Seeing the Monte Carlo moves

[Martin]

I think the most visual approach to Quantum Gravity is the one called dynamical triangulations

 

the specific type being most worked on is called CDT (causal dynamical triangulations) where the researchers can simulate in a computer the evolution of the geometry of the universe----they do randomized runs so as to explore the idea of a typical geometry or history.

 

Every geometry is specified by a triangulation, so you have to have a way for one triangulation to change into the next one and this happens by what they call, in their model, Monte Carlo moves

 

these are local changes in the triangulation

and the way the computer simulation works is that at some moment in time it has a triangulation of space and at every spatial point it considers the list of possible MOVES which would be local changes in the triangulation (erase the connection between two points and draw a new connection between two other points, but slightly more systematic than that) and with each move it consults a RANDOM number generator ("tosses a coin") to see whether to make that move or not.

 

So I can imagine the triangulation of space as kind of "flickering" as everywhere it is changing by these montecarlo moves

 

As an analogy think of 2D

if you put a point in the middle of a triangle and connect out to the vertices then you now have 3 triangles....thats a 2D move, call it (1,3) for 1 triangle changes to 3.

 

And there is the reverse move (3,1) where 3 triangles fuse into 1, and a point (the vertex they shared) is removed

 

And another 2D move is (2,2) where you have two triangles sharing a side (think of the side as horizontal) and you erase that side and connect the two vertices that werent connected before (think of drawing a vertical)

so again you have two triangles sharing a side.

 

In the 2D version of dynamical triangulations, these moves just described are enough to get from any triangulation to any other----so they are a complete set of Monte Carlo moves for the simple 2D situation.

[Martin]

Feynman introduced a method of studying the motion of a particle by averaging together all the possible ways the particle could take to get from point A to point B

 

in quantum theories things do not have a known trajectory where you know the things location all the way along-----I guess you can think of the particle as jittering around and trying all conceivable paths----each gets weighted with an amplitude and you get a weighted average over all possible paths and this lets you calculate the amplitude of getting from here to there.

 

in a Feynman path integral the wilder motions generally cancel out so that what it averages to may be fairly smooth and tame---it is a way of quantizing that gives right answers so people use it (tho nobody can say if the particle really does explore all the crazy possibilities included in the average)

 

Well a triangulation of space, at some moment, is analogous to the position of a particle at some moment. And the jittering around that the particle does is analogous to MonteCarlo moves, that the triangulation makes to get to the new way it is in the next moment.

 

so old triangles are always getting broken and new ones are always forming and the triangulation (which describes the geometry of space) is

constantly FLICKERING like that

 

and the CDT approach (causal dynam. triang.) is a "path-integral" approach where you do a weighted average of ALL the possible changes in spatial geometry that get you from geometry A to geometry B

 

so we need to crank the analogy up one more dimension and think of a 3D world, where instead of being triangulated in triangles it is triangulated in TETRAHEDRONS which are little pyramids with triangles for sides.

[Severian]

I don't understand what you mean by a 'traingulation' in this context. What is the traingulation supposed to represent?

 

I understand Feynman's path integrals (I think I may have to teach this next year) but your triangulation have me confused. Are the triangles somehow supposed to be a discretization of a 2D space time? i.e. building a 2D surface out of little triangles?

 

If so, wouldn't this be 'cheating' in that one would be regulating the ultraviolet divergence by introducting a finite triangulation size? I suppose approaching the continuum limit would bring back the divergence in the usual way (since the whole problem with Quantum GR is that it is non-renormalizable) - are they advocating that the finite triangulation is real, thus regulating the divergence automatically?

[Martin]

I don't understand what you mean by a 'triangulation' in this context. What is the triangulation supposed to represent?

 

It will be hard to reply concisely but bear with me

the original paper was by Tullio Regge in 1961 and called something like

"General Relativity without Coordinates"

at the time he was working with John Archibald Wheeler at Princeton

 

Regge's approach to Gen Rel divides the 4D spacetime manifold up into 4-simplexes

this is called "triangulation" because a 4-simplex is the 4D analog of a triangle

Regge was able to express the Einstein equation in discrete form in terms of the dimensions of the 4-simplex-----the lengths of the edges etc.

