Marseille workshop on loops and spin foams

[selfAdjoint]

Does the graviton represent a quantum of geometry? Certainly not in string physics, where it is a spin 2 particle in a "flat" background spacetime, whose interactions mimic Einstein gravity at a certain level of approximation.

If spacetime ever becomes quantized, surely the quanta will not be gravitons. They may emit and absorb gravitons, though, just as the known quanta emit and absorb various bosons.

 

[Mike2]

Does the graviton represent a quantum of geometry? Certainly not in string physics, where it is a spin 2 particle in a "flat" background spacetime, whose interactions mimic Einstein gravity at a certain level of approximation.

So String theory treats gravity like any other force and ignores spacetime warping of Einstein, is that what you are saying?

If spacetime ever becomes quantized, surely the quanta will not be gravitons. They may emit and absorb gravitons, though, just as the known quanta emit and absorb various bosons.

It seems to me that a quanta of geometry cannot interact with a particle any more than particles can interact with paths.

 

[selfAdjoint]

So String theory treats gravity like any other force and ignores spacetime warping of Einstein, is that what you are saying?

That is exactly right. String theory lives in a 26 or 10 dimensional flat Minkowski space, and the graviton simulates Einstein's equations without any space warping. (There are advanced descendents of string theory where the action determines the spacetime, but I don't know how they work out with gravitons).

It seems to me that a quanta of geometry cannot interact with a particle any more than particles can interact with paths.

Sorry, I don't quite see what this means.

 

[Mike2]

That is exactly right. String theory lives in a 26 or 10 dimensional flat Minkowski space, and the graviton simulates Einstein's equations without any space warping. (There are advanced descendents of string theory where the action determines the spacetime, but I don't know how they work out with gravitons).

It would seem impossible for string theory, then, to explain the background it works in, and so it cannot be a TOE. Nor does it seem likely that the flat space of string theory can explain things at the level of such a tiny universe that the dimensions are curled up. So at what level of energy or expansion is string theory supposed to address? Thanks.

 

[selfAdjoint]

It would seem impossible for string theory, then, to explain the background it works in, and so it cannot be a TOE. Nor does it seem likely that the flat space of string theory can explain things at the level of such a tiny universe that the dimensions are curled up. So at what level of energy or expansion is string theory supposed to address? Thanks.

The energy level is close to, but not at, the Planck level. Do pay attention the the caveat I put in my post. There are newer versions of stringy physics that do address the background space question. I just don't know anything about them.

 

[Mike2]

The energy level is close to, but not at, the Planck level. Do pay attention the the caveat I put in my post. There are newer versions of stringy physics that do address the background space question. I just don't know anything about them.

I understand strings are suppose to explain some of the constants in the Standard Model and leave only the string tension and speed of ligh still unexplained. But that's about it, isn't it?

 

[selfAdjoint]

I don't think SST can really explain the constants in the SM. Supersymmetry is supposed to expain some of them (like the generations of quarks) and at least some of the stringy constructions have low energy forms that look something like supersymmetrical extensions of SM, but that's as close as it gets.

 

[marcus]

I just got back from the Marseille conference on loop quantum gravity and spin foams:

http://w3.lpm.univ-montp2.fr/~philippe/quantumgravitywebsite/

It was really great, so I devoted "week206" of my column This Week's Finds entirely to this conference:

http://math.ucr.edu/home/baez/week206.html

In particular, I spend a lot of time giving a very simple nontechnical introduction to the recent work of Ambjorn, Jurkiewicz and Loll in which they seem to get a 4d spacetime to emerge from a discrete quantum model - something that nobody had succeeded in doing before!

http://www.arXiv.org/abs/hep-th/0404156

I hope this lays to rest certain rumors here that I'd burnt out on quantum gravity.

I want to use the may conference as a window on the important developments that have happened in the first half of 2004 in Quantum Gravity.

there are some papers in the May lineup to notice and also the informal message we got about Lee Smolin's interest in what I think Moffat would call a "Nonsymmetric Gravitational" theory or NGT---a modification of GR's lowenergy Newtonian limit. John Baez referred to it as "MOND" but I think what they were really talking about is the latest version of a mondic-type thing that isn't the crude old mond.
The new thing, let's call it NGT which is Moffat's term, does the same thing about explaining rotation curves without dark matter and handling the cosmological constant---and it connects with a version of DSR Smolin is working on with KowalskiGlikman--the TSR or triply special relativity socalled.
So there is some scuttlebut background from the May conference as well as the formal presentations. I am only guessing about the informal gossip but there is a lot of related stuff at Baez website now that came from people's response to his TWF 206

I want to try to put these things together and get some kind of picture to jell out of it---a picture of what is going on in Quantum Gravity in first half of 2004. A lot is

 

[marcus]

First thing is to follow the link Baez gave to his TWF 206 and read his account of what Smolin was talking about mond-wise, and then
read all the responses that Baez got about mond-ish stuff including critiques and a recent Bekenstein article.

but then look at a few scheduled talks
(not intended as a representative sample!)
-------------------------

Monday, May 3rd


J. Pullin (Consistent discretization)


------------------------------

Tuesday, May 4th

R. Loll (Dynamical triangulations)


---------------------------------------

Wednesday, May 5th


R. Gambini (Relational time in consistent discrete quantum gravity)


------------------

Friday, May 7th

J. Kowalski Gliksman : (DSR as a possible limit of quantum gravity)
F. Girelli (Special Relativity as a non commutative geometry: Lessons)

-----------------
Notice the merging of lines of research as they mature. DSR is not a theory of gravity it is just a modification of minkowski-space to make one more quantity invariant (besides c). but analogous to how old minkowski space was the tangent space or local streetmap for old GR, if we have a new quantum GR maybe it could have DSR as its local approximation. or
maybe an even better modification of minkowski space like TSR (that jerzy k-g and smolin are working on) or the DDSR that girelli and livine and oriti just posted on----so these Friday talks by Jerzy K-G and by Girelli are about that

and the other interesting thing about them is that they are not only merging DSR with QG, they are putting out feelers to Moffat's mond-ish Nonsymmetric Gravitational Theory (with its comprehension of dark matter and dark energy)-----because Girelli/Livine/Oriti said that explicitly in the paper they just posted, and they are working somewhat parallel with Smolin and JerzyKG and Smolin is talking about mondish stuff.

