Marseille workshop on loops and spin foams [Archive]

Mike2

May13-04, 11:58 AM

Was this paper an attempt to justify or perhaps even derive the very overall topology of space-time?

I wonder if causality is the key to the topology of space-time. For from the simplest logic, causality is one event producing another event. Those events would have to be represented by some region. Even if the events were a single point, one "event" producing another, would require one point to produce another. I suppose that at such a differential state as one or two points that the properties between points or small regions would not change between points, since the changes in these properties cannot be instantaneous at such a differential scale. This causality, one point producing another, would mean that the number of points (or regions) would all increase at the same rate. This would give an expansion of the universe proportional to its size, or an exponential expansion as is predicted in an inflationary universe.

I wonder if the number of dimensions and the metric can be determined from this topology. Since the closes points (or regions) would be responsible for the next point produced, and since all these points in conjunction implies that all are the cause of the others and the next, it would figure that the universe at this scale would be tightly curled up, not elongated into a line for example. This would seem to indicate a tightly curved metric for space-time. The first point would produce a second, and you would have a 1D line, these two, or one of them, would produce a 3 point, and since that next point would have to be about equally close to the other 2, you would have a 2D plane. These three would produce a forth, and since it would have to be about equally as close to the other 3 points, it would be in a 3D volume, etc. Where does this process lead?

I suppose this would conflict with the idea of a specific amount of space-time dispersing as it expands with time. I wonder if the two views can be reconciled

marcus

May13-04, 12:19 PM

again on the subject of the May 2004 Marseille quantum gravity conference, I wonder if anything can be learned from the list of talks. Rovelli was probably the main organizer, so the lineup would reflect somewhat how he sees the field:

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

Monday, May 3rd: Loop quantum gravity
am
Opening remarks
A. Ashtekar (Quantum geometry)
T. Thiemann (Dynamics and low energy)
L. Smolin (Overall results)
T. Jacobson, as devil's advocate (Some questions to loop quantum gravity)

pm
L. Doplicher (Propagation kernel techniques for loop quantum gravity)
W . Fairbairn (Separable Hilbert space in loop quantum gravity)
J. Lewandowski (Quantum group deformations of the holonomy-flux algebra)
B. Dittrich (Master constraint program for loop quantum gravity)
J. Pullin (Consistent discretization)
S. Alexandrov (Lorentz covariant loop quantum gravity)
H. Salhmann (Uniqueness of the Ashtekar-Isham-Lewandowski representation)

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

Tuesday, May 4th: Spinfoam formalism
am
J. Baez (Spinfoams)
L. Freidel (Group field theory and sum over 2-complexes)
J. Barrett (BC models)
R. Loll (Dynamical triangulations)

pm
A. Perez (Spin-foam representation of the physical scalar product in 2+1 gravity)
R. Oeckl (Boundary formulation of quantum mechanics and application to spin foams)
A. Starodubtsev (Definition of particles in 4d quantum gravity)
F. Markopoulou (Quantum information theory and particles in spinfoam)
E. Livine (Instantons in GFT and continuum limit)
H. Pfeiffer (Quantum gravity smooth manifold and triangulation)

18:30 Campus Colloquium (open to external participation)
A Ashtekar (Space and Time: From Antiquity to Einstein and Beyond)

20:00: Lunar eclipses

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

Wednesday, May 5th: Miscellaneous
am
T. Jacobson (Mode creation: quantum field theory on a growing lattice)
L. Bombelli (Statistical framework for the continuum approximation to quantum gravity)
R. Gambini (Relational time in consistent discrete quantum gravity)
G. Mena Marugan (Perturbative and nonperturbative cylindrical gravity)
O. Winkler (Singularity avoidance or how compact is the world?)
J. Swain (Spin-Networks and Approximations of Diffeomorphism Groups)
E. Buffenoir (Quantum radar time in 2+1 dimensions)

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

Thursday, May 6th: Applications of loop quantum gravity, cosmology, black holes and quasinormal modes
am
M. Bojowald (Loop cosmology)
D. Sudarsky (Phenomenology)
A. Corichi (Black holes)
K. Krasnov (Quasi normal modes of black holes)

pm
O. Dreyer (Quasinormal modes)
P. Forgacs (Quasi normal modes of the t'Hooft-Polyakov monopole)
K. Noui (Hamiltonian analysis in Plebansky theory)
S. Parampreet (Some applications of loop cosmology)
K. Vandersloot (A path integral representation of loop quantum cosmology)
S. Major (Observations on a lorentzian model)
P. Majumdar (Universal canonical black hole entropy)

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

Friday, May 7th
am: Related approaches
M. Niedermaier (Asymptotic safety)
R. Percacci (Is Newton's constant essential?)
J. Klauder (Affine Quantum Gravity: An All-Scale Theory)
D. Minic (Modification of quantum mechanics and quantum gravity)
J. Kowalski Gliksman : (DSR as a possible limit of quantum gravity)
F. Girelli (Special Relativity as a non commutative geometry: Lessons)

pm
Panel and general discussion
Conclusion

marcus

May13-04, 02:21 PM

In the thread started earlier about the AJL paper "Emergence of a 4D World" two people (arivero and Mike2) asked about what model of quantum gravity are they using.

I found what I think is a good link. It turns out that it is one John Baez already recommended in "Week 206" when he was talking about the same paper. It is a pedagogical lecture by Renate Loll, with a lot of pictures.
Dated 13 January 2003.
I printed it out. It seemed worth keeping and studying. and to have easy parts.

"A discrete history of the Lorentzian path integral"
http://www.arxiv.org/hep-th/0212340
38 pages

Notice that R. Loll's quantum gravity is not the same as Ashtekar's or Rovelli's or Smolin's, at least on the surface. Here is a brief exerpt from the beginning of the paper:

----exerpt---
In these lecture notes, I describe the motivation behind a recent formulation of a non-perturbative gravitational path integral for Lorentzian (instead of the usual Euclidean) space-times, and give a pedagogical introduction to its main features.

At the regularized, discrete level this approach solves the problems of (i) having a welldefined Wick rotation, (ii) possessing a coordinate-invariant cutoff, and (iii) leading to convergent sums over geometries. Although little is known as yet about the existence and nature of an underlying continuum theory of quantum gravity in four dimensions, there are already a number of beautiful results in d = 2 and d = 3 where continuum
limits have been found.

They include an explicit example of the inequivalence of the Euclidean and Lorentzian path integrals, a non-perturbative mechanism for the cancellation of the conformal factor, and the discovery that causality can act as an
effective regulator of quantum geometry.

1 Introduction
The desire to understand the quantum physics of the gravitational interactions lies at the root of many recent developments in theoretical high-energy physics. By quantum gravity I will mean a consistent fundamental quantum description of space-time geometry whose classical limit is General Relativity...
--------end quote----

what seems to have just changed is that the results for d = 2 and d = 3 begin to extend to d = 4

Mike2

May13-04, 05:15 PM

This causality, one point producing another, would mean that the number of points (or regions) would all increase at the same rate. This would give an expansion of the universe proportional to its size, or an exponential expansion as is predicted in an inflationary universe.
...
I suppose this would conflict with the idea of a specific amount of space-time dispersing as it expands with time. I wonder if the two views can be reconciled
The question becomes how fast do new points appear. I suppose that as long as the rate was not infinite, it would all be just a matter of scale. And since there is nothing else to compare with, it would all seem the same to us.

So can someone tell me how such a scenario would modify the usual point set topology studies?

marcus

May14-04, 07:00 PM

She and her co-workers have been calling their QG approach various
names and have not quite settled on one.

Recently it is "Causal Quantum Gravity"
(causal denotes Lorentzian rather than Euclidean")

Earlier on it was "Dynamical Triangulations" and "Lorentzian Q.G."
or the "Lorentzian Path Integral" approach to Q.G.

Back in 1992 when Ambjorn and Jurkiewicz were doing it
they called it "Simplicial Quantum Gravity"
as in their 1992 article in Phys Lett B.

It is a commonsense notion and maybe goes back to Regge in the 1950s (?)
or further----you let spacetime build itself out of regular blocks (simplices)
and get the dynamics out of a "partition function" of sorts that tells you how likely some transition is to happen by counting the number of ways it can happen. A rudimentary combinatorics of space.

The new thing is that AJL got it to work and 4D spacetimes started emerging from it
somehow Loll began collaborating with Ambjorn
maybe around 1998, then in 2001 she moved from AEI (MPI-Potsdam) over to
Utrecht,
and they must have decided at some point that the Euclidean approach
to Simplicial QG
wasnt working (despite at least 10 years of trying) and they
would try the Lorentzian approach, where you distinguish time-like legs of the simplex from space-like legs, so there is a past and future idea and the possibility of cause and effect.

Strictly speaking this approach to QG is one of the newest since
the "Lorentzian" or "Causal" simplicial QG papers seem mainly post-2000

I may be wrong about these details---still trying to sort this business out.
Loll is probably the best historian of this approach to QG.
She has an invited LivingReviews article on it which gives the history
going back to 1976 and citing some 200 papers.

http://arxiv.org/gr-qc/9805049

I have been trying to understand Loll's background and looked in spires, where I saw a large number of papers published since 1988 from a series of places:
1988 Imperial College London (postdoc working with Isham?)
1990 Bonn University
1992 Syracuse
1993 Penn State
1995 Florence (and MPI Potsdam)
1996 MPI Potsdam
...
...
2001 move from MPI Potsdam to Utrecht

Baez Week 69 (1995) describes his meeting Loll in 1991 in Seattle and also
meeting Isham and Ashtekar at the same conference----Baez introduction to LQG. Week 69 has thumbnails of the first three LQG researchers Baez encountered
http://math.ucr.edu/home/baez/week69.html

as a wild guess if she was a postdoc in 1988 she could have been born roughly around 1962. that could be way off of course. here's a snapshot:
http://www1.phys.uu.nl/wwwitf/fotopagina's/Medewerkers/Renate.htm
the URL needs to be copy/pasted in

pelastration

May15-04, 01:47 AM

I have been trying to understand Loll's background
Lecture of spring 1999. Lost of graphics.
http://cgpg.gravity.psu.edu/online/Html/Seminars/Spring1999/Loll/Slides/s01.html

meteor

May15-04, 05:55 AM

It is a commonsense notion and maybe goes back to Regge in the 1950s (?)

I think that both dynamical triangulations and Regge calculus are different approaches both encompassed by a wider theory called simplicial quantum gravity. While dynamical triangulations keep the edges of the sinplices fixed and varies the triangulations, Regge calculus do the opposite, varies the edges of the simplices and
maintain fixed the triangulation. Baez 122 explains a bit about all this stuff

http://math.ucr.edu/home/baez/twf.ascii/week122

I've found this review of Regge calculus by Giorgio Immirzi, curiously the same person that the famous/infamous Immirzi parameter takes name from
http://arxiv.org/abs/gr-qc/9701052
"Quantum gravity and Regge calculus"

marcus

May15-04, 10:11 AM

Lecture of spring 1999. Lots of graphics.
http://cgpg.gravity.psu.edu/online/Html/Seminars/Spring1999/Loll/Slides/s01.html

Yes there are a lot of sketches, and it helps. I hadnt seen these lecture slides and am glad you pointed to them.