 

======

as you know there has been a huge struggle to quantize Gen Rel

in the 1980s an approach was worked on which used Regge formulas but approximated the 4D manifold by a bunch of 4-simplexes that were

all the same size.

then' date=' to calculate curvatures and such, for the einstein equation, one would COUNT stuff like number of vertices, number of edges, faces, volumes.

that works just as well as measuring the dimensions of an individual 4 simplex.

======

also in the 1980s Stephen Hawking developed his approach to quantum gravity called "path integral"

 

a solution to the einstein equation that gets you from a 3D manifold A to a 3D manifold B is LIKE a path that a particle takes from point A to point B

 

except that instead of a path it is a 4D spacetime manifold with boundary A and B.

======

I am not sure why, but the Hawking path integral approach seems to have gotten bogged down

 

However someone else, Jan Ambjorn, in the 1980s and 1990s was working on an alternative path integral approach using Dynam. Triangulations.

 

He also called it "Simplicial Quantum Gravity" because it uses simplexes.

=======

 

It probably has a lot of kinship with Lattice QFT, except that there is no fixed lattice. there is a unpredetermined cluster of 4 simplexes making a 4D piecewise linear manifold

 

in fact there is an infinite number of these-----like the infinite number of paths for the particle to get from A to B in the feynman path integral.

 

==========

I am sorry to have such a sketchy knowledge of the history.

this is all the murky past.

 

==========

Now about 1998 Jan Ambjorn found a way out of the woods.

 

For some years he and his coworkers had been doing montecarlo computer simulations of 2D, 3D and 4D simplicial gravity (or dynam. triang.) and all thru the 1990s it was NOT WORKING------they kept getting crumpled or fractally things

 

but around 1998, I think, they got an idea.

===========

 

this finally culminated in 2004 when the computer simulation started to produce nice 4D worlds

 

and then they did some study of these 4D spacetimes---averaging their statistics and stuff----and they found that it reproduces a semiclassical

cosmology behavior of hawking, hartle, villenkin etc. going back to the 1980s.

So there was some proper largescale behavior in 4D

 

they had already gotten good classical or semi-classical behavior in lower dimensions but they finally got it in 4D this year.

 

-----------

The history of Dynam. Triang. as an approach to quantizing Gen Rel is the

important thing to get down first of all. I have been trying to sketch it.

But one way would be to list some Ambjorn papers in chronol. order.

 

i will do this and let the titles speak for themselves.

 

the last two papers will be from 2004:

[b']Emergence of a 4D world from Causal Quantum Gravity[/b]

 

A Semiclassical Universe from First Principles

 

many of the earlier papers describe things NOT working,

like a proliferation of "baby universe" blemishes, or

crumpled worlds, or fractally feathery worlds

anyway it took Ambjorn and friends a long time

but now it is one of the more promising quantum gravity approaches

 

I will list some of the papers over the past 4 or 5 years

[Martin]

Here are some early computer animation studies of lower dimension dynamical triangulation. this is a catalog, you click on the one you want to see animated.

http://www.nbi.dk/~ambjorn/lqg2/

 

here are a couple of good samples, saves you the trouble of choosing:

http://www.nbi.dk/~ambjorn/lqg2/d8p16384.gif

http://www.nbi.dk/~ambjorn/lqg2/d8p32400.gif

 

http://arxiv.org/hep-th/0001124

Lorentzian and Euclidean Quantum Gravity - Analytical and Numerical Results

Authors: J. Ambjorn, J. Jurkiewicz, R. Loll

Comments: 69 pages, 16 figures, references added

 

http://arxiv.org/hep-th/0002050

A non-perturbative Lorentzian path integral for gravity

Authors: J. Ambjorn (Niels Bohr Institute), J. Jurkiewicz (Jagellonian Univ.), R. Loll (Albert-Einstein-Institut)

Comments: 11 pages, LaTeX, improved discussion of reflection positivity, conclusions unchanged, references updated

Journal-ref: Phys.Rev.Lett. 85 (2000) 924-927

 

http://arxiv.org/hep-lat/0011055

Computer Simulations of 3d Lorentzian Quantum Gravity

Authors: J. Ambjorn, J. Jurkiewicz, R. Loll

Comments: 4 pages, contribution to Lattice 2000 (Gravity and Matrix Models), typos corrected

Journal-ref: Nucl.Phys.Proc.Suppl. 94 (2001) 689-692

 

http://arxiv.org/hep-th/0011276

Non-perturbative 3d Lorentzian Quantum Gravity

Authors: J. Ambjorn (NBI, Copenhagen), J. Jurkiewicz (U. Krakow), R. Loll (AEI, Golm)