We arent going to have separate fields, it seems, because quantum gravity is making contact with and beginning to absorb things like DSR and MOND or versions or decendants of them.

And then it happened today that Jorge Pullin posted that paper on resolving the Black Hole Information puzzle---by Gambini and Porto (at Carnegie Mellon) and Pullin (at Louisiana)
I think it is an important paper because that puzzle has NOT till now been resolved, it is a real puzzle and GP and P are proposing a really simple solution.
And they were at the Marseille conference talking about relational time
and it is exactly thus they resolve the puzzle----absolute time is not real!
Absolute time does not exist in nature, all we have is whatever clocks we can manage to build or observe and they relate conditional quantum-fashion to other observables. OK they say, let us be realistic and use actual observable material clocks. Let us not pretend there is an absolute perfect clock that God winds up every day for all eternity, but only various imperfect clocks like your wife has.

then, Lo and Behold, the black hole information puzzle vanishes
(but there seems to be a nontrivial calculation to show this---two years ago they tried but didnt get it, then just now they got it)

with realistic (relational) time, evolution is just very slightly nonunitary!

(maybe our PF member called "Nonunitary" will like this)

and because of the very slight nonunitariness, information is not forever, it gradually fades out, but very very slowly

however black holes evaporate very slowly

so by the time the BH has evaporated all the information would have
faded into nonunitary oblivion ANYWAY
therefore no information is lost by the BH evaporating

Those friends and associates of Susskind who speculated about black holes leaving remanants or the information "teleporting" out of them by stringy business, they did not have to worry themselves about it. Theirs may merely have been a deluded effort to save perfect absolute-time unitarianism.

Why do I think Rovelli will be amused by Gambini Porto Pullin's paper
resolving the BH info paradox? Didnt he suspect already that understanding time better would do that?

 

[marcus]

the poet Borges said (and Wilbur, a great translator, translated)


"One thing does not exist: oblivion
God saves the metal and he saves the dross
and his prophetic memory guards from loss
the moons to come, and those of evenings gone..."

it is the first four lines of one of the most wonderful sonnets
ever written in english

but if relational time destroys the unitariness of time-evolution and
pure quantum states gradually lose coherence
and informations fades, even as dewdrops and black holes evaporate,
then Borges vision is incorrect.

he wouldn't have liked that, he more than any 20th century poet
tried to make sonnets and stories which were true to the general theory
of relativity and to quantum mechanics. especially his short stories which are true to quantum mechanics. he wanted his poetry to be correct.

 

[marcus]

I was typing from memory, here is a longer exerpt of Borges poem
the Letralia website has the complete poem in both languages
http://www.letralia.com/58/en02-058.htm

Everness

One thing does not exist: Oblivion.
God saves the metal and he saves the dross,
And his prophetic memory guards from loss
The moons to come, and those of evenings gone.
Everything is: the shadows in the glass.
Which, in between the day's two twilights, you
Have scattered by the thousands, or shall strew
Henceforward in the mirrors that you pass.
And everything is part of that diverse
Crystalline memory, the universe:
...

Everness

Sólo una cosa no hay. Es el olvido.
Dios, que salva el metal, salva la escoria
Y cifra en Su profética memoria
Las lunas que serán y las que han sido.
Ya todo está. Los miles de reflejos
Que entre los dos crepúsculos del día
Tu rostro fue dejando en los espejos
Y los que irá dejando todavía.
Y todo es una parte del diverso
Cristal de esa memoria, el universo;
...

Everything is: the shadows in the glass.

that is, no information is ever lost.

And everything is part of that diverse
Crystalline memory,

that is, 4D spacetime is a static eternity with all our worldlines
and the worldlines of all the particles which momentarily interweave to make us

 

[marcus]

...


Tuesday, May 4th

R. Loll (Dynamical triangulations)


---------------------------------------

Wednesday, May 5th


R. Gambini (Relational time in consistent discrete quantum gravity)


------------------

Friday, May 7th

J. Kowalski Gliksman : (DSR as a possible limit of quantum gravity)
F. Girelli (Special Relativity as a non commutative geometry: Lessons)

...

several of the talks at the Marseille symposium have subsequently appeared as papers

Girelli, Livine
"Special Relativity as a non-commutative geometry: Lessons for Deformed Special Relativity"
http://arxiv.org/gr-qc/0407098

Kowalski-Glikman, Smolin
"Triply Special Relativity"
http://arxiv.org/hep-th/0406279

Gambini, Porto, Pullin
"Realistic clocks, universal decoherence and the black hole information paradox"
http://arxiv.org/hep-th/0406260

BTW wasnt it great having Baez drop into PF and report from the Marseille conference, starting this thread!

I hope he makes it a habit. I would very much like to hear what he has to say about September's London conference in honor of Chris Isham.
Renate Loll will be one of the speakers.

 

[marcus]

moment of truth for simplex gravity (dynamical triangulations)

... I would very much like to hear what he has to say about September's London conference in honor of Chris Isham.
Renate Loll will be one of the speakers.

tomorrow the Isham 60th birthday conference at Blackett Lab Imperial College London

http://www.imperial.ac.uk/research/theory/about/isham60/schedule.htm

10AM tuesday is Renate Loll talk.

they posted a paper in April, computer study results,
"Emergence of a 4D world..."