Meteor thanks! I will look at Week 122---hopefully it will sort simplicial QG out into its various types and I will understand it better

Baez posted again on SPR yesterday about the new simplicial QG work

http://physicsforums.com/showthread.php?p=210452#post210452

marcus

May15-04, 12:14 PM

As an aside, by his own account Kepler discovered the third law
on 15 May 1618

a few days later he finished writing "Harmonice Mundi"---the book was published that year

his finding those three laws set a 400 year agenda of figuring out
why gravity acts like that.
why do the planets go in ellipes with sun at focus, sweeping out constant area per unit time, and why does the period-squared vary as the distance-cubed? or why, as he put it, is a planet's period the "sesquipotence" ( 3/2 power) of its average distance from the sun?

marcus

May15-04, 12:40 PM

a day for thoughts concerning the shape of the world
(Kepler's mundus, he too was trying to explain its proportions)
and the hope that starting from those first laws of gravity
mankind may come to grasp the world's geometry.

it would be curious if Ambjorn Jurkiewicz and Loll were on the right track
and that the world's 4D shape including the 1915 Einstein equation
actually does arise from the "causal dynamical triangulation" (referred to in their abstract) which seems merely to be the random sticking together of simplices----with sensible rules that make it possible to simulate in a computer

the fascinating thing is they generate pictures of 4D geometries,
and Baez applied the "a picture is worth..." adage, appropriately

marcus

May16-04, 09:38 AM

... 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 think that both dynamical triangulations and Regge calculus are different approaches both encompassed by a wider theory called simplicial quantum gravity. While dynamical triangulations keep the edges of the simplices fixed and varies the triangulations, Regge calculus do the opposite, varies the edges of the simplices and
maintain fixed the triangulation. Baez 122 explains a bit about all this stuff

http://math.ucr.edu/home/baez/twf.ascii/week122

...

Meteor gives the right perspective. Simplicial Quantum Gravity (SQG?) is the overall line of research and Dynamical Triangulations is one of two or three main approaches within SQG.

Simplicial Quantum Gravity seems to have a lot in common with spinfoam research (Barrett-Crane models could be mentioned). The boundaries of these research areas seem able to shift. Maybe Dynamical Triangulations will turn out to merge with Spinfoams.

At the Marseille conference there was just one DT paper and it was put with the Spinfoam bunch.

In his reply to Larsson on SPR
http://physicsforums.com/showthread.php?p=210452#post210452
Baez says he hopes to work on DT with Dan Christensen at UBC
and that he talked to Fotini M. who is also planning some DT
research with a grad student of hers.

The impressive thing to me about DT is that you can put a million identical simplices in a computer and simulate the universe and see it happen:
the whole story---beginning middle and end

since there is a finite number of simplex blocks it has to be a closed universe that bangs, swells up, collapses, and then crunches
but that's a detail
what's nice is the prospect of a 4D spacetime---a history of the geometry of the world---that you can simulate and see

Baez has already done some heavyduty spinfoam computer stuff with Dan Christensen
apparently they have a large computing facility at UBC
so it's a good bet that they will run MonteCarlo DT simulations
and get pictures

I hope they get about it soon, wd very much like to see graphic results
of others besides Ambjorn Jurkiewicz Loll.

marcus

May17-04, 10:06 AM

Animations

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

Ambjorn's homepage

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

marcus

May17-04, 10:42 AM

http://www.phys.lsu.edu/mog/mog19/node12.html

this includes a bibliography (mostly on line) of what Visser says are key papers

----exerpts----
Quantum gravity: progress from an unexpected direction

Over the last few of years, a new candidate theory of quantum gravity has been emerging: the so-called ``Lorentzian lattice quantum gravity'' championed by Jan Ambjorn [Niels Bohr Institute], Renate Loll [Utrecht], and co-workers [1]...

...On the one hand, "Lorentzian lattice quantum gravity" has grown out of the lattice community, itself a subset of the particle physics community. In lattice physics spacetime is approximated by a discrete lattice of points spaced a finite distance apart. This "latticization" process is a way of guaranteeing that quantum field theory can be defined in a finite and non-perturbative fashion. (Indeed currently the lattice is the only known non-perturbative regulator for flat-space quantum field theory. This technique is absolutely essential when carrying out computer simulations of quantum field theories, and in particular, computer simulations of quarks, gluons, and the like in QCD.)

In addition to these particle physics notions, "Lorentzian lattice quantum gravity" has strongly adopted the geometric flavour of general relativity; it speaks of surfaces and spaces, of geometries and shapes.

On the other hand, "Lorentzian lattice quantum gravity" has irritated both brane theorists and general relativists (and more than a few lattice physicists as well): It does not have, and does not seem to require, the complicated superstructure of supersymmetry and all the other technical machinery of brane theory/string theory. (A critically important feature of brane theory/ string theory which justifies the amount of time spent on the model is that in an appropriate limit it seems to approximate key aspects of general relativity; and do so without the violent mathematical infinities encountered in most other approaches. Of course, there is always the risk that there might be other less complicated theories out there that might do an equally good job in this regard.) Additionally, "Lorentzian lattice quantum gravity" irritates some members of the relativity community by not including all possible 4-dimensional geometries: The key ingredient that makes this Lorentzian approach different (and successful, at least in a lower-dimensional setting) is that it to some extent enforces a separation between the notions of space and time, so that space-time is really taken as a product of "space" with "time". It then sums over the resulting restricted set of (3+1)-dimensional geometries; not over all 4-dimensional geometries (that being the traditional approach of the so-called Euclidean lattice quantum gravity).

... The result of this topological/ geometrical restriction is that the model produces reasonably large, reasonably smooth patches of spacetime that look like they are good precursors for our observable universe. ...

The good news is that once reasonably large, reasonably flat, patches of spacetime exist, the arguments leading to Sakharov's notion of "induced gravity" almost guarantee the generation of a cosmological constant and an Einstein-Hilbert term in the effective action through one-loop quantum effects [3]; and this would almost automatically guarantee an inverse-square law at very low energies (large distances).

The bad news is that so far the large flat regions have only been demonstrated to exist in 1+1 and 2+1 dimensions -- the (3+1)-dimensional case continues to pose considerable technical difficulties.

All in all, the development of "Lorentzian lattice quantum gravity" is extremely exciting: It is non-perturbative, definitely high-energy (ultraviolet) finite, and has good prospects for an acceptable low-energy (infra-red) limit. It has taken ideas from both the quantum and the relativity camps, though it has not completely satisfied either camp. Keep an eye out for further developments.

References:

Key papers on Lorentzian lattice quantum gravity:
J. Ambjorn, A. Dasgupta, J. Jurkiewicz and R. Loll, ``A Lorentzian cure for Euclidean troubles,'' Nucl. Phys. Proc. Suppl. 106 (2002) 977-979 arXiv:hep-th/0201104

J. Ambjorn, J. Jurkiewicz and R. Loll, ``3d Lorentzian, dynamically triangulated quantum gravity,'' Nucl. Phys. Proc. Suppl. 106 (2002) 980-982 arXiv:hep-lat/0201013.

J. Ambjorn, J. Jurkiewicz, R. Loll and G. Vernizzi, ``Lorentzian 3d gravity with wormholes via matrix models,'' JHEP 0109 (2001) 022 arXiv:hep-th/0106082

J. Ambjorn, J. Jurkiewicz and R. Loll, ``Dynamically triangulating Lorentzian quantum gravity,'' Nucl. Phys. B610 (2001) 347-382 arXiv:hep-th/0105267.

A. Dasgupta and R. Loll, ``A proper-time cure for the conformal sickness in quantum gravity,'' Nucl. Phys. B 606 (2001) 357-379 arXiv:hep-th/0103186.

J. Ambjorn, J. Jurkiewicz and R. Loll, ``Non-perturbative 3d Lorentzian quantum gravity,'' Phys. Rev. D 64 (2001) 044011 arXiv:hep-th/0011276.

R. Loll, ``Discrete Lorentzian quantum gravity,'' Nucl. Phys. Proc. Suppl. 94 (2001) 96-107 arXiv:hep-th/0011194.

J. Ambjorn, J. Jurkiewicz and R. Loll, ``Computer simulations of 3d Lorentzian quantum gravity,'' Nucl. Phys. Proc. Suppl. 94 (2001) 689-692 arXiv:hep-lat/0011055.

J. Ambjorn, J. Jurkiewicz and R. Loll, ``A non-perturbative Lorentzian path integral for gravity,'' Phys. Rev. Lett. 85 (2000) 924-927 arXiv:hep-th/0002050.

[2] A survey of brane theory and quantum geometry:
G. Horowitz, ``Quantum Gravity at the Turn of the Millennium'',
MG9 -- Ninth Marcel Grossmann meeting, Rome, Jul 2000,
arXiv:gr-qc/0011089.

[3] Sakharov's induced gravity:
A.D. Sakharov, ``Vacuum quantum fluctuations in curved space and the theory of gravitation'', Sov. Phys. Dokl. 12 (1968) 1040-1041; Dokl. Akad. Nauk Ser. Fiz. 177 (1967) 70-71.

------end quote from Visser----
this is from Jorge Pullin's newsletter "Matters of Gravity"
Pullin gives this address for the author:
Matt Visser, Washington University visser@wuphys.wustl.edu

The point of the recent AJL paper, where they get extended normal-looking 4D regions, is crumbling of what Visser calls the bad news
("The bad news is that so far the large flat regions have only been demonstrated to exist in 1+1 and 2+1 dimensions -- the (3+1)-dimensional case continues to pose considerable technical difficulties."). This barrier has now to some extent been penetrated by AJL.

This opens the way, if Matt Visser is right about this, to what he calls the good news, namely that the model
"almost automatically guarantees an inverse-square law at very low energies (large distances)."

obviously a breaking story, to be continued

marcus

May18-04, 01:00 PM

my hunch is that the Livine/Oriti paper applies to this
http://arxiv.org/gr-qc/0405085
"About Lorentz invariance in a discrete quantum setting"

Livine and Oriti just posted this, and have also announced two
papers (with Girelli) in preparation:

"Deformed Special Relativity as an effective flat limit of Quantum Gravity"

and

"A quantum clock in a quantum causal set"

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

In the first paper "About Lorentz invariance in a discrete quantum setting"
they set about disposing of an objection that could be raised to any
discrete spacetime geometry model

If it is discrete then it probably has some characteristic length---like Planck length. And what happens to this length when you boost?

Relatively moving observers are all presumed to see this same constant length
and this (naively at least) seems paradoxical.

This objection could conceivably be raised to Ambjorn's and Loll's approach.
And it is also the main cause of the stir over DSR.
So Livine and Oriti are talking about something that applies rather widely: not only to Loop gravity and to Spinfoams, but also to DSR and the lattice-like or simplicial quantum gravity models that interest AJL---the "dynamical triangulation" models that Baez called our attention to in this thread.

marcus

May18-04, 01:46 PM

I'm hoping that looking over the program for the Marseille conference
will give us some ideas of the direction QG is going.
I copied the program of talks here a few posts back.
Loll's talk on Dynamical Triangulations is of course what Baez focussed on
but here are a few others with evocative titles:

J. Kowalski Gliksman (DSR as a possible limit of quantum gravity)
F. Girelli (Special Relativity as a non commutative geometry: Lessons)
E. Livine (Instantons in GFT and continuum limit)

marcus

May18-04, 02:29 PM

some Livine/Oriti quotes:

"Geometric quantities are the observable properties of the gravitational field."

"Gravitation is geometry and measurements of distances are measurements of properties of the gravitational field."

"Does a quantum gravity theory with an invariant length and a discrete spectrum for geometric observables necessarily break Lorentz symmetry or necessarily require some sort of modification/deformation of it? The answer, as we will see, is simply 'no' ”.

marcus

May19-04, 09:42 AM

key quote from Larsson's most recent post on SPR:

----quote from Thomas Larsson---
Thus, I believe that it is a fair chance that AJL have indeed succeeded in quantizing gravity. They do so not by assuming a lot of experimentally unconfirmed new physics, but rather by strictly implementing the time-honored principles of old physics, especially causality.
---end quote---

My bolding.
Larsson makes important points in this post.
I defer to his judgement and generally agree, but have a couple of
comments to make in the context of this thread.
Here is the text of his post, which was in reply to Baez.

