run). well maybe that is too impressionistic an impression.

I promised in Quantum Graffiti thread to say something about Wick rotation and Einstein Hilbert action

Cinquero

Is there a general theorem available for that purpose? I believe to remember that the mathematics behind that is usually quite non-obvious.

selfAdjoint

It's a procedure in complex variables, called analytic continuation.

Cinquero

So Wick rotation = analytic continuation?

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 [tex]e^{ut}[/tex], 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 [tex]i\tau = t[/tex], or euclidean four space. Then the integrals convege because the factors now read [tex]e^{ui\tau}[/tex] which is bounded for all tau. Then after you evaluate the integrals (they mostly reduce to a gaussian quadrature) you can rotate back.

CarlB

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 [tex]i\hbar[/tex] in the exponential) to a statistical mechanics (which has a [tex]-k/T[/tex] in the exponential). Your note is essentially saying that [tex]+k/T[/tex] 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.

But that's not why I was reading the thread.

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

marcus

"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."

marcus

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.

marcus

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.

marcus

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

marcus

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)

marcus

A major chronological bibliography for LQG
Over a thousand papers (arranged by date) often with arxiv numbers making online access easy
Over forty books and PhD dissertations.
Plus miscellaneous other useful sources of information.

http://www.arxiv.org/abs/gr-qc/0509039
Bibliography of Publications related to Classical Self-dual variables and Loop Quantum Gravity

Alejandro Corichi, Alberto Hauser
45 pages
"This bibliography attempts to give a comprehensive overview of all the literature related to what is known as the Ashtekar-Sen connection and the Rovelli-Smolin loop variables, from which the program currently known as Loop Quantum Gravity emerged..."

Corichi gives some guidance as to his own judgement of what are good introductions, primers, surveys, mathematical treatments.

======================================
Dan Christensen's SpinFoam website at U Western Ontario
is another resource for people wanting to get acquainted with LQG and related QG

http://jdc.math.uwo.ca/spin-foams/

he has links to things sorted out by topic, and level and different users' needs and purposes, and he has some links to some Greg Egan JAVA applets. Seeing how he organizes things gives you a practical overview of QG from his perspective.

Dan says he has room for some more grad students and postdocs in his QG/computation program. It looks like anybody who might want to study QG (or massive parallel computation applied to QG) should probably check this out.

----------------------------
EDIT TO REPLY TO CINQUERO
Hi Cinquero, since i can still edit this I will reply this way and save making a new post. Please go to Dan Christensen site. He has many links in an organized convenient form. If there is anything that you need a further PDF link for, tell me what it is is and I will try to find it. I am not certain I understand your request for links to PDF----was it links to things found at Dan's UWO page or for something else?

Cinquero

Thx!

But could someone please add hyperlinks to the PDF output? :-)))

marcus

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

Cinquero

Actually, mys request for hyperlinks was in regard to:

"Bibliography of Publications related to Classical Self-dual variables and Loop Quantum Gravity"

:)

marcus

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.