Ambjorn Jurkiewicz Loll
"Emergence of a 4D World from Causal Quantum Gravity"
http://www.arXiv.org/abs/hep-th/0404156

and Renate presented the results in May at the Marseille conference.

It caused some stir because it seems there is some chance of real progress in that area. It came after about 15 years of people trying this approach with success only in lower dimensions.
Simplicial quantum gravity had seemed reasonable but had never generated a normal healthy 4D world in computer modeling, until the AJL paper.

that was May, now it is 4 months later, September. Has there been further progress or not?

John Baez is attending tomorrow's conference, but not giving a paper IIRC.
Maybe we will hear some word from him



The speakers include Hawking, Rovelli, Ashtekar, Penrose, Loll...

Here are some talks

K. Kuchar: Spacetime Covariance in Canonical Relativity.

J. Hartle: Arrows of Time and Generalized Quantum Theory

R. Penrose: What is Twistor-String Theory?

G. Gibbons: The First Law of Thermodynamics for Kerr-Anti-de-Sitter Black Holes in Arbitrary Dimensions

R. Loll: Emergence of a 4d World from Causal Path Integrals

S. Hawking: The Information Paradox for Black Holes

R. Sorkin: Is a Past Finite Order the Inner Basis of Spacetime?

C. Rovelli: How to Extract Physical Predictions from a Diffeomorphism Invariant Quantum Field Theory

A. Ashtekar: Recent Advances in Loop Quantum Gravity

 

[selfAdjoint]

Marcus, have you heard any more about this? Any of the talks posted online?

 

[marcus]

Marcus, have you heard any more about this? Any of the talks posted online?

I am glad that you are back sA,
I was expecting that John Baez, since he attended, would post something about it, but so far he didnt.

Maybe he would if we asked him nicely.

I am sorry to say that I have no lead on any of the London talks.

 

[marcus]

To a large extent the discussions in this thread were around the first AJL paper
Ambjorn Jurkiewicz Loll
"Emergence of a 4D World from Causal Quantum Gravity"
http://www.arXiv.org/abs/hep-th/0404156

and some of the "sidebar" material may be worth recalling.

Renate Loll presented the paper at the May conference.
We got some of John Baez perspective on it from him in this thread,
and in his TWF#206
and is parallel conversations with Larsson and others on SPR.

This thread has some links to some of that parallel discussion, and
also to an article about Simplicial Gravity---or Dynamical Triangulations---
that Matt Visser had in Jorge Pullin's newsletter Matters of Gravity

 

[marcus]

To a large extent the discussions in this thread were around the first AJL paper
Ambjorn Jurkiewicz Loll
"Emergence of a 4D World from Causal Quantum Gravity"
http://www.arXiv.org/abs/hep-th/0404156

John Baez introduced the Dynamical Trianglulation (DT) quantum gravity approach to us at PF by starting this thread and highlighting the above paper in his report from the May 2004 conference.

I guess this is our main DT thread. I'm going to do an "introduction to DT"
here. I will put links to tributary threads, and to biblio.

For me, after the April 2004 paper, there was a waiting period to see how things would go. I think DT now looks stronger than ever, as a proposed QG.

the best introduction to DT that I have been able to find in the literature is
sections of a 96 page paper by AJL, posted January 2000
LORENTZIAN AND EUCLIDEAN QUANTUM GRAVITY– ANALYTICAL AND NUMERICAL RESULTS
http://arxiv.org/hep-th/0001124

this is the closest thing to the introductory chapters of a textbook, as yet. but it has non-essential sections that deal with problems they were having back in 1999 and 2000.

there are some later AJL papers that carry on the introductory exposition,
after this one. I want to map out how to piece together a kind of beginning text.

BTW I think DT is turning out to be a serious rival to any quantum gravity theory you can name. thanks to John Baez for alerting us to it.

 

[marcus]

some PF threads discussing DT

https://www.physicsforums.com/showthread.php?t=54262

https://www.physicsforums.com/showthread.php?t=53974

https://www.physicsforums.com/showthread.php?t=52958
(I didnt do the greatest job explaining in that thread,
and want to remedy that and give a better account)

https://www.physicsforums.com/showthread.php?t=55806
(this has some bibliography)

 

[marcus]

the history of DT

DT is based a modified version of Regge calculus
So it goes back originally to Tullio Regge's landmark 1961 paper
General Relativity without Coordinates
which showed how to do a discrete Einstein equation
in a triangulated 4D space (a space divided up into 4simplexes)

Regge's method involved knowing the lengths of the edges of the simplices and doing arithmetic with them. He could get a substitute for curvature without ever taking the derivative.

the first distinctively DT approach was around 1985 in 3 separate papers:
Ambjorn et al, by F.David, and by V.A. Kazakov, I.K. Kostov and A.A. Migdal.

What made DT different was you made all the 4simplexes be identical, or all of a small number of types. Then all that matters is COUNTING. counting numbers of simplexes, and vertices, and edges etc.

that is, DT is different from Regge style because Regge allowed for individual variation in the size and shape of simplexes, so everything depended on measuring the individual simplexes in some locale. but
DT just uses some stock simplexes and counts. But it also works.

So starting around 1985, Ambjorn et al got into trying to do quantum gravity with DT.

Particularly they wanted to do a path integral approach, the idea of which had been made popular by Stephen Hawking. And they started doing Monte Carlo computer runs with random 4D triangulations (and lower dimensional analogs) to evaluate the path integral.

DT suffered from a lot trouble and the random triangulated spacetimes were always crumpled or fractal-feathery, or plagued by budded-off "baby" universes. So for over 10 years it seemed discouraging.