-----Larsson post, for possible comment-----
baez@galaxy.ucr.edu (John Baez) wrote in message news:<c82uao$i34$1@glue.ucr.edu>...
> In article <24a23f36.0405112105.6569f265@posting.google.com>,
> Thomas Larsson <thomas_larsson_01@hotmail.com> wrote:
>
> >baez@math.removethis.ucr.andthis.edu (John Baez) wrote in message
> >news:<c7pbsa$p9$1@glue.ucr.edu>...
>
> >> Given all this, I'm delighted to see some real progress on getting 4d
> >> spacetime to emerge from nonperturbative quantum gravity:
> >>
> >> 3) Jan Ambjorn, Jerzy Jurkiewicz and Renate Loll, Emergence of a 4d world
> >> from causal quantum gravity, available as hep-th/0404156.
>
> >This is pretty exciting.
>
> I'm glad you think so! I sure do!
>
Maybe I was overreacting. It was becoming boring to be negative all the time, so when I realized that somebody had made tangible progress towards some kind of quantum gravity, I got carried away.

Anyway, I would like to discuss to what extent AJL really have succeed in constructing a model of QG in 4D. As I see it, there are three things that could go wrong: that the model isn't quantum, that it isn't gravity, or that the measure is wrong.

1. Is the AJL model really quantum? Some time ago, Urs Schreiber argued that LQG, or at least the LQG string, fails to be a true quantum theory, and I tend to agree. However, the AJL model can be viewed as a statistical lattice model, and if such a model has a good continuum limit, it is AFAIK always described by some kind of QFT. What else could it be?

2. Is the AJL model really gravity? The action is a rather straightforward discretization of the Einstein action with a cosmological term:

\int R => sum over (d-2)-simplices

\int det g = volume => sum over d-simplices.

What is perhaps somewhat unusual is that all edges have the same length, which is different from Regge calculus. Nevertheless, I don't think that this really matters, but one could check if the results look different if you allow for variable edge lengths.

3. Is the measure right? Here is the place where AJL differ significantly from previous simulations. AFAIU, the crux is that AJL insist on a strict form of causality: they exclude spacetimes where the metric is singular, even at isolated points. This may seem like an innocent restriction, but it rules out things like topology change and baby universes, which require that the metric be singular somewhere.

It is not obvious to me whether one should insist on such a strong form of causality or not, but this assumption leads at least to better results, e.g. a reasonably smooth 4D spacetime. Thus, I believe that it is a fair chance that AJL have indeed succeeded in quantizing gravity. They do so not by assuming a lot of experimentally unconfirmed new physics, but rather by strictly implementing the time-honored principles of old physics, especially causality. That is cool.
---------end quote------------
http://physicsforums.com/showthread.php?p=212669#post212669

Notice that he says
"What is perhaps somewhat unusual is that all edges have the same length, which is different from Regge calculus."

This is the "dynamical triangulation" approach which has been extensively pursued since around 1985. In the 1990s it has seemingly replaced Regge calculus as the main focus of attention, or so is my impression. Here is a very good historical account from 1992 by Ambjorn Jurkiewicz and Kristjansen

"Quantum gravity, dynamical triangulations and higher derivative regularization"
http://arxiv.org./hep-th/9208032

I have put keywords "dynamical triangulation" in arxiv search and come up with 155 papers mostly since 1995----this includes some search-engine mistakes, not all are dynamical triangulation quantum gravity.

Surveys of quantum gravity typically list DT along with Regge approach on equal footing in the "Discrete Approaches" category. For example in Rovelli's
1998 survey "Strings Loops and Others" (gr-qc/9803024) plenary talk given at the GR15 conference, the approaches are listed:
string, loop, Regge, dynamical triangulation, Ponzano-Regge, euclidean quantum gravity (a Hawking favorite),....,etc,....

the history makes no difference to Larsson's excellent and well-qualified points but I want to know it anyway

DT has been there all along, since 1985 work by David and by Ambjorn, and maybe earlier. But I at least simply did not notice! There is a lot of work, a lot of computer simulations, review papers, interesting graphix including spacetime animation. We should have this stuff assembled and be aware of it. Here are some 1985 papers that I think are DT

[12] F. David, Nucl. Phys. B 257 (1985) 45.
[13] J. Ambjørn, B. Durhuus and J. Froehlich, Nucl. Phys. B 257 (1985) 433.
[14] F. David, Nucl. Phys. B257 (1985) 543.
[15] V. A. Kazakov, I. K. Kostov and A. A. Migdal, Phys. Lett. 157B (1985) 295.

these are from the good survey by Ambjorn, Jurkiewicz, Kristjansen

Renate Loll also has a 2003 introduction
"A discrete history of the Lorentzian path integral"
http://arxiv.org/hep-th/0212340
this is of course DT, but also more----it is foliated
and this goes to another point Larsson made, his point 3, about
the "strict causality"
This has been the variant of the DT approach prevalent since 1998.
Loll gives somewhat of the history of this "Lorentzian" or "causal" DT.
Lots to discuss here

john baez

May19-04, 07:30 PM

[...] there are some previous papers by AJL which foreshadow the current one and illuminate what is going on. I think John Baez gave these links to lead-ups.

http://arxiv.org./hep-th/0002050
http://arxiv.org/hep-th/0105267

As far as I can tell the ideas that bear fruit in the recent paper (and generate an extended 4D world) are already three years old. I can't find anything conceptual that wasn't already suggested in the paper dated 27 May 2001:

"A Nonperturbative Lorentzian Path Integral for Gravity"

so right now I'm trying to understand the lead-up papers.

Good! You're right, the concepts were all there in those earlier papers, and the concepts are simple and elegant. But as you probably know, these earlier papers were just warmup exercises. They tackled quantum gravity in 1+1 and 2+1 dimensions. Lots of approaches work in those low dimensions; the physically realistic 3+1-dimensional case is much harder. So, the first really strong evidence that Ambjorn, Loll and Jurkiewicz are on the right track came from their new calculations in the 3+1-dimensional case. The reason it took them a while to do these new calculations is that they require some heavy-duty computer work.

In case anyone out there doesn't know: classically, in 1+1 dimensions every metric is a solution to the equations of general relativity without matter (with vanishing cosmological constant). In 2+1 dimensions, only flat metrics are solutions to these equations. Only in 3+1 and higher do the equations become interesting... with gravitational waves, black holes and so on.

So, while there are millions of papers on quantum gravity in 1+1 dimensions and 2+1 dimensions, and many of them are actually interesting, it's always incredibly risky to extrapolate any conclusions about higher dimensions from those special cases.

john baez

May19-04, 08:06 PM

Was this paper an attempt to justify or perhaps even derive the very overall topology of space-time?

Not really. The main goal was to get a theory of quantum gravity in 3+1 dimensions that works - meaning that it reduces to general relativity at length scales much larger than the Planck scale. They didn't prove their model works, but they produced some impressive evidence that it might.

But, there is something to say about topology here.

In their model, you can take space at a given time to have any topology you want - any compact 3-dimensional manifold, that is. The model then ensures that the topology of space will remain the same at all other times.

In other words, the model forbids "topology change".

They wanted this, because in very similar models (dynamical triangulation models) that don't forbid topology change, there's a strong tendency for all hell to break loose: typical spacetimes are either "crumpled" or "branched polymers". This problem had afflicted the subject for decades! This is what Ambjorn, Jurkiewicz and Loll seem to have gotten around!

I wonder if causality is the key to the topology of space-time.

I'd prefer to say it's the key to preventing topology change. This is well-known in classical general relativity, where one can prove there's no topology change if spacetime is "globally hyperbolic" - that is, very roughly, if it has a well-behaved concept of causality.

john baez

May19-04, 08:28 PM

Mike2 on the other thread raised the question of whether the quantum gravity model assumed by AJL had been rigorously developed.

As far as I can tell, it's completely rigorous. And I'm a mathematician by training, so I'm more fussy about these things than most. :devil:

I know of rigorous (1,1)-dimensional theories and maybe some (1,2)-dimensional ones, but I don't know of any fully (1,3) relativistic quantized ones.

It's easier to make discrete models rigorous than models that assume spacetime is a continuum. That's the main reason I like discrete models.

In particular, all the 3+1-dimensional spin foam models of quantum gravity I've worked on - various versions of the Barrett-Crane model - are mathematically rigorous and background-free.

The problem is, we haven't gotten good evidence that these spin foam models "work" - namely, that they reduce to general relativity in the limit of distance scales that are large compared to the Planck length.

See my Marseille talk for a taste of the problems:

http://math.ucr.edu/home/baez/spin_foam_calculations.ps

Since we don't have any experimental evidence concerning quantum gravity, mathematical rigor is one way to make sure we're not playing tennis with the net down. I will be very happy when we get any rigorously well-defined background-free quantum theory of gravity that works in the sense defined above.

More precisely: I will be very happy if we get numerical evidence that it works, and ecstatic if we can mathematically prove that it works. But since such a model is likely to be nonperturbative, a mathematical proof of this sort might be very difficult. Nobody has even proved confinement in lattice QCD, even though numerical calculations have convinced everyone it's true.

john baez

May19-04, 09:35 PM

While we are on possibly important papers (of which e.g. AJL's)
does anyone have any guidance or comment about
Marni Dee Sheppeard's recent
http://arxiv.org/gr-qc/0404121
unless for some reason it is tactless of me to ask.

Hmm, I just see this paper has been withdrawn! Good! I read it while flying to Marseille. She's young, she made a mistake, she's smart, she did the right thing. 'Nuff said.

Read this one, it's much better:

L. Crane, M.D. Sheppeard
2-categorical Poincare Representations and State Sum Applications
http://www.arxiv.org/abs/math.QA/0306440

Also I wish we could hear more about the Marseille conference
since Week 206 merely whetted my appetite.

Well, what can I say? The scenery was great:

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

but the wine was awful. Did you know the French Mathematical Society makes their own wine? I'm not sure that's what we were drinking, but it might explain it. At least it got the job done... so even though I was jet-lagged and completely exhausted, I stayed up late every night talking to all my favorite quantum gravity folks: Ashtekar, Barrett, Christensen, Jacobson, Krasnov, Lewandowski, Markopoulou, Loll, Rovelli, Smolin, and others - carefully listed in alphabetical order to avoid any appearance of favoritism.

Seriously, us "old-timers" were all very impressed by the large numbers of bright new young folks moving into the field and doing good things.

We talked about pretty much everything, but I was always trying to get everyone to tell me what calculations Dan Christensen and I should do in the Barrett-Crane model to either get more "physical" results than we've gotten so far, or kill the model dead. Also, a bunch of us were pestering Renate Loll to make sure we understood her model in detail. Ashtekar is doing a lot of work on semiclassical states, so he was talking about that a bunch....

But if you want to see quantum gravity folks in action, it's probably not too late to register for Isham's conference at Imperial College this September - a lot of the same people will be there, too. I'm spending most of the summer in Cambridge, and I'd been planning to go down and spend a week in London starting September 8th, but I may go down a bit early to catch this - or just take the train down for the day a couple of times!

marcus

May19-04, 09:55 PM

your account of the late night conversations and bad homemade wine is heartwarming. thanks for these posts

pelastration

May20-04, 03:11 AM

The scenery was great:

http://math.ucr.edu/home/baez/calanque.html
Indeed beautiful.

You said: I saw these sights, but I didn't take these photographs! I got them off the web, but now I can't find where I got them. If you know, please tell me so I can credit the photographer.

Check George Gollin's website: http://www.hep.uiuc.edu/home/g-gollin/graphics/france.html

marcus

May20-04, 10:06 AM

Indeed beautiful.
...[/URL]

You said it! As long as we are doing pictures, here's the awful truth of what some of these people look like.

http://www.edge.org/3rd_culture/bios/baez.html

Carlo Rovelli:
http://www.cpt.univ-mrs.fr/~rovelli/rovelli.html

Since Ambjorn Loll Jurkiewicz may have finally succeeded in quantizing GR
(necessarily in a Backgr. Indep. fashion since GR's spacetime geometry is B.I.)
we should include a picture of Loll. [edit: this link is courtesy selfAdjoint]

http://phys.uu.nl/wwwitf/fotopagina's/Medewerkers/Renate.htm (http://www1.phys.uu.nl/wwwitf/fotopagina's/Medewerkers/Renate.htm)

and bring forward the link to the picture of Jan Ambjorn

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

More pictures---a graphic realization of the Background Independence of SQG
(Ambjorn has proposed to call the "dynamical triangulations" approach by the name Simplicial Quantum Gravity---yes I know some people think of SQG as a more general term that includes DT as a subfield) can be seen in these computer animations:

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

clearly these spacetimes have no fixed geometry! they squirm and ripple.