Comments: 35 pages, 17 figures, final version, to appear in Phys. Rev. D (some clarifying comments and some references added)

Journal-ref: Phys.Rev. D64 (2001) 044011

 

http://arxiv.org/hep-th/0105267

Dynamically Triangulating Lorentzian Quantum Gravity

Authors: J. Ambjorn (NBI, Copenhagen), J. Jurkiewicz (U. Krakow), R. Loll (AEI, Golm)

Comments: 41 pages, 14 figures

Journal-ref: Nucl.Phys. B610 (2001) 347-382

 

http://arxiv.org/hep-lat/0201013

3d Lorentzian, Dynamically Triangulated Quantum Gravity

Authors: J. Ambjorn, J. Jurkiewicz, R. Loll

Comments: Lattice2001(surface)

Journal-ref: Nucl.Phys.Proc.Suppl. 106 (2002) 980-982

 

http://arxiv.org/gr-qc/0201028

Simplicial Euclidean and Lorentzian Quantum Gravity

Authors: J. Ambjorn

Comments: 23 pages, 4 eps figures, Plenary talk GR16

 

http://arxiv.org/hep-th/0201104

A Lorentzian cure for Euclidean troubles

Authors: J. Ambjorn, A. Dasgupta, J. Jurkiewicz, R. Loll

 

http://arxiv.org/hep-th/0307263

Renormalization of 3d quantum gravity from matrix models

Authors: J. Ambjorn (NBI, Copenhagen), J. Jurkiewicz (U. Krakow), R. Loll (Spinoza Inst. and U. Utrecht)

Comments: 14 pages, 3 figures

Journal-ref: Phys.Lett. B581 (2004) 255-262

 

http://arxiv.org/hep-th/0404156

Emergence of a 4D World from Causal Quantum Gravity

Authors: J. Ambjorn (1 and 3), J. Jurkiewicz (2), R. Loll (3) ((1) Niels Bohr Institute, Copenhagen, (2) Jagellonian University, Krakow, (3) Spinoza Institute, Utrecht)

Comments: 11 pages, 3 figures; some short clarifying comments added; final version to appear in Phys. Rev. Lett

Journal-ref: Phys.Rev.Lett. 93 (2004) 131301

 

http://arxiv.org/hep-th/0411152

Semiclassical Universe from First Principles

Authors: J. Ambjorn, J. Jurkiewicz, R. Loll

Comments: 15 pages, 4 figures

 

You can see various things, like Renate Loll used to be at Max Planck Institute Potsdam (AEI) and now she is at Gerard 't Hooft's Institute at Utrecht.

And as time goes on they call their approach various things----a path integral, simplicial QG, dynamical triangulations.....

you can get a certain amount of the history just by clicking on

the links and glancing at the abstracts.

I have boldfaced the titles of 3 or 4 papers that I found helpful, will try to summarize some of the main points

[Martin]

 

If so' date=' wouldn't this be 'cheating' in that one would be regulating the ultraviolet divergence by introducting a finite triangulation size? I suppose approaching the continuum limit would bring back the divergence in the usual way (since the whole problem with Quantum GR is that it is non-renormalizable) - are they advocating that the finite triangulation is real, thus regulating the divergence automatically?[/quote']

 

No they dont impute to nature a minimum lattice size

they let the triangulation size go to zero

and nevertheless still things dont blow up, they dont have trouble with divergences!

 

you really have to read their account of it, but one reason that there is

no infinitities is that it is a nonperturbative approach.

there is no background space, and no perturbation series

 

here is page 4 from http://arxiv.org/hep-th/0411152

"A necessary condition for obtaining a well-defined continuum limit from this regularized setting is that the lattice spacing... goes to zero while the number N of four-simplices goes to infinity in such a way that the continuum four-volume V := Na^4 stays fixed. Let us emphasize that the parameter a therefore does not play the role of a fundamental discrete length."

[Severian]

I am still not following. So where do the divergences actually go then? The lattice should just give a regularisation of the divergences (just like putting in a momentum cut-off) but unless you cancel them with something they will just come back again when you take the continuum limit. Since the big problem with quantum gravity is that it is non-remormalizable (ie that there is nothing to cancel the terms once they are regularized) I don't see how this works. Presumably there must be some other feature of the theory itself which restores renormalizability, but then why not regularize in the usual way?