It seems to have been around 1998 that Ambjorn and Loll got the notion of restricting DT to a kind of FOLIATED triangulation which would have some causal or Lorentzian structure.

they began a program of working up from 2D to 3D to 4D
and it worked at each stage and got better all the time
and this finally led to the two papers that posted this year.
which kind of put this approach on the map

 

[selfAdjoint]

Marcus, I want to thank you for going through all this and keeping us up to date, and particularly the fine explanations you have worked up about DT. As your discussion of Oriti's latest paper suggests, this DT program may be about to converge with other approaches to quantum gravity - sort of the way K-Mart merged with Sears, where the stores will all become Sears named but the management will all be K-Mart.

 

[marcus]

Nightcleaner,
the Letralia website has the complete poem in both languages
http://www.letralia.com/58/en02-058.htm
and many more besides this one

Everness

One thing does not exist: Oblivion.
God saves the metal and he saves the dross,
And his prophetic memory guards from loss
The moons to come, and those of evenings gone.
Everything is: the shadows in the glass.
Which, in between the day's two twilights, you
Have scattered by the thousands, or shall strew
Henceforward in the mirrors that you pass.
And everything is part of that diverse
Crystalline memory, the universe:
Whoever though its endless mazes wanders
Hears door on door click shut behind his stride,
And only from the sunset's farther side
Shall view at last the Archetypes and Splendors.


Everness

Sólo una cosa no hay. Es el olvido.
Dios, que salva el metal, salva la escoria
Y cifra en Su profética memoria
Las lunas que serán y las que han sido.
Ya todo está. Los miles de reflejos
Que entre los dos crepúsculos del día
Tu rostro fue dejando en los espejos
Y los que irá dejando todavía.
Y todo es una parte del diverso
Cristal de esa memoria, el universo;
No tienen fin sus arduos corredores
Y las puertas se cierran a tu paso;
Sólo del otro lado del ocaso
Verás los Arquetipos y Esplendores.

...

here is one that Letralia doesn't have:

to see a world in a grain of sand
and a heaven in a wild flower,
hold infinity in the palm of your hand,
and eternity in an hour

I don't know what you are talking about
I know what you are talking about

pick one

 

[marcus]

sorry everybody
I got off topic
it is probably better to start a separate thread for poetry et al. and
let this one stay focused on what John Baez called attention to:
the AJL paper
Dynamical Triangulations

 

[selfAdjoint]

I don't know where you get that about the Planck scale. In the general relativity view, spacetime at any scale is one unified thing.

Go back to Marcus' earlier post about Regge Calculus. Years ago Tullio Regge triangulated GR spacetime and by doing combinatorial things with the edge-lengths of the triangulation he was able to do all the GR curvature math that is usually done with tensors and differential forms and second derivatives. Then Ambjorn and coworkers made all the lengths the same size and revised the combinatorial shuffle to an even simpler form, but they had problems and it was a years-long slog to get to their present causal triangulations which work so splendedly.

Now if you want to use packed spheres instead of triangulations, go to it, but you have to show as Regge did and Ambjorn et al did that it reproduces the world we know, not just at the handwaving level but in the details where god and the devil duke it out.

 

[marcus]

I don't know where you get that about the Planck scale. In the general relativity view, spacetime at any scale is one unified thing.

Go back to Marcus' earlier post about Regge Calculus. Years ago Tullio Regge triangulated GR spacetime and by doing combinatorial things with the edge-lengths of the triangulation he was able to do all the GR curvature math that is usually done with tensors and differential forms and second derivatives. Then Ambjorn and coworkers made all the lengths the same size and revised the combinatorial shuffle to an even simpler form, but they had problems and it was a years-long slog to get to their present causal triangulations which work so splendidly.

Now if you want to use packed spheres instead of triangulations, go to it, but you have to show as Regge did and Ambjorn et al did that it reproduces the world we know, not just at the handwaving level but in the details where god and the devil duke it out.

classic epigrammatical account, wanted to email it to Ambjorn as a kind of
maximally concise statement of their work's place in the q.g. story.
won't though, since they must have plenty to think about without
e-fanmail

 

[marcus]

photo of Tullio Regge

http://www.icra.it/MG/awards/Images/regge.jpg

born 1931 Torino
discovered Regge calculus 1961 while at Princeton
where he worked with John Archibald Wheeler

 

[marcus]

I still think the best detailed introduction to AJL dynamical triangulations
is "Dynamically Triangulating Lorentzian Quantum Gravity"
http://arxiv.org/hep-th/0105267

more than one person at PF has indicated they'd found it useful,
printed it out, etc. Also AJL refer back to it as a basic reference
several times in their recent (2004) papers.

would also be nice to have an online source giving an
introduction to Regge calculus----hopefully would have pictures
since the subject could be presented visually

for basic path integral terminology, here is the Wiki article
on "path integral"
http://en.wikipedia.org/wiki/Path_integral_formulation

if you wish, this will lead you back to contributory Wikis on "action", "Lagrangian" etc.

Here's a brief introduction to Regge calculus (esp. as applied to numerical relativity) by Adrian Gentle
http://arxiv.org/abs/gr-qc/0408006
it really has barely a page actually explaining R's discrete gen. rel.

hope we find more. I will keep looking

 

[nightcleaner]

I don't know where you get that about the Planck scale. In the general relativity view, spacetime at any scale is one unified thing.

Now if you want to use packed spheres instead of triangulations, go to it, but you have to show as Regge did and Ambjorn et al did that it reproduces the world we know, not just at the handwaving level but in the details where god and the devil duke it out.

Yes, in GR spacetime is one unified thing, but my reading has led me to think that GR isn't used to describe the world at the Planck scale, but is considered to be a poor model of events in the very small, very high energy realm. GR is a cosmological paragigm, while the standard model of particles in flat space is more often used in discussions of the very small. Did I miss something?