Baez says that Fotini M plans to start some research in this area with a grad student of hers. I would like to find a photo of Fotini. Does anyone have a better link?
http://www.sciam.com/article.cfm?articleID=0007E95C-9597-1DC9-AF71809EC588EEDF
http://www.perimeterinstitute.ca/people/researchers/longterm.cfm

Here's Lee Smolin:
http://www.edge.org/3rd_culture/bios/smolin.html

selfAdjoint

May20-04, 02:07 PM

Why not link to Renate Loll (http://www1.phys.uu.nl/wwwitf/fotopagina's/Medewerkers/Renate.htm) ?

marcus

May22-04, 12:01 AM

What is the area spectrum in Simplicial Quantum Gravity?

if Ambjorn and Loll-style "dynamical triangulation" approach to quantum gravity works out (a possibility John Baez seemed to be allowing for) then a major unsolved question concerns the area spectrum

will it come out discrete, as in Loop gravity

will it come out the same as in Loop gravity, the same multiples of the Planck unit area, the same set of numbers for eigenvalues of the operator

how can the area operator in Simplicial Quantum Gravity be constructed

can computer (monte carlo) simulations be used to calculate areas?

jeff

May22-04, 09:04 AM

What is the area spectrum in Simplicial Quantum Gravity?...a major unsolved question concerns the area spectrum

will it come out discrete, as in Loop gravity

will it come out the same as in Loop gravity, the same multiples of the Planck unit area, the same set of numbers for eigenvalues of the operator

how can the area operator in Simplicial Quantum Gravity be constructed

You should think about the fact that the simplicial edges here may be viewed as having been pre-assigned one quantum of length. What we need to know is if GR is produced. Anyway, how seriously should we take quantum theories requiring all but a very special family of quantum fluctuations be ignored? (This may explain why such an obvious idea wasn't previously pursued).

pelastration

May22-04, 09:22 AM

Baez says that Fotini M plans to start some research in this area with a grad student of hers. I would like to find a photo of Fotini. Does anyone have a better link?
http://www.sciam.com/article.cfm?articleID=0007E95C-9597-1DC9-AF71809EC588EEDF

Not a better photo. You can email here and ask. :wink:
http://www.perimeterinstitute.ca/people/researchers/view_bio.cfm?id=18.

Added: I asked here a link to some photo's.

marcus

May22-04, 09:37 AM

maybe we have enough mugshots for the moment :wink:

issues about the history of the "dynamical triangulations" line of research
(also Ambjorn calls it SQG simplicial quantum gravity, treating
that as a synonym for DT in one paper I saw) are bound to come up.

so the history of this (1985, 1992, 1998 are important years) is another thing to keep in mind
as a reminder here are some links I posted a while back

http://www.physicsforums.com/showthread.php?p=213245#post213245

for the notes on history, scroll down just past the end of the Larsson quote

john baez

May22-04, 02:55 PM

Indeed beautiful.

[JB] said: I saw these sights, but I didn't take these photographs! I got them off the web, but now I can't find where I got them. If you know, please tell me so I can credit the photographer.

Check George Gollin's website: http://www.hep.uiuc.edu/home/g-gollin/graphics/france.html

Thanks! That's it! I'll credit him.

By the way, Marcus raised an interesting question about the spectrum of the area operator in the Ambjorn-Jurkiewicz-Loll model:

Will it come out the same as in Loop gravity, the same multiples of the Planck unit area, the same set of numbers for eigenvalues of the operator?

That seems very unlikely! The model is based on the assumption that all the tetrahedra of which space is built are regular tetrahedra, all with the same basic edge-length. So, every face of every tetrahedron is an equilateral triangle of the same size. This size is the "quantum of area" in this model - call it a. The model doesn't specify what this number a is, but the obvious area operator - I claim there's an obvious area operator on the Hilbert space of this theory - will have as its spectrum the numbers

0, a, 2a, 3a...

This is more like what Bekenstein and Mukhanov claim in their famous paper on black hole spectroscopy than anything one gets in loop quantum gravity.

How can the area operator in Simplicial Quantum Gravity be constructed?

There are lots of ways, and we could have a fun argument about which one is "right", just like people have had in loop quantum gravity, with the Ashtekar-Lewandowski area operator battling the Rovelli-Smolin area operator. The problem is that we can't tell which area operator is "right" until we find some calculations that only give nice answers with the "right" area operator. Or, do an experiment and measure areas at the Planck scale - not very practical, and this will only work if not only our area operator but also our whole theory is also right!

It seems pretty easy to cook up a nice area operator in the AJL model: there's a Hilbert space whose orthonormal basis consists of all ways of triangulating a given 3-manifold into tetrahedra. Picking a surface for each triangulation, with the surface made of triangles in that triangulation, we get an area operator such that each state in the above basis is an eigenstate with eigenvalue na, where n is the number of triangles in the surface for the given triangulation.

That was a bit terse, but it was a complete description of the "obvious" area operator. It even takes into account the fact that which surface we're talking about can only be specified after we say which state of the universe we've got! In other words, this is a physical observable, not a "kinematical" one.

Can computer (monte carlo) simulations be used to calculate areas?

I don't see the need for this, since the dynamics of the theory don't affect the area spectrum in any obvious way.

(This is what people hope in loop quantum gravity, which is why people dare talk about the area spectrum even before solving the Hamiltonian constraint. In other words, they're computing the spectrum of a kinematical observable and hoping that'll be the spectrum of a physical observable. But the AJL model is gauge-fixed, so there's no Hamiltonian constraint! - so it's much easier to construct physical observables.)

(Where of course "physical" is a technical term that doesn't imply any of this stuff is relevant to the real physical world!)

sol2

May22-04, 04:31 PM

I have a question about topology.

If we accept the ideas of LQG as a basis of of these triangulations in quantum gravity, can we say that these are discrete.

As part of the mathematical structure (http://superstringtheory.com/forum/extraboard/messages12/666.html) of string theory based on Kaluza and Klein's ordering of geometries this would seem consistent to me, while LQG might be lacking in this discription and less pervasiveness?

Looking at the monte carlo for better comprehension of the energy bending plot, helped to define the structure for me in visualization.s If we can get better pictures then It goes a long way for me:)

I was looking for a consistent geometrical basis. Can this be done?

Also string frequencies can be calculated (http://www.superstringtheory.com/forum/stringboard/messages18/1.html) why can LQG not?

marcus

May23-04, 11:33 AM

getting back to the main topic of the thread (Simplicial Quantum Gravity---AJL model---Baez comments after the Marseille conference) here are exerpts from JB post from yesterday, 22 May:

... interesting question about the spectrum of the area operator in the Ambjorn-Jurkiewicz-Loll model:

"Will it come out the same as in Loop gravity, the same multiples of the Planck unit area, the same set of numbers for eigenvalues of the operator?"

That seems very unlikely! The model is based on the assumption
that all the tetrahedra of which space is built are regular tetrahedra, all
with the same basic edge-length. So, every face of every tetrahedron is an equilateral triangle of the same size. This size is the "quantum of area" in this model - call it a. The model doesn't specify what this number a is, but the obvious area operator - I claim there's an obvious area operator on the Hilbert space of this theory - will have as its spectrum the numbers

0, a, 2a, 3a...

This is more like what Bekenstein and Mukhanov claim in their famous paper on black hole spectroscopy than anything one gets in loop quantum gravity.

"How can the area operator in Simplicial Quantum Gravity be constructed?"

There are lots of ways, and we could have a fun argument about which one is "right", just like people have had in loop quantum gravity, with the Ashtekar-Lewandowski area operator battling the Rovelli-Smolin area operator. The problem is that we can't tell which area operator is "right" until we find some calculations that only give nice answers with the "right" area operator. Or, do an experiment and measure areas at the Planck scale - not very practical, and this will only work if not only our area operator but also our whole theory is also right!

It seems pretty easy to cook up a nice area operator in the AJL model: there's a Hilbert space whose orthonormal basis consists of all ways of triangulating a given 3-manifold into tetrahedra. Picking a surface for each triangulation, with the surface made of triangles in that triangulation, we get an area operator such that each state in the above basis is an eigenstate with eigenvalue na, where n is the number of triangles in the surface for the given triangulation.

That was a bit terse, but it was a complete description of the "obvious" area operator. It even takes into account the fact that which surface we're talking about can only be specified after we say which state of the universe we've got! In other words, this is a physical observable, not a "kinematical" one...

at first sight, at least, it strikes me as strange that there's been no mention of matter participating in defining the surface whose area is to be measured (maybe this would be grist for the "fun arguments" JB says could be possible)

but I am not looking for arguments! I just find it unintuitive that one should have an area operator that measures the area of a perfectly abstract immaterial surface apparently chosen arbitrarily---a different one for each quantum state. I would feel more comfortable if the surface was the surface of my desk or something tangible like that (or at least defined by the gravitational field like a BH horizon would be). Maybe this is qvetching.

sol2

May23-04, 02:00 PM

Marcus I would like to remove my post to preserve continuity between you and JB, Should this be done? I like your questions about what is real as well, but tangibles are not always easy moving to hyperspace realizations and quantum gravity? We'll have to see what JB saids to your response.

selfAdjoint

May23-04, 02:51 PM

getting back to the main topic of the thread (Simplicial Quantum Gravity---AJL model---Baez comments after the Marseille conference) here are exerpts from JB post from yesterday, 22 May:

at first sight, at least, it strikes me as strange that there's been no mention of matter participating in defining the surface whose area is to be measured (maybe this would be grist for the "fun arguments" JB says could be possible)

but I am not looking for arguments! I just find it unintuitive that one should have an area operator that measures the area of a perfectly abstract immaterial surface apparently chosen arbitrarily---a different one for each quantum state. I would feel more comfortable if the surface was the surface of my desk or something tangible like that (or at least defined by the gravitational field like a BH horizon would be). Maybe this is qvetching.

Note that in their work on two and three dimensional "toy models' they did fit them with a toy model of matter, specifically an Ising spin lattice. And they showed that in the continuum limit the parameters of the model which they could derive in their quantum gravity form matched the ones derived in flat Minkowski space. I would presume the next thing they are going to do with their four dimensional mosel is to repeat this calculation.

Note also that this is a "get our claim out there in print" kind of paper, which sets out to do no more than prove their claim to derive four dimensional space from their local quantization. And bells and whistles will come later.

marcus

May23-04, 06:11 PM

Marcus I would like to remove my post.

Sol please don't feel you should remove your "I have a question about topology" post of yesterday. Everyone's (positive) expressions of interest adds to the welcome with which we honor a visiting expert.
It is considerate and sensitive of you to worry about on topic/off topic issues.

the fact is for me I have to focus (which is just my situation relative to this)
but that does not mean you have to do the same!
there is a place for intellectual leaping in these discussions
(and for starting new threads on related tangents too)

however i must say that SQG is still at a rudimentary stage and
dealing with the most basic nuts and bolts
or so it seems to me

and accordingly I think you will find more possibilities for tie-ins with
loop gravity foam and the rest later on when
SQG has been cooking a while longer

john baez

May23-04, 09:25 PM

at first sight, at least, it strikes me as strange that there's been no mention of matter participating in defining the surface whose area is to be measured (maybe this would be grist for the "fun arguments" JB says could be possible).

Actually I don't find this particular type of argument very fun - it's based too much on philosophical taste and not enough on the details of the model being considered. The AJL model has no matter in it, so we can't use matter to locate a surface in spacetime in this model. We can include matter, but then we have a different model.

Some people have strong philosophical objections against models of quantum gravity that don't include matter, but I've never understood these, since classical gravity is a perfectly sensible theory without matter, and nobody ever explains why the philosophical objections are supposed to kick in only when you quantize this theory!