[Martin]

So where do the divergences actually go then?

 

apparently that is something they intend to study in a forthcoming paper!

 

here is the next-to-last paragraph of their most recent paper ("Semiclassical Universe from First Principles")

 

"A number of open issues remain to be addressed, including the details of the renormalization mechanism. Here Causal Dynamical Triangulations gives us the possibility to study Weinberg’s scenario of “asymptotic safety” [21] in the context of an explicit quantum-gravitational model. As indicated in earlier work on Causal Dynamical Triangulations in space-time dimension three, the renormalization may be non-standard [22], which in a way would be welcome. This is also supported by the present computer simulations, in the sense that no fine-tuning of the bare gravitational coupling constant seems to be necessary to reach the continuum limit. Details of this will be discussed elsewhere [18]."

 

[18] is a paper by them, to appear.

[22] is a paper by them that deals with these issues in the simpler 3D case, namely:

http://arxiv.org/hep-th/0307263

Renormalization of 3d quantum gravity from matrix models

J. Ambjorn (NBI, Copenhagen), J. Jurkiewicz (U. Krakow), R. Loll (Spinoza Inst. and U. Utrecht)

14 pages, 3 figures

Journal-ref: Phys.Lett. B581 (2004) 255-262

[Martin]

Since the big problem with quantum gravity is that it is non-renormalizable ...

 

you are helping me get this clear, thanks for the questions!

Can we say that the big problem with a perturbative approach to quantum gravity is that it is non-renorm'ble-----the divergences in other words?

 

there is a big fork in the road between pert. approach and non-pert. approach.

 

Nonperturbative QG does not use a prearranged geometric background, it is harder to calculate, it is harder to prove the semiclassical limit or the largescale gross behavior is right, but now they have gotten some good results

 

they have a background independent nonperturbative QG that has some good largescale behavior

 

NOTE THAT IT IS NOT A LATTICE APPROACH in the sense that they do not start with a given pre-arranged lattice. What looks like a lattice is actually something that spontaneously assembles itself

these are discrete spacetimes which are solutions of discrete analog of Einstein equation. What they should look like is not predetermined and every computer run produces a different triangulated spacetime.

 

You might like to look at some of the graphics in their most recent papers, or at the early (circa 2000) computer animations in low dimension cases.

 

so although the outcomes look like triangulated odd-shape universes and thus close KIN to lattices, they are not conventional latticework.

 

============

the question you are pointing out seems to be how does it happen that when you go over to the perturbative treatment corresponding to this that it does not blow up?

 

if the nonperturbative model works, as it appears to do, then you should be able to go over to a perturbative model with a fixed background space and that should work too-----so how come?---why dont you get incurable infinities?

 

For the 4D case, they just say wait for the next paper.

 

I will look at the 3D case (reference [22]) and see if I can get any clue.

[Martin]

Here is a quote from [22]

http://arxiv.org/hep-th/0307263

Renormalization of 3d quantum gravity from matrix models

J. Ambjorn (NBI, Copenhagen), J. Jurkiewicz (U. Krakow), R. Loll (Spinoza Inst. and U. Utrecht)

14 pages, 3 figures

Journal-ref: Phys.Lett. B581 (2004) 255-262

 

----quote from page 2---

Quantum gravity in three space-time dimensions represents an interesting case in between dimensions two and four. On the one hand, it contains no propagating gravitational degrees of freedom and can be reduced classically to a finite-dimensional physical phase space, both in a metric [5] and a connection (Chern-Simons) formulation [6]. Nevertheless, the unreduced theory in terms of the metric g_mu nu appears to be non-renormalizable when one tries to expand around a fixed background geometry, just as in four dimensions. A definition of three-dimensional quantum gravity via a “sum over geometries” therefore seems to require a genuinely non-perturbative construction, and in turn may shed light on the problem of non-renormalizability of the full, four-dimensional theory, where an explicit classical reduction is not available.

---endquote---

 

it seems clear that the straightforward answer to when something is non-renormalizable (in perturubative treatment) is to take a non-pert. approach! but then, if it works, one has a mystery to explain or at least something to shed light on (namely the nonrenormalizability in the original fixed background-cum-perturbation context)

 

Is there something I am missing here?