Now, it is very nice of you to invite me to try to match the work of PH.D's at two major European universities and The Max Planck research institute, backed up by all the departmental machinery and academic freedoms they have available to them, while I am nothing but a nightcleaner in a tourist restaurant. Actually I am gratified by the fact that AJL has done work in the very field I have been unsuccessfully trying to draw attention to here and in previous years on SST.com.

I feel somewhat as a bean farmer must who finds a fertile plot of ground, scratches at it with a stick and makes a little crop for a few years, then finds himself and his tender garden uprooted by the massive machinerey of agribusiness. It seems they want to build a driveway for their factories on top of my little digs, and I may as well get out of the way or get paved over. Huh.

Well, it is no real surprise. And I have the small satisfaction of saying that I, at least, knew where to dig. And I take away something else as well. I may not be able to apply the Regge calculus (at least not yet) but my model is prettier.

nc

 

[marcus]

... I may not be able to apply the Regge calculus (at least not yet) ...

hello NC, i am still groping around for introductory material on Regge calculus and the closely related DT approach.

here are some online page references, if for no other use than my own!
I found parts of these helpful.

Loll 98----pages 8-13 are about standard Regge
pages 14-17 are about DT
http://arxiv.org/gr-qc/9805049
Discrete approaches to quantum gravity in four dimensions
this is a "LivingReviews" survey article that the AEI invited Loll to contribute
it surveys current (1998) research in several related areas and give
a large bibliography. She includes a halfdozen or so introductory sources on Regge calculus but none are online. her own treatment is quite concise.


For a more elemenary discussion: try Loll 02, pages 8-16
see also the summary at the end pages 34 and 35.
http://arxiv.org/hep-th/0212340
A Discrete History of the Lorentzian Path Integral

this is a pedagogical article, to help get graduate students involved.
It is historical, describing difficulties as they were encountered. I find this often helps me understand.
this essay is willing to waste words explaining some simpler points that a normal research article would not explain

However the rest of the article----pages 1-7 and 17-33
is much concerned with the problems that were being encountered in 2002!
since they have gotten past some of that, it is less interesting now IMO.

Maybe as a sample I will quote some from around page 8.

 

[marcus]

Here is a sample from around page 8 of Loll 20 survey
http://arxiv.org/hep-th/0212340
A Discrete History of the Lorentzian Path Integral

this is just to give the flavor. I will not bother to reproduce the math symbols exactly but will simply drop symbols in some cases---leaving whatever copies easily: the words.

---sample---
“Lorentzian dynamical triangulations”, first proposed in [13] and further elaborated in [14, 15] tries to establish a logical connection between the fact that non-perturbative path integrals were constructed for Euclidean instead of Lorentzian geometries and their apparent failure to lead to an interesting continuum theory. Is it conceivable that we can kill two birds with one stone, ie. cure the problem of degenerate quantum geometry by taking a path integral over geometries with a physical, Lorentzian signature? Remarkably, this is indeed what happens in the quantum gravity theories in d < 4 which have already been studied extensively. The way in which Lorentzian dynamical triangulations overcome the problems mentioned above is the subject of the Sec. 5.

4 Geometry from simplices
The use of simplicial methods in general relativity goes back to the pioneering work of Regge [16]. In classical applications one tries to approximate a classical space-time geometry by a triangulation, that is, a piecewise linear space obtained by gluing together flat simplicial building blocks, which in dimension d are d-dimensional generalizations of triangles. By “flat” I mean that they are isometric to a subspace of d-dimensional Euclidean or Minkowski space. We will only be interested in gluings leading to genuine manifolds, which therefore look locally like an R^d. A nice feature of such simplicial manifolds is that their geometric properties are completely described by the discrete set ... of the squared lengths of their edges. Note that this amounts to a description of geometry without the use of coordinates. There is nothing to prevent us from reintroducing coordinate patches covering the piecewise linear manifold, for example, on each individual simplex, with suitable transition functions between patches. In such a coordinate system the metric tensor will then assume a definite form. However, for the purposes of formulating the path integral we will not be interested in doing this, but rather work with the edge lengths, which constitute a direct, regularized parametrization of the space Geom(M) of geometries. How precisely is the intrinsic geometry of a simplicial space, most importantly, its curvature, encoded in its edge lengths? A useful example to keep in mind is the case of dimension two, which can easily be visualized. A 2d piecewise linear space is a triangulation, and its scalar curvature R(x) coincides with the so-called Gaussian curvature. One way of measuring this curvature is by parallel-transporting a vector around closed curves in the manifold. In our piecewise-flat manifold such a vector will always return to its original orientation unless it has surrounded lattice vertices v at which the surrounding angles did not add up to 2[pi], but [formula omitted]
see Fig.4. The so-called deficit angle [delta] is precisely the rotation angle picked up by the vector and is a direct measure for the scalar curvature at the vertex. The operational description to obtain the scalar curvature in higher dimensions is very similar, one basically has to sum in each point over the Gaussian curvatures of all two-dimensional submanifolds. This explains why in Regge calculus the curvature part of the Einstein action is given by a sum over building blocks of dimension (d-2) which are simply the objects dual to those local 2d submanifolds
---end quote---

Notice how new Causal DT (Lorentzian DT) is! She says it was first proposed only in 1998-----the reference [13] is to a paper by Ambjorn
and her.

I have bolded "...which constitute a direct, regularized parametrization of the space Geom(M) of geometries..."

you have a formless continuum M, and you make a "space" consisting of all the possible geometries you could have on M. this is where the quantum state of the geometry is going to live, or be defined. In another of her writings Loll calls this space of geometries, this Geom(M) the "mother of all spaces" or something like that.

this is a more direct "quantization-ready" approach than some others (e.g. think of the Ashtekar variables). In the straight Regge, there is just this long list of EDGE LENGTHS and that effectively describes a geometry and coordinatizes Geom(M)

now DT insists that all the edges are standard lengths and so instead of a list of edgelengths you have a list of what is next to what, recording the "connectivity"---it should be simple enough: some ways of writing it down would be more efficient than others----some computer data structure that names the vertices and says which ones are vertices of what tetrahedron etc., enough information so you can tell what is a side of what.