Of course our universe has matter and we're striving for a theory of that. Also of course, there may be technical reasons why a theory of quantum gravity without matter can't possibly work. Nobody knows: this is a big open question. But the vague philosophical argument that "you can't tell where anything is without matter" just seems wrong to me. In curved spacetime, different places are different, so you can tell where features are. The vacuum Einstein equations make perfect sense classically; they don't become ambiguous due to the lack of matter. So, we can try to quantize this theory and see what happens.

The "fun" arguments I was alluding to are those that start with the AJL model, accept the fact that this theory has no matter in it, write down some well-defined operators, and then argue about which one is "the right area operator". Here we are dealing with a tough problem that might actually have a solution.

but I am not looking for arguments!

Don't worry, I'm not really arguing - just explaining why certain arguments don't seem fun to me, while others do. :smile:

I just find it unintuitive that one should have an area operator that measures the area of a perfectly abstract immaterial surface apparently chosen arbitrarily---a different one for each quantum state.

It may seem weird, but observables of this sort exist in the classical theory, so it should not really be shocking that they exist in the quantum theory. All we're saying here is that if you have a particular state of quantum gravity, you've got a particular "spacetime" of a quantum sort, and as in a classical spacetime you can talk about surfaces and their area.

I would feel more comfortable if the surface was the surface of my desk or something tangible like that (or at least defined by the gravitational field like a BH horizon would be).

I can see why these make you feel comfortable, but your comfort will turn to terror when you try to do calculations with an operator whose definition relies upon a mathematically precise definition of "the surface of my desk" - or even a "black hole horizon", which is much simpler to define, but still rather complicated. It's easier to first define the area of an arbitrary surface, and worry later about whether it's the surface of your desk. We do this in classical gravity, so I think we should do it in quantum gravity too.

marcus

May23-04, 09:41 PM

very glad to have this lengthier discussion expanding on what you said earlier, which now seems quite reasonable or at least less strange.
so I should now imagine a hilbert space of (linear combinations of) all possible triangulations of a certain 3-manifold using a uniform set of tetrahedra

I have to stop and think if this is separable.

I think so.

for each N, all the possible ways of snapping together a set of N tetrahedra
the union of that has to be countable.

what is the inner product?

selfAdjoint might have an intelligent question (having proceeded ahead a ways), I am still getting my bearings.

pelastration

May24-04, 06:40 AM

It's interesting to see what other people actually do on Elastic Interval Geometry.

Some movies showing dynamical triangulations in 3D: http://www.beautifulcode.nl/fluidiom/index.php?pagename=Main.FluidiomMovies The second mpg-movie (first image) (14Mb) is impressive.

http://www.beautifulcode.nl/gallery/

The server seems a little bit slow. Take your time.

marcus

May24-04, 09:15 AM

what is the inner product?

Would it be nice if the inner product could somehow respect
dual triangulations.

Maybe it says in one of these papers we have links to how one
defines the inner product on a linear space consisting of linear combinations
of trglns of a 3-manifold---and I just missed it---or maybe it is a well-known proceedure I dont know about.

anyone other than JB have a page reference or link for this?

those "moves" that get you from one trgln to another turn into linear operators----possibly fun---any special properties?

marcus

May24-04, 09:21 AM

I have a vague feeling of dejavu that the *star-category (from Q.Quandaries paper) idea could relate to this dynamical triangulations business. Was this spelled out somewhere and i just forgot about seeing it?
Maybe should just drink my coffee and not worry about this.

marcus

May29-04, 10:51 AM

----from sA post on "Rovelli program" linkbasket thread----

...The two great historical exemplars of beauty first were Einstein and Dirac. In each case their approach achieved a great success early but then led them into unproductive wastelands. And it is at least arguable that both string physics and LQG research in the Ashtekar tradition are right now spinning their wheels. Maybe it's time for a younger generation, playing Feynman and Dyson to the Witten - Ashtekar version of Einstein-Dirac to have their say. Which is why I am very interseted in the AJL paper, a possibly rough hewed (remember Feyman's early rep?) but undoubtedly novel approach to the problem of background independent quantum mechanics (and THAT, not just quantum gravity is the big kahuna)...

------end exerpt----

some provocative idea(s) or seeds thereof here

AJL dynamical triangulations approach seems very close in spirit and practice to spinfoam
But also LQG and spinfoam are closely allied lines of research with people moving back and forth between them---even erecting theoretical bridges as in Livine' thesis.

we lack a good general classification-----all these research lines are aiming at a background independent quantum gravity----no official name but could call the goal a quantum general relativity
and there is a tailwagsdog effect that the background independence feature of GR is so massive that when you try to "quantize GR" it begins to look as if you are "bacgroundindependencing quantum mechanics".

two people on the ice, who is pulling whom, that kind of thing

well I didnt quite respond to your point about beauty and the historical parallels, but I want to see where it leads and also this big kahuna idea

BTW here's todays post on SPR by Thomas Larsson about the AJL paper:

http://physicsforums.com/showthread.php?p=221756#post221756

-----sample from Larsson----
Dear Zirkus,

Motl has of course completely missed the main point. Distler's
objection from 3 years ago was that he didn't believe in a good
continuum limit in 4D; a "miracle" as he puts it. This may have
been good point at that time; I thought so myself, although I
would have been much less pessimistic if I had known that
Ambjorn and Loll had already succeeded in 2 and 3D.

The new thing is that AJL have presented rather compelling
numerical evidence for a good continuum limit in 4D, thus making
Distler's objection obsolete. It is the fact that AJL have
apparently succeeded in quantizing gravity numerically that
people are so excited about....

------end quote-----

marcus

May29-04, 07:06 PM

In his 1998 review Rovelli says that both spinfoam and dynamical triangulation simplicial QG can be seen as developing from Hawking's "Euclidean QG" which he discusses in the section called "Old Hopes turning into Approximate Theories".

-------quote from Rovelli gr-qc/9803024--------

B. Old hopes --> approximate theories

1. Euclidean quantum gravity
Euclidean quantum gravity is the approach based on a formal sum over Euclidean geometries (6):

Z = N \int D[g] e^{-\int d^4x \sqrt {g} R[g]}

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 IV-A) and the spin foam approach (Section V-C2) 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 V-C1).

Second, this approach can be used as an approximate method for describing certain regimes of nonperturbative spacetime physics...

------end exerpt----

Mike2

May31-04, 07:56 PM

I hate to be late to the discussion, but I just got through reading Three roads to Quantum Gravity and it amazes me that I still have basic questions such as: how is the discrete spacetime connected in a topologal sense. I mean, does each "cell" of spacetime share a side with adjacent cells? Or are there infinitesimal edges that connect regions together? What are the loops in quantum gravity? And how does this differ from the AJL picture of a spacetime?

Thanks.

marcus

May31-04, 08:47 PM

how is the discrete spacetime connected in a topological sense.

I dont think a discrete spacetime model needs to be topologically connected.

At least I never heard it said that one needed a space to be connected before one could define fields and waves and stuff on it.

A lattice of points isnt connected but you can define stuff on it
that looks and acts like waves.

Computers do that all the time, like waves in computer animations are defined on a finite array of points.

Mike2 maybe you are asking the wrong question. Instead of how is the underlying space connected
maybe you should be asking whether and whether there is any reason it needs to be.

I just took a bath in a deep tub of hot water. it looked continuous to me, the water. It acted continuous and connected. It conducts heat and sound and water-waves and is transparent to lightwaves. It would conduct electricity if I was unlucky enough to be struck by lightning while in the bathtub
But it wasnt topologically connected or anythinglike a differentiable manifold.

It was actually a finite set of molecules, behaving like a continuum.

[edit: clarification, I infer from your next post that you thought I was making a reference to LQG, but without mentioning LQG, when i was mentioned lattices! As far as i know LQG is not a lattice theory and does not model space by discrete points. It has an underlying manifold, just no pre-specified geometry. We really need a general classifier word for
the various background indep. approaches to quantizing General Relativity.
they have a lot of resemblances but differ in details. What shall we call them, maybe "Loop etc. gravity" so that it is clear we are including the spin foam and simplicial models?]

Mike2

Jun1-04, 08:02 AM

I dont think a discrete spacetime model needs to be topologically connected.

At least I never heard it said that one needed a space to be connected before one could define fields and waves and stuff on it.

A lattice of points isnt connected but you can define stuff on it
that looks and acts like waves.

Computers do that all the time, like waves in computer animations are defined on a finite array of points.

Mike2 maybe you are asking the wrong question. Instead of how is the underlying space connected
maybe you should be asking whether and whether there is any reason it needs to be.

I just took a bath in a deep tub of hot water. it looked continuous to me, the water. It acted continuous and connected. It conducts heat and sound and water-waves and is transparent to lightwaves. It would conduct electricity if I was unlucky enough to be struck by lightning while in the bathtub
But it wasnt topologically connected or anythinglike a differentiable manifold.

It was actually a finite set of molecules, behaving like a continuum.
It seems necessary to me that there be some sort of connected space or connected items in order to transmit any kind of signal through a medium. Otherwise, how does information travel from one point to the next if there is absolutely no medium of any kind between the points? So I wonder how the lattice of LQG is connected. Perhaps information travels through the edges. But then can information travel through an infinitesimally thin line? Is LQG creating point particles of space-time, with action at a distance through no medium at all? What?

marcus

Jun7-04, 01:06 PM

... So I wonder how the lattice of LQG is connected...

Mike2, as selfAdjoint has explained to you in another thread, LQG is based on a continuum, on a differentiable manifold, not on a lattice. You dont have to worry about it being connected.
As sA also remarked the simplicial AJL model, which is really more the topic of this thread, is also a continuum. Think of it as a diff. manif that has been "triangulated" ----built up out of simplices----fused glued welded together from simplices----partitioned into simplices without actually splitting them (they touch).
We need to get on with following developments around the SQG (simplicial quantum gravity) or "dynamical triangulations" model of Ambjorn Jurkiewicz Loll

marcus

Jun7-04, 01:13 PM

the AJL paper and Simplicial Gravity is a fast moving story so I think we should try to keep up on it
Yesterday Baez posted on SPR---some strong statements about AJL approach in response to Charlie Stromeyer

--------Baez post Sunday, quote----
In article <61773ed7.0405240822.1c7108de@posting.google.com>,
Charlie Stromeyer Jr. <cstromey@hotmail.com> wrote:

>Here are three other reasons to be skeptical of discretized approaches
>to gravity:
>
>1) How are such approaches to be made compatible with vector
>supersymmetry (or vsusy) which is a topological type of symmetry that
>appears in both gravity and topological gauge theories [1].

This "vector supersymmetry" is a mathematical feature of certain
field theories - not something that anyone has observed experimentally.

Nobody has yet constructed a background-free quantum theory that has
general relativity as its limit at large distance scales. The Ambjorn-
Jurkiewicz-Loll model is the closest anyone has come. If they succeed,
this will be of interest regardless of whether their model displays
mathematical features that appear in certain other theories!

>2) How are such approaches to be made compatible with Bell-like
>correlations, non-locality and non-causality which are each present in
>the experiment described in this brief four page paper [2].

As a quantum theory, the Ambjorn-Jurkiewicz-Loll model automatically
has Bell-like "entanglement" and all that jazz.

>3) To paraphrase a sentence that Stephen Hawking once wrote, to not
>believe in the beauty and unity of the dualities of M-theory is like
>believing that evolution did not occur because instead God placed by
>hand all the fossils in the Earth just to play a joke on the
>paleontologists :-)

We resort to theological arguments in physics only when better arguments
are lacking. If a scintilla of experimental evidence for M-theory is
ever found, people will instantly stop making arguments of the sort
you mention here.

Please understand what I'm saying:

I'm not saying that M-theory is "wrong" or that the Ambjorn-Jurkiewicz-Loll
model is "right". M-theory makes too few definite predictions to be wrong.
The AJL model does not include matter, so it cannot be right. But the
AJL model is *interesting*, because it represents the best attempt so far
to find a background-free quantum theory that reduces to general relativity
in the large-scale limit!