 

[nightcleaner]

hello NC, i am still groping around for introductory material on Regge calculus and the closely related DT approach.

here are some online page references, if for no other use than my own!
I found parts of these helpful.

...

Hi Marcus, and thanks for posting your finds here. I do think they are helpful.

However, I have trouble reading the math, and the papers are full of jargon which make them difficult for ordinary English speakers such as myself. I have been reading physics for a couple years and trying to improve my math skills, so I think I have some idea of what AJL are trying to convey. Still, I find myself taking a drubbing on the forehead when trying to read Loll and her collegues.

Are you willing to entertain questions on the math and physics here?

For example, here is a web link from Mathworld. It seems to be relevant, but I am not sure, and will withdraw it from the forum if it is not to the point of this thread.

http://mathworld.wolfram.com/Simplex.html

nc

3790

 

[selfAdjoint]

Cleaner, in 0 dimensions a simplex is a point. In one dimesion, take a 0-simplex and another point not on it, and draw all possible straight lines from the one to the other (there's only one line in this case). The result is the 1-simplex, which is just a line segment, right? Now on to 2 dimensions; take a 1-simplex and a point in the plane not on the 1-simplex, and draw all possible straight lines from one to the other. BTW this construction is called "taking the cone" over the 1-simplex. The result is the 2-simplex, which you should see to be a triangle. For the 3-simplex take the cone over the 2-simplex in 3 dimensions. The result is a 4 sided pyraimid on a triangular base.

And so on, although you can't visualize it. Notice that the 0-simplex had 1 vertex, 0 edges, and 0 faces. The 1-simplex had 2 vertices, 1 edge, and 0 faces. The 2-simplex had 3 vertices, 3 edges, and 1 face, and the 3-simplex has 4 vertices, 6 edges, and 4 faces. Evidently an n-simplex has n+1 vertices. How many edges, faces, and higher dimensional faces does it have? Well each k-dimensional face is a k-simplex itself, and so it has k+1 vertices, as we just said. And the number of k-faces in an n-simplex is the number of combinations of n+1 vertices taken k=1 at a time; the total number in the n-simplex taken the number in a typical k-face at a time.

Mathematicians use simplexes instead of cubes or whatever to triangulate spaces because they have this simple facial property, which leads to simple formulas for combining them.

 

[nightcleaner]

My reading of Regge Calculus: a unique tool for numerical relativity, Adrian P. Gentle, arXiv:gr-qc/0408006 v1 2 Aug 2004 today has brought me to the following notions of what Regge calculus is, and how it is applied to the structure of space time at the Planck scale.

Regge calculus uses objects called simplices , which in general are lower dimensional structures applied to approximate higher dimensional surfaces. For example, a three spatial dimensional object like the event horizon of a black hole seen at an instant of time appears spherical, and has the dimensions given by spherical geometry at the Schwartzchild radius. The positions of points on the Schwartxchild surface of the event horizon can be approximated by applying a large number of triangles to the surface. Each triangle is two dimensional, making it easier to calculate positions of points on the three dimensional surface of the event horizon as approximations to positions of the vertices of the triangles. Given that the triangles have straight edges and that these edges are very close to the curvature at any point, it is possible to calculate the vertices, and these calculations are very close to the values of the more difficult calculations required to obtain the exact three dimensional position on the surface. To get a better fit, the edges of the triangles can be made shorter, so giving a better approximation to the surface.

Triangular simplices in two dimensions can be applied as described above to a three dimensional surface. In general, then , the same procedure can be applied to a four dimensional spacetime by applying three dimensional simplices to the four dimensional manifold. The three dimensional analog of the two dimensional triangle is the tetrahedron. So, by building a three dimensional lattice of tetrahedrons, we can model events in four dimensions.

The paper referenced above applies two dimensional triangulations to an embedding diagram of a black hole. The embedding diagram uses artistic perspective on a two dimensional surface to represent the black hole as a gravitational depression in an elastic sheet, the familiar “whirlpool” or vortex shape. The triangles are shown as fitted to the curvature of the surface of the deformed sheet. Far from the edges of the hole, the surface is flat and the triangles make a near perfect fit. At the lip of the hole, the surface is highly curved and the triangles are made smaller to better approximate the curvature. How does this process translate into higher dimensional analyses, so that we can use a three dimensional lattice to model four dimensional space-time events?

If we build an undistorted lattice of tetrahedrons all of which have edges of identical length, we have a three dimensional model of four dimensional spacetime in a region where there is no mass or energy to distort the lengths of the edges. This lattice can be extended to infinity and points in the lattice can easily be calculated using any coordinate system, so the lattice can be said to be background independent.

If we introduce a gravitational field to the lattice, we should see the edges in the vicinity of the gravitational object made shorter as the curvature of spacetime increases near the object. This process breaks the symmetry of the lattice, since the original conditions of the lattice, as seen far from the gravitational object, gradually transition to the reduced conditions near the object. This transition cannot be made in the three dimensional model in a smooth way, but must rely on our making certain choices in the representation in regard to edge lengths and interior angles of the triangles which make up the tetrahedra. These choices are not background independent, since the lattice itself has now become the background for the model.

I will now recommend a solution to this difficulty, which seems to me to be inherent in the Regge calculus, at least as far as I have been able to understand it.