---------end quote-------

for me the key point in this post is a mathematician's or mathematical physicist's judgement call:

Nobody has yet constructed a background-free quantum theory that has
general relativity as its limit at large distance scales. The Ambjorn-
Jurkiewicz-Loll model is the closest anyone has come.

marcus

Jun7-04, 01:45 PM

Another recent Baez post on the AJL paper, this time in response to Thomas Larsson:
-------quote from Sunday 6 June SPR post----

In article <24a23f36.0405170344.69e74067@posting.google.com>,
Thomas Larsson <thomas_larsson_01@hotmail.com> wrote:

>1. Is the AJL model really quantum?

Yes! It has a Hilbert space of states, observables described
as noncommuting self-adjoint operators on this Hilbert space,
and discrete time evolution described by unitary operators on
this Hilbert space.

>Some time ago, Urs
>Schreiber argued that LQG, or at least the LQG string,
>fails to be a true quantum theory, and I tend to agree.

I disagree, but it's not really relevant here: we're not
talking about those other theories.

>However, the AJL model can be viewed as a statistical
>lattice model, and if such a model has a good continuum
>limit, it is AFAIK always described by some kind of QFT.
>What else could it be?

Right!

>2. Is the AJL model really gravity. The action is a rather
>straightforward discretization of the Einstein action with
>a cosmological term:
>
> sum over (d-2)-simplices
>
> det g = volume => sum over d-simplices.
>
>What is perhaps somewhat unusual is that all edges have
>the same length, which is different from Regge calculus.
>Nevertheless, I don't think that this really matters, but
>one could check if the results look different if you
>allow for variable edge lengths.

Right! But, the test of whether the model "is really
gravity" is to carefully examine its behavior in the limit
of large distance scales (i.e. lots of 4-simplices). One
can't easily guess this from looking at the action.
Nonperturbative effects are too important! So, in the
absence of good analytical techniques, one really needs
to run computer simulations - as AJL are doing.

>3. Is the measure right? Here is the place where AJL differ
>significantly from previous simulations. AFAIU, the crux is
>that AJL insist on a strict form of causality: they exclude
>spacetimes where the metric is singular, even at isolated
>points. This may seem like an innoscent restriction, but it
>rules out things like topology change and baby universes,
>which require that the metric be singular somewhere.
>
>It is not obvious to me whether one should insist on such a
>strong form of causality or not, but this assumption leads
>at least to better results, e.g. a reasonably smooth 4D
>spacetime. Thus, I believe that it is a fair chance that
>AJL have indeed succeeded in quantizing gravity.

The issue of the "right measure" is very tricky, so tricky
in fact that I again think the most efficient way to begin
tackling it is to run computer simulations and see if the
AJL model acts like general relativity at large length scales.

>They do so not by assuming a lot of experimentally unconfirmed
>new physics, but rather by strictly implementing the
>time-honored principles of old physics, especially
>causality. That is cool.

Yes! Very cool!

----end quote----

For me, there are two key statements here:

---exerpts---

>1. Is the AJL model really quantum?

Yes! It has a Hilbert space of states, observables described
as noncommuting self-adjoint operators on this Hilbert space,
and discrete time evolution described by unitary operators on
this Hilbert space.

...

>They do so not by assuming a lot of experimentally unconfirmed
>new physics, but rather by strictly implementing the
>time-honored principles of old physics, especially
>causality. That is cool.

Yes! Very cool!

----end exerpts---

the last is again a professional mathematician's judgement call. It may be time to quantize the theory of gravity we all use and to do that in a way
that does not "assume a lot of experimentally unconfirmed new physics".
It looks cool to these guys to get GR quantized by conservatively implementing the tried-and-true established principles. In other words spare us the fairy tales about extra dimensions and just get the job done.

You pay mathematicians in part to make educated guesses about what is cool and not cool, what is interesting and not interesting, and what might work. Part aesthetic and part a kind of laboriously enhanced common sense. I'm listening to both these guy's judgement.

http://physicsforums.com/showthread.php?p=227813#post227813

Mike2

Jun8-04, 08:01 PM

Mike2, as selfAdjoint has explained to you in another thread, LQG is based on a continuum, on a differentiable manifold, not on a lattice. You dont have to worry about it being connected.
As sA also remarked the simplicial AJL model, which is really more the topic of this thread, is also a continuum. Think of it as a diff. manif that has been "triangulated" ----built up out of simplices----fused glued welded together from simplices----partitioned into simplices without actually splitting them (they touch).
You seem to be missing the fundamental delemma. Or perhaps I'm hard of hearing. To quantize gravity IS to quantize the spacetime metric. But quantizing spacetime would of necessity make spacetime discrete and makes impossible any propagation of signals. This is too fundamental of a delemma. What could possibly fix it? So at the moment is seems impossible that you will ever quantize gravity.

selfAdjoint

Jun8-04, 08:37 PM

Suppose we did have a complete quantization of spacetime. Then we would have interacting spaceons, no doubt exchanging gravitons, and communicating thus across distances. Where's the problem?

Olias

Jun9-04, 05:36 AM

sol2

Jun9-04, 10:26 AM

Spacetime is relevent to 3+1 Dimensionsn and has a metric which gives constant results.

Space-Field, where 'TIME' is detached(exchanged) from the above metric follows certain values in GR, the major factor it is a 2 dimensional arena, and not 3+1, as one 'lose's' the time component in Einsteins field equations, this compactifies and restrains all measures into a non-time dependant arena.

The simplistic overview is that there are only Directional values of motion, all directions are based on 'back-to-back' interactions, like with all Field Equations the action, re-action are similtainious, for every Positive Action there is a corresponding Negative reaction.

When you get to one dimension(string) what happens then :smile:

Gr had to be consistently expressed, but it is surrounded, before and after ?:smile:

Gravity and electromagnetism are now one( you can't see it but the one is white )?

marcus

Jun9-04, 11:47 AM

You seem to be missing the fundamental delemma. Or perhaps I'm hard of hearing. To quantize gravity IS to quantize the spacetime metric. But quantizing spacetime would of necessity make spacetime discrete and makes impossible any propagation of signals. This is too fundamental of a delemma. What could possibly fix it? So at the moment is seems impossible that you will ever quantize gravity.

Suppose we did have a complete quantization of spacetime. Then we would have interacting spaceons, no doubt exchanging gravitons, and communicating thus across distances. Where's the problem?

want to try to respond
no time now since i have to go out briefly

will bring in this quote
-------quote from JB post on SPR Sunday 6 June------
Please understand what I'm saying:

I'm not saying that M-theory is "wrong" or that the Ambjorn-Jurkiewicz-Loll
model is "right". M-theory makes too few definite predictions to be wrong.
The AJL model does not include matter, so it cannot be right. But the
AJL model is *interesting*, because it represents the best attempt so far
to find a background-free quantum theory that reduces to general relativity
in the large-scale limit!

---------end quote-------

It is important to realize that quantizing the geometry of a continuum (a manifold) does not necessarily mean to chop up the manifold into little bits.
the manifold can stay continuous and smooth and connected while its
geometry-observables----areas, volumes, angles---become operators on a hilbertspace.

quantization is a way of representing observables, measurements.
it does not necessarily divide everything in sight into discrete quanta.

Mike2 is right in saying that to quantize gravity means to quantize the metric----in that the metric is one common mathematical representation of the geometry. It does not necessarily mean to divide the metric into little bits or force it to have discretized values. Above all it does not mean one necessarily pulverizes space into little bits! I guess that is one possibility (as selfAdjoint suggests) but it is not the necessary outcome.

Going back to the Seventies (and probably earlier) I think what seemed to a lot of people to be an obvious approach to quantizing GR was to have a smooth manifold and take the space of all (smooth) metrics on that manifold and make a hilbertspace which was
L2 functions on that space of geometries. And then you define operators on that hilbert space.
that is, dont think you have to discretize space and dont think you have to discretize the metric. what you want is to have the measurement of geometric properties like areas correspond to operators on a hilbertspace.
and they might turn out to have discrete spectra.

this approach did not work in the Seventies, although later Rovelli and Smolin did get area and volume operators with discrete spectra. by then (the Nineties) they were using the connection, instead of the metric, to represent the geometry.

None of these approaches recognizes a necessity to divide space up into isolated bits.

And the AJL approach which is the focus of this thread does not either.
differential geometers have been triangulating manifolds for ages (over a hundred years I guess) it is a standard thing
and AJL take a manifold---called S3 in their paper---and
triangulate it in a "dynamical" changing way

So Mike2 you are mistaken when you say:
"But quantizing spacetime would of necessity make spacetime discrete and makes impossible any propagation of signals."

It simply isn't true that quantizing spacetime (or more precisely the geometry of spacetime) would make spacetime discrete.

Mike2

Jun9-04, 12:17 PM

-------quote from JB post on SPR Sunday 6 June------
Please understand what I'm saying:

I'm not saying that M-theory is "wrong" or that the Ambjorn-Jurkiewicz-Loll
model is "right". M-theory makes too few definite predictions to be wrong.
The AJL model does not include matter, so it cannot be right. But the
AJL model is *interesting*, because it represents the best attempt so far
to find a background-free quantum theory that reduces to general relativity
in the large-scale limit!

---------end quote-------
So it sounds like they are saying that GR does not hold up at very, very small distances. Then quantizing gravity is not equivalent to quantizing (discretizing) spacetime itself. Nevertheless,... discrete causality? That is a contradiction of terms. If a change at a point does not even have a start to an effect on a neighbor, then there is no "immediate" reason why it should have any effect at all.

selfAdjoint

Jun9-04, 01:42 PM

Mike2

Jun9-04, 01:57 PM

Mike, you continue to equate quantizing to discretizing. You really need to study more about what quantizing really is. States, operators, and uncertainty, superposition and entanglement. Not "separate chunks".
Admittedly, I am not as informed as many in this field. I am trying to develop a better intuition about all this. And I know that QM does not lend itself to any kind of intuition. That said, I have studied sum higher math and physics. And I don't know of any variables/observables that are quantized that do not take on discrete values. What I am trying to understand is how gravity/spacetime can be "quantized" without being made discrete. And if it is discrete, what does that mean. Your response, of course, is no answer to that. Thank you.

setAI

Jun9-04, 04:12 PM

What I am trying to understand is how gravity/spacetime can be "quantized" without being made discrete.

hint: fractal structures are in a way both discrete and continous- they represent hierarchies of quantized structures which can seem discrete but are really fundamentally continuous- and vice versa!

interestingly enough fractal structures are what always emerge from chaos- and any truly fundamental view of the ontology of Existence itself suggests that the spacetime/forces/energy/matter of a universe must emerge and crystalize out of an "initially" chaotic state-

ultimately you can either have Existence or Non-existence- if you have existence it must be absolute Chaos because if it existed and wasn't random it must have resulted from some more fundamntal ordered process which excluded an infinity of possible forms- you have to "start" with Chaos-

so the ultimate ontology of existence is Chaos> annihilation of equal-opposite interacting structures > remaining structures seeking entropic equilibration [the fundamental origin of Motion itself] crystalizing into a fractal hierarchy > emergence of seemingly discrete matrices/foams/graphs/lattices that emerge as spacetime vacua/branes > particles/forces

um- but don't listen to me- I think I went off topic- sorry for the crazytalk :tongue2: :uhh: :yuck: :redface:
___________________________

/:set\AI transmedia laboratories

http://setai-transmedia.com

selfAdjoint

Jun9-04, 05:22 PM

Admittedly, I am not as informed as many in this field. I am trying to develop a better intuition about all this. And I know that QM does not lend itself to any kind of intuition. That said, I have studied sum higher math and physics. And I don't know of any variables/observables that are quantized that do not take on discrete values. What I am trying to understand is how gravity/spacetime can be "quantized" without being made discrete. And if it is discrete, what does that mean. Your response, of course, is no answer to that. Thank you.

Consider a quantized field. The field is continuous, although discrete packets can be exchanged. Remember that a photon is not just a particle; it also manifests as a continuous wave. In basic quantum mechanics the discreteness comes in the measurement or observation. There can be a discrete set of outcomes (eigenvalues) when the Hermitian operator acts on the continuous state function.

Mike2

Jun9-04, 05:56 PM

Consider a quantized field. The field is continuous, although discrete packets can be exchanged. Remember that a photon is not just a particle; it also manifests as a continuous wave. In basic quantum mechanics the discreteness comes in the measurement or observation. There can be a discrete set of outcomes (eigenvalues) when the Hermitian operator acts on the continuous state function.
It's easy to visualize these things for quantized fields with respect to spacetime variables, the wave function squared tells you the probability of finding the particle at a certain location and time, etc. But I have difficulty imagining what it would even mean to quantize spaetime itself. Is it like the metric is tells you the probability of finding a particle of spacetime? And what happens to the validity of QED and QCD in a world of quantized gravity/spacetime? Wouldn't QED and QCD have to be reformulated with respect to something other than spacetime so that all quantization procedures are with respect to the same variables? If photons, gluons, and gravitons all must interact, then you'd expect their quantization procedure to be based on some commonality; evidently, spacetime/gravity is NOT that commonality. What then is?

marcus

Jun9-04, 08:37 PM

It's easy to visualize these things for quantized fields with respect to spacetime variables, the wave function squared tells you the probability of finding the particle at a certain location and time, etc.

Mike2, do you mind if I answer----you asked this of selfAdjoint and he can also answer, anyway I only refer to a part of your question (and dont mean to horn in)

an important analogy. think of a very simple space of locations, like the unit interval or the real axis. you say:
"...the wave function squared tells you the probability of finding the particle at a certain location..."

now think of the set of all metrics on some manifold
that is analogous to the unit interval
the set of all possible geometries on this manifold
can be imagined as itself a mathematical space
and wave functions can be defined on it

"...the wave function squared tells you the probability of finding the geometry of the universe in a certain configuration..."

In practice things may be done differently but this gives you
a rough idea of what quantizing the geometry can mean
the wavefunctions are a hilbert space and
then one has operators on that hilbertspace corresponding to
measuring particular observable facts about the quantum state or wavefunction of the geomtry.

but one never totally nails down the geometry, just as one never nails down the position of a particle on the unit interval or the real axis.
does this make it more understandable?

Mike2

Jun9-04, 09:46 PM

Mike2, do you mind if I answer----you asked this of selfAdjoint and he can also answer, anyway I only refer to a part of your question (and dont mean to horn in)
You should never appologize for contributing to an open forum. That's what it's here for. Just put your 2 cents in, please.

an important analogy. think of a very simple space of locations, like the unit interval or the real axis. you say:
"...the wave function squared tells you the probability of finding the particle at a certain location..."

now think of the set of all metrics on some manifold
that is analogous to the unit interval
the set of all possible geometries on this manifold
can be imagined as itself a mathematical space
and wave functions can be defined on it

"...the wave function squared tells you the probability of finding the geometry of the universe in a certain configuration..."

In practice things may be done differently but this gives you
a rough idea of what quantizing the geometry can mean
the wavefunctions are a hilbert space and
then one has operators on that hilbertspace corresponding to
measuring particular observable facts about the quantum state or wavefunction of the geomtry.

but one never totally nails down the geometry, just as one never nails down the position of a particle on the unit interval or the real axis.
does this make it more understandable?

That's beginning to make sense, thank you. So would our universe then be a particular one of the geometries (a collapsed wave function), or is it always a superposition, and what we see is a classical limit of a type of "geodesic" average?

This all sounds like a 3rd level of quantization. And just as the 2nd level of quantization cannot be used to describe the 1st level (or can it?), the 3 level cannot be considered on par with the results of the 2nd level? Paths cannot be considered the same as particles, and particles cannot be considered the same as geometries, right? How then can the geometries (gravitons?) interact with particles?

Olias

Jun10-04, 12:32 AM

You should never appologize for contributing to an open forum. That's what it's here for. Just put your 2 cents in, please.

That's beginning to make sense, thank you. So would our universe then be a particular one of the geometries (a collapsed wave function), or is it always a superposition, and what we see is a classical limit of a type of "geodesic" average?

This all sounds like a 3rd level of quantization. And just as the 2nd level of quantization cannot be used to describe the 1st level (or can it?), the 3 level cannot be considered on par with the results of the 2nd level? Paths cannot be considered the same as particles, and particles cannot be considered the same as geometries, right? How then can the geometries (gravitons?) interact with particles?

Mike you want to review this recent paper, it has an interesting angle of relevence:

http://uk.arxiv.org/abs/quant-ph/0406028
http://uk.arxiv.org/abs/quant-ph/0406029

A previous paper: http://uk.arxiv.org/abs/quant-ph/0308101

Mike2

Jun13-04, 02:32 PM

This all sounds like a 3rd level of quantization. And just as the 2nd level of quantization cannot be used to describe the 1st level (or can it?), the 3 level cannot be considered on par with the results of the 2nd level? Paths cannot be considered the same as particles, and particles cannot be considered the same as geometries, right? How then can the geometries (gravitons?) interact with particles?
What confuses me is that you are treating a graviton as a particle within some background geometry. But it is suppose to represent a quanta of geometry itself. It seems a particle assumes a backgound geometry used to describe its feature such as where and when it is and how big it is and how fast it is vibrating, etc. So how can one possibly describe a particle of "backgound", what non-background measures can be used to describe it? If the graviton is just another mode of vibration of a string, and strings assume a background, then a graviton cannot be a description of that background geometry, and so it does not describe gravity. I need a better picture because this sound like a contradiction. If quantum gravity means quantum spacetime, how do I visualize this? So all of space is a superposition of various quantum geometries? What does that mean? Does that mean that our particular spacetime is just one of the possible states of quantum geometry/spacetime/gravity? Or if there are other observations of a different quanta of spacetime, then how are the boundaries manifest between the different quanta of spacetimes? Thanks.

selfAdjoint

Jun13-04, 04:30 PM

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

Jun14-04, 04:35 PM

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

Jun14-04, 09:19 PM

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

Jun15-04, 11:53 AM

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

Jun15-04, 07:26 PM

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

Jun15-04, 08:09 PM

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

Jun15-04, 09:36 PM

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

Jun30-04, 10:36 PM

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. :devil:

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 isnt the crude old mond.
The new thing, lets 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

Jun30-04, 11:16 PM

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

Jun30-04, 11:29 PM

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 wouldnt 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

Jun30-04, 11:47 PM

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

Jul28-04, 02:00 PM

...

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 in to 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

Sep5-04, 05:18 PM

... 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

Sep12-04, 09:16 PM

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

marcus

Sep12-04, 09:59 PM

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

Nov16-04, 09:04 PM

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

Dec8-04, 06:45 PM

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

Dec8-04, 06:55 PM

http://physicsforums.com/showthread.php?t=54262

http://physicsforums.com/showthread.php?t=53974

http://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)

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

marcus

Dec8-04, 07:16 PM

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

Dec8-04, 11:11 PM

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

Dec9-04, 11:56 PM

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 doesnt 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 dont know what you are talking about
I know what you are talking about

pick one

marcus

Dec10-04, 12:02 AM

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

Dec10-04, 10:16 AM

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

Dec10-04, 11:04 AM

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

Dec12-04, 01:20 PM

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

Dec12-04, 01:36 PM

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

Dec15-04, 01:23 PM

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

Dec15-04, 09:35 PM

... 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

Dec15-04, 09:49 PM

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 Rd. 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

Dec16-04, 01:16 PM

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

Dec16-04, 02:21 PM

Cleaner, in 0 dimesions 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

Dec16-04, 04:26 PM

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

Dec16-04, 05:04 PM

marcus

Dec16-04, 07:45 PM

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/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/1998/0/1

http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/1999/0/1

http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/2000/0/1

http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/2001/0/1

http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/2002/0/1

http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+gravity+AND+Lorentzian+quantum/0/1/0/2003/0/1

http://arxiv.org/find/grp_physics/1/OR+OR+abs:+AND+triangulations+AND+Lorentzian+dynam ical+abs:+AND+triangulations+AND+causal+dynamical+ ti:+AND+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 shouldnt 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

Dec17-04, 12:48 AM

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

nightcleaner

Dec17-04, 12:50 AM

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

3850

nightcleaner

Dec17-04, 01:10 AM

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

these are not perfect, they get some they shouldnt 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

Marcus! This is really great work! If I didn't have an eyestrain, I'd stay up all night reading. Oh well, tomorrow is tomorrow. Thank you very much for these searches.

Richard

marcus

Dec17-04, 10:21 AM

Richard, do you know about the "N-choose-k" number?

it is relevant here.

it is often written
\left(\begin{array}{cc}N\\k\end{array}\right)

and sometimes called the "combinations" of size k taken from a set of size N, the language is awkward but the idea is very simple

if you have a set of N things then how many subsets of size k are there?
N-choose-k

If you have a set of 3 things (N=3) then how many subsets of size 2 (k=2) are there?

3-choose-2 is equal to 3

the N-choose-k numbers are those appearing in the "Pascal triangle"

\left(\begin{array}{cc}4\\0\end{array}\right) = 1

\left(\begin{array}{cc}4\\1\end{array}\right) = 4

\left(\begin{array}{cc}4\\2\end{array}\right) = 6

\left(\begin{array}{cc}4\\3\end{array}\right) = 4

\left(\begin{array}{cc}4\\4\end{array}\right) = 1

nightcleaner

Dec17-04, 10:47 AM

No, Marcus, my math is pretty limited. I've had some college calc, but didn't do well in it, and that's been long ago. I was pretty good at physics, in the Life Sciences version, which was light on calclulus. I thought yesterday's problem was interesting, and just tried to think my way through it.

I have considered returning to school to improve my maths. University of Minnesota Duluth is closest, but does not offer much of a curriculum.

Meanwhile I have been reviewing, using a GRE text, and trying to get what I can from internet. Any suggestions would be welcome.

Thanks,

Richard

marcus

Dec17-04, 01:41 PM

\left(\begin{array}{cc}N\\k\end{array}\right) = \frac{N!}{k!(N-k)!}

....

for this, you have to know what N! the factorial of N is,
and you should know the convention that 0!, the zero factorial, equals one.

.........

(but these, after all, are not terribly hard facts to learn
or, more precisely, to accept)

the N-choose-k numbers are those appearing in the "Pascal triangle"

\left(\begin{array}{cc}3\\0\end{array}\right) = 1

\left(\begin{array}{cc}3\\1\end{array}\right) = 3

\left(\begin{array}{cc}3\\2\end{array}\right) = 3

\left(\begin{array}{cc}3\\3\end{array}\right) = 1

\left(\begin{array}{cc}4\\0\end{array}\right) = 1

\left(\begin{array}{cc}4\\1\end{array}\right) = 4

\left(\begin{array}{cc}4\\2\end{array}\right) = 6

\left(\begin{array}{cc}4\\3\end{array}\right) = 4

\left(\begin{array}{cc}4\\4\end{array}\right) = 1

\left(\begin{array}{cc}5\\0\end{array}\right) = 1

\left(\begin{array}{cc}5\\1\end{array}\right) = 5

\left(\begin{array}{cc}5\\2\end{array}\right) = 10

\left(\begin{array}{cc}5\\3\end{array}\right) = 10

...

about GRE review, sounds smart, but I cant advise
Duluth likewise.
maybe selfAdjoint, who also lives in midwest, can give wise and kind counsel
I really cant. all what you say sounds sensible and intelligent
(but in our tangled web of hardship and difficulty how can anyone give advice or encouragement to anyone else besides to say take care)

However whatever you do or do not do in your life, you should understand Pascal triangle and N-choose-k. I seriously insist on this.

the number of triangle faces of a tetrahedron is the number
of THREEPOINTED simplices belonging to a FOURPOINTED
four choose three

[edited to moderate language]

nightcleaner

Dec17-04, 06:55 PM

Hi Marcus.