If the lattice is built using a type of simplex which has a naturally occurring and easily calculable expansion symmetry, the choice of which lattice edge and so which internal angle to reduce can be automated, restoring a degree of background independence. For example in the triangular fit to a curved surface we could choose to divide each triangle by a reduced triangle inscribed between the midpoints of the edge lines. Each time we choose to make this division, there is a discontinuity in the accuracy of the fit of the simplex to the surface, but locally at the center of the triangle the fit is improved.
I will now suggest a better way to make the said choice. It is better because it does not rely on the human intervention of someone to decide where to apply the division into smaller triangles.

First, instead of a tetrahedral simplex, use a spherical one. Then, by stacking the spheres in close contact, and using the contact points as vertices, a type of lattice structure is formed. In this case the edges of the lattice are virtual, since they cut through the spherical surfaces in the same way a cord cuts a circle. However the geometry of the stack ensures that the lengths of the edges are the same as the length of the radius of the circle, just as a hexagram can be inscribed in a circle by dividing the circumference with a compass set to the length of the radius. This structure (I imagine groans from some readers) is none other than the isomatrix.

It is to be noted now that there are twelve spheres of equal radius which can be fitted around a central sphere of the same radius. These twelve spheres can be encompassed by a new, larger sphere, with a new radius equal to three times the radius of the original sphere. I suggest then that this three dimensional naturally occurring model be chosen as the most appropriate lattice structure for representing events in four dimensions. The lattice structure can be refined to any number of iterations to fit any curved space, even down to the singularity.

Then, returning to the three dimensional lattice meant to represent a gravitational object in four dimensions, as one approaches the object, the spheres and their contact lattice naturally and smoothly contract without human interference or background dependence. This model has to provide a better fit than any artificially constructed tetrahedral lattice, and I recommend that persons interested in spacetime geometry investigate the relatively simple mathematics of this type of nested spherical stack. Not only is it simpler and more beautiful than the broken glass edges used in AJL, but it is free of the complications introduced by human choice of when and where and by how much to reduce triangular edges in order to conform to increasing curvature.

I am now able to offer a simplified model conforming to the above suggestion which involves four interwoven circles of equal diameter which can be laid on a two dimensional flat surface, then given any degree of deformation up to the three dimensional sphere, without discontinuities. But that will have to wait until I can return to this, and depends upon my not being hit by any trucks.

Be well,

Richard

 

[nightcleaner]

Cleaner, in 0 dimensions a simplex is a point. In one dimesion, take a 0-simplex and another point not on it, and draw all possible straight lines from the one to the other (there's only one line in this case). The result is the 1-simplex, which is just a line segment, right? Now on to 2 dimensions; take a 1-simplex and a point in the plane not on the 1-simplex, and draw all possible straight lines from one to the other. BTW this construction is called "taking the cone" over the 1-simplex. The result is the 2-simplex, which you should see to be a triangle. For the 3-simplex take the cone over the 2-simplex in 3 dimensions. The result is a 4 sided pyraimid on a triangular base.

And so on, although you can't visualize it. Notice that the 0-simplex had 1 vertex, 0 edges, and 0 faces. The 1-simplex had 2 vertices, 1 edge, and 0 faces. The 2-simplex had 3 vertices, 3 edges, and 1 face, and the 3-simplex has 4 vertices, 6 edges, and 4 faces. Evidently an n-simplex has n+1 vertices. How many edges, faces, and higher dimensional faces does it have? Well each k-dimensional face is a k-simplex itself, and so it has k+1 vertices, as we just said. And the number of k-faces in an n-simplex is the number of combinations of n+1 vertices taken k=1 at a time; the total number in the n-simplex taken the number in a typical k-face at a time.

Mathematicians use simplexes instead of cubes or whatever to triangulate spaces because they have this simple facial property, which leads to simple formulas for combining them.

Ok, thanks. Then a 4-simplex has five vertices. This could be shown as five vertices on a circle, a pentagram. It has ten edges. To count the faces, one must look at the structure as a pair of four sided pyramids joined by a common three edged base? One edge is lost, or obscured, by the representation of a higher dimensional object in a lower dimensional space?

then,

n,v,e,f
0,1,0,0
1,2,1,0
2,3,3,1
3,4,6,4
4,5,10,10
5,6,15,20

n an integer
v=n+1
e=nv/2=[(n^2)+n]/2
f=e_(n-1) + f_(n-1)
={ [(n-1)^2]+n-1]/2} + f_n-1
= n^2-n + f_(n-1)?

(working. have to go do evening chores. Be back, iidghbat )

3,815

 

[marcus]

I agree with NC about these numbers of vertices, edges, faces (and they agree with the combinatorics that sA said)

then,

n,v,e,f
0,1,0,0
1,2,1,0
2,3,3,1
3,4,6,4
4,5,10,10
5,6,15,20

...

I had something else to mention. Although I think this is a very promising line of research there are still very very few papers in it.
In fact the idea of the LORENTZIAN or causally ordered approach to DT was only first proposed in 1998! so lorentzian or causal DT is newer than several other approaches (e.g. string and loop)

But even tho there are very few papers, why not have a regular way of keeping track? So here is an arxiv search engine table for this line of research:

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/1998/0/1

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/1999/0/1

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/2000/0/1

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/2001/0/1

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/2002/0/1

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/2003/0/1

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/2004/0/1

1998   3
1999   3
2000   5 
2001   4
2002   6
2003   4
2004   4

these are not perfect, they get some they shouldn't and probably miss some, but I've found the links are a good way to check for existence of papers I didnt previously know about in this area. As you can see there is little or no growth as yet. Will be interesting to run the same keyword search in 2005 and see if there's any change

 

[nightcleaner]

Ok, I think I see the pattern now.

A zero dimensional simplex is a vertex (or point).
A one dimensional simplex is an edge (or line), composed of two vertices (zero dimensional simplices).
A two dimensional simplex is a triangle (or plane), composed of three edges (one dimensional simplices).
A three dimensional simplex is a tetrahedron, composed of four triangles (two dimensional simplices)
A four dimensional simplex is a hyper-tetrahedron, and should be composed of five tetrahedral ( three dimensional simplices).