Thanks for trying to cheer me up. The world has been, uh, dissappointing. I think it was Dr. Seuss whose last words were "we could have done much better."

I have been sleeping and doing chores all day, and havn't gotten back to this except for a few minutes and then I was interupted. Tonight I have to work. But I will work on the n-choose-k thing.

I am enjoying this conversation.

Pascal triangle, I have read of this somewhere. This board's format makes it hard to show here, but you line the numbers up in centered rows, as I recall. The numbers in each row are the sum of the numbers immediately above them, I see, so (1,1,)(1,2,1)(1,3,3,1)(1,4,6,4,1)(1,5,10,10,5,1) (1,6,15,20,15,6,1) and so on. I don't remember what the connection was but I do know this is familiar.

Four choose three? is four? Four ways to choose three things from a set of four things. I am still looking at the triangle, but my first thought is that if I choose three dimensions to be spatial, then there is one left, which we use for time. So, blessed be, there should be four ways to do this, so four different choices of time line in each instant. Of course, we are coming from one line, so we really have a choice of three time lines at any instant, since going back to the previous node is the same as not going anywhere at all. So the three we can choose from become our three spatial dimensions. The critical point here seems to me to be that the multiverse extends beyond what we have to choose from.

The sceptic view of this idea is that if we cannot choose, cannot even know anything about it, why should we believe it exists? Actually you can get along just fine without all those unwieldy extra universes cluttering up the scenery. Does the Donald know or care about the multiverse?

Well, here is the thing. In the isomatrix model, any instant is surrounded by and in contact with twelve other instants. Three of them are in the immediate future, three in the immediate past, and six co-inhabit the present instant. The expansion pushes out in all twelve directions. We only measure the past-future push. That leaves ten other directions to account for the extra dimensions of string theory. That means that when looking at infinities, like the total expansion of the universe, we find a factor of ten masses unaccounted for. Huh. The missing matter is co-instantaneous but found along other time lines, ones that lead, instantaneously, to other time lines, other dimensions, other universes in the multiverse.

I have to tell you that I have found Rees et al and their knob twiddling to be rather trivial. Please don't be mad. There may be other universes where such things as the electron-muon distance are different from ours, but none of those universes is anywhere close to our region of the multiverse. Those choices were made so long ago that we do not have to make them ever again. All the universes that branch out from here, and all the others that have branched out for several billion years now, are nearly identical to our own in choice of these fundamental numbers.

Of course, one may learn a great deal about how the chosen numbers affect our universe, but the anthropological principle is correct, in so far as we only have to be concerned about the universal constants we find ourselves inhabiting.

Anyway, the reason I have been insisting that you and the others look into the isomatrix is because not all the directions are through the triangles. There are also directions that move in a space-like fashion. The isomatrix has as its fundamental simplex the triangle, but then there are also sets of rectangular planes. In fact, there are four sets of triangular planes, but three sets of rectangular planes. The rectangular planes look just like our usual interpretation of 3space, the Cartesian coordinate system.

By the way, I agree that life is torment, or at least, pain. The Budhists point out that the source of pain is desire. Eliminate desire, and the pain is, well, not gone, but not so important. I would not be here today typing at this screen if I had not fallen off a tall roof and shattered my right femur. That was a painful experience, but it gave me pause for reflection. I decided, while laying in a frozen swamp at the bottom of a hill below the building which was so kind as to try to kill me, that I should not keep all these thoughts about the multiverse to myself any more. After all, the Dalai Lamma, who may well be the only real spiritual leader in the world today, has called for all the libraries to be opened. He has played the drum, he has given breath to the flute, and we in the public have heard these things. It is time to open the libraries, which have been kept so close to preserve them, but now must be flung to the winds if they are to be saved.

In our hundred years of solitude, we must rely, at last, on the wings of butterflies to come and carry us away. Gaia is waiting for us.

Anyway, I am still working on the meaning of the factorial formula. I have to go get ready for the menial labor which keeps me paying my bills, now. But I will copy the factorial formula onto paper and carry it with me to think about while scrubbing. I believe in the virtue of physical work. Especially physical work in service at the root of things. That is where the real differences are made. But I am getting tired. I don't know how much longer I can keep this up. For a while, at least, I guess. Only I feel sorry for my hands, which are gradually becoming lumps covered with a net of scars. They were beautiful, once, I guess.

Be well, Marcus, whoever you are. Why don't you email me and tell me about yourself? I am mostly harmless.

Richard.

Lacy33

Dec17-04, 09:17 PM

Hi Marcus.

After all, the Dalai Lamma, who may well be the only real spiritual leader in the world today, has called for all the libraries to be opened. It is time to open the libraries, which have been kept so close to preserve them, but now must be flung to the winds if they are to be saved.
Richard.

3899

Richard I hope that one of the books you intend to "fling into the wind" , I'm guessing is teach to the world is not the Sefer Yetzirah which you took from this library.
Anyone suffering from mental illness such as the "deep, serious, permanent depression" you have now shared with the world is a most dangerous combination to the Kabbalah and any works and attempts to comprehend these concepts as I see much of your offerings are directly connected can cause irreversible damage.
Before you go any further with your calculations in the areas you are exploring now and have been. For fear of danger, do what ever you can to make yourself permanently positive and reasonably happy. You are exploring things of a potentially explosive nature.
It is true, "they are out to get you" ...But I am not one of them.
I believe in your work.
Your work however far from being organized is something that may one day make a substantial breakthough in physical science simply because it IS beginning to reveal the mysteries and this I state as your arguments are revealing the secrets of the ancient books of Kabbalah thousands of years old, that I know you had not seen previously.
Suzanne

marcus

Dec18-04, 01:14 AM

...
... mostly harmless.

a pleasure to hear Douglas Adams quoted. the HH's Guide verdict on our planet

...
I have to tell you that I have found Rees et al and their knob twiddling to be rather trivial. Please don't be mad.

:biggrin:
certainly not angry
it is a bit trivial from my perspective too, but I would like some better
articles found or written about everyday life interpretation of basic constants-----havent seen Rees: he may be watered down which would make it blah

...
This board's format makes it hard to show here, but you line the numbers up in centered rows, as I recall. The numbers in each row are the sum of the numbers immediately above them, I see, so (1,1,)(1,2,1)(1,3,3,1)(1,4,6,4,1)(1,5,10,10,5,1) (1,6,15,20,15,6,1) and so on.
...

You should learn the "code" format for writing TABLES and MATRICES here at PF. it is elementary and easy
just look at this post, where I will write a table, and press "quote" and it will show you how it is typed

in essence you just say at the beginning and at the end, but spell code right, with a c.

1 1
1 2 1
1 3 3 1

all that happens is that it is forced to take it seriously when you type spaces between things. It hears the spacebar.

you can use "code" format to make pascal triangle, for example

1
1 1
1 2 1
1 3 3 1

marcus

Dec18-04, 02:17 AM

from the Babylonian Talmud:

"It is never advisable for anyone to speculate on these four questions:

What is above?
What is below?
What was there before the world?
What will there be after it?

It would have been better for him had he never been born."

this is from the part of the Babylonian Talmud called
Hagigah 2:1, I have just seen this as the reference. I am not an expert about this and have not read it.

this is a bitter wisdom

if everybody would pay attention to this there would be no cosmologists because cosmologists are always asking these unwise questions

One of the young Quantum Gravity researchers has chosen this quote from the babylonian talmud to begin his thesis.
He is French and he has used the French version of this Hagigah 2:1 passage.
His name is Etera Livine and his thesis is here:
http://arxiv.org/gr-qc/0309028

the french version is this:
Quiconque s’est jamais avisé de spéculer sur ces 4 questions :
– Qu’y a-t-il au-dessus ?
– Qu’y a-t-il en-dessous ?
– Qu’y avait-il avant le monde ?
– Qu’y aura-t-il après ?
Il aurait mieux valu pour lui qu’il ne fut jamais né.

Talmud de Babylone

the title of Livine's thesis, in case anyone is curious about that, is
"Boucles et Mousses de Spin en Gravité Quantique"

nightcleaner

Dec18-04, 04:54 AM

Hi Marcus. Hi S.

Well, I tried to work out the factorial formula, but muffed it. I'll have a look at it again in the daylight. Thanks, Marcus, for the code. That will no doubt turn out to be useful. And thanks, Shoshanna, for the book. That is still protected, unopened, under covers three. It is a dirty world, after all. Lots of work for a night cleaner.

So I should be afraid that asking these questions will make me more depressed? Not likely. Entertainments don't lift my spirit, and focused thought does not drag it down. It is what it is. I am not trying to change it, but to accept it as a gift. And it is not a matter of being negative or even of being unhappy. I am happy enough. And I always try to take what I find ugly and make it better somehow. Not to hide it. Just to make it better.

I am not ashamed of my depression. It is pain. Should I be ashamed of my broken leg? Thanks to medicine and surgery, I can walk, and still want to. Without help, I would not be here. Should I be ashamed of that? Then who would be here if not for help? Only a blind fool thinks he or she stands alone in this world. None of us got here on our own. I am neither proud nor ashamed of myself. On the whole, however, I am ashamed of the Human species. We could have done much better. For human individuals, I feel a great love, and a great pride, but it is not for their humanity, it for their transcendence. We find ourselves in the muck. We try to rise. It is no sin to fail to rise, but it is terrible not to try.

So, onward and upward. To paraphrase Oscar Wilde, 'I may be lying in the gutter, madam, but while you are looking down on me, I am looking upward, at the stars.' And of course, the next day, he was sober, although I am sure he wanted to get drunk again, while she, never having been drunk, was still stupid and ugly.

Thank you for the warnings, but I think I know the danger of star gazing. I already think it would have been better, for me at least, not to have been born, so what have I to lose? I am here. I will do the best I can with what I have been given. And among the things given to me was a curiosity about what is above the above, and what is beneath that which is below. That curse, if it is one, has already been cast.

As for the explosions, there was a time when I thought I should keep what I know to myself, for fear of the big W. Not Dubya, no, but Weaponization. I thought it would be terrible if someone found out how to use my ideas to make a more terrible bomb with which to destroy the world. But what use was that thought to me? And, what use was it to the world? I am a lowly servant of idiots, so it has done me no good to hide. And as for the world, there is already enough explosive power to destroy us all. If we are going to blow ourselves up, then it will happen without my little bit of wisdom. What we need now is not to hide the light under a basket, but to bring it out and use it to make tools for our common survival.

My mind is still open, so is my heart. I will not let fear shut either down. If you want me to shut up, give me good reason, and put away your bug-a-bears, I am not frightened. Time will shut me up soon enough. While we are here, let's converse a while. Let's learn what we can from each other. Tell me if I am wrong, I will not be angry.

Let us be well,

in the name of Love,

Richard

Lacy33

Dec18-04, 07:14 AM

Hi Marcus. Hi S.
My mind is still open, so is my heart. I will not let fear shut either down. If you want me to shut up, give me good reason, and put away your bug-a-bears, I am not frightened. Time will shut me up soon enough. While we are here, let's converse a while. Let's learn what we can from each other. Tell me if I am wrong, I will not be angry.

Let us be well,

in the name of Love,

Richard

Good day Marcus and Richard,
Marcus, The Talmud says the things you say but it is also written that if a person seeks out the mysteries and can not leave it alone than "They belong to him". Seeking the mysteries are discouraged to test a persons strength and need to engage. It is not the Rabbis, who wrote the Talmud, who discourage the seeking of the mysteries, but the nature of the mysteries themselves. Certianaly we could fill pages of Talmud here arguments for both.
The Mystic will find their way through while the Scholars of the Talmud go back and forth and the Scholars of Blessed Memory know this!
Richard's work, by his own intuitition has led him to the same findings of the Mystics of long ago, (in that it hints at what is written in the Kabbalah). Again it is not complete in the way it resembles the Kabbalah nor do I think it is a complete work in physics. (That part I can not comment on as I am not a physicsist). BUT it does have Merit.