The table then would look like this:


N s0 s1 s2 s3 s4
0 1 0 0 0 0
1 2 1 0 0 0
2 3 3 1 0 0
3 4 6 4 1 0
4 5 5 1

Where N is dimension, s0 is number of vertices, s1 is number of edges, s2 is number of triangles, s3 is number of tetrahedrons, and s4 is number of hyper-tetrahedrons. So, to get the right s2 and s1 for n=4, we have to adjoin five 3-simplices, that is five tetrahedrons. We can do this simply by surrounding one tetrahedron with four more, each adjoined to one of the original tetrahedron’s faces. Then we have a three dimensional model of a four dimensional structure, consisting of five co-joined tetrahedrons.

All four of the original tetrahedron’s faces are now interior to the new structure, and are co-joined to one face each of the outer four tetrahedrons. Since five tetrahedrons would have twenty faces, but eight are now co-joined on the interior of the new structure, we are left with twelve exterior s2 triangular faces.

Five tetrahedrons would have thirty edges, but the lines of the central tetrahedron are all co-joined with two other exterior tetrahedrons. We can neglect these interior lines in our count. Then four exterior tetrahedrons would have twenty-four edges, but twelve of these edges are co-joined with one other exterior tetrahedron each, leaving a count of twenty-four minus six, or eighteen exterior edges. Twelve of these edges are acute, and the other six, the co-joined ones, are oblique.

So the completed table would look like this:


N s0 s1 s2 s3 s4
0 1 0 0 0 0
1 2 1 0 0 0
2 3 3 1 0 0
3 4 6 4 1 0
4 5 18 12 5 1


However I now notice that there are stated to be five s0 vertices in the four dimensional simplex. This comes from the rule that the number of vertices is n+1. The described joining of four tetrahedrons to one tetrahedron in the center does not result in a three dimensional structure with five points. Instead, the structure has four external vertices and four internal vertices. Do we throw out the structure or modify the rule?

Is there any way to co-join four tetrahedrons to end up with five points? I haven’t thought of any.

Is there any reason to change the rule? Where did the rule come from? Let’s look at the rule.

We began with a single vertex in otherwise empty space, and noted that it had zero dimension. To obtain one dimension, we had to add another vertex, so that space was no longer empty. This new vertex had to be constrained to be exterior to the original vertex, so that a line was formed.

Then, when we added a third vertex, we had to constrain the placement again, so that the new vertex was placed in the plane, and not on the edge previously constructed. Our new vertex was non-co-linear with the two that formed the definition of the edge.

Then, when we added the fourth vertex, to make a three dimensional simplex, we had to specify that the new vertex was non-co-planar with the existing three. So what is the rule for the fifth vertex? It cannot occupy the same three dimensional space as the existing three vertices. How might this be done?

We might place the new vertex as an offset in time. In other words, we might take our three dimensional simplex, a tetrahedron, and map it onto a tetrahedron extended one instant in time. Each vertex of our t=0 tetrahedron corresponds with a vertex on our t=1 tetrahedron. This would then be eight vertices, four in one instant and four in the offset instant. But we only want five vertices.

Five vertices can be achieved by setting the time interval to infinity. Doing this reduces one of the tetrahedrons to a point. Then, we see, we have five points, four in one instant mapped onto a fifth which, being in another instant, is not in the same space, and so obeys our constraint.

The result is a space-time structure with five vertices. It has four space-like vertices and six space-like edges, and it has four time-like edges which map from the four space-like vertices onto the one time-like vertex. So we see that there are now, as originally predicted, four dimensions, five vertices, and ten edges. But what happened to the five predicted s3 tetrahedral simplices? Are we justified in saying that they are somehow interspersed along the time line, so that we really have a set of five, one in 3space, one at infinite time, and then one each at the half and quarter marks? What is half or a quarter of infinity?

I have, as usual, an alternative proposition, which I think is more elegant. It is this. Any vertex in 4 space has at least four positions in any 3space. It exists at the origin. It exists at infinity. And it exists at least at two points somewhere in between. Those would be the spaces which contain the original tetrahedron we built in 3space at t=0, and the offset tetrahedron we built in another 3space at t=1.

Of course t=1 is not t=infinity, but again, what is half of infinity? From 3space, when we try observe the 5-vertice structure which exists in 4space, all we can see at one glance is ten edges between five points. That is the whole structure, as far as our familiar 3space geometry will allow. But we must conclude that it is not the entire 4space simplex. Parts of it are hidden from our 3space view.


Now, (groan) to return to the isomatrix model. There is one sphere in the center, representing any universe you choose. There are twelve spheres around it, representing the fundamental unit of spacetime in multiple dimensions. This structure is extended to infinity, in, out, and in every conceivable direction, both in time and in space. When an observer notes that there has been a change in the universe, it is not because the spacetime structure of the multiverse, that is the frozen river of 4d, has changed, but it is because the observer has moved through the unmoving spacetime structure. When the observer moves, it is in a direction, just one direction, not every direction possible. This movement results in a loss of information from the direction opposite to the motion. Information from the direction opposite to motion in the multiverse cannot catch up to the observer. Instead, the observer can obtain information only about the universe current to the observers instantaneous position (one sphere, the center sphere of the observer) and the three spheres which are annexed to that central sphere and still in the line of motion of the observer, and then one more bit, from the sphere that is just beyond the three spheres that form the possible next instant. The universe as you know it in this instant, the three universes that are possible in the next instant, and the one universe that has to be beyond those three. That’s five, the very same five vertices, or origins, that make up the structure we view as fourth dimensional from our familiar three dimensional universe.

Be well. Comments appreciated. And, yes, Love always,


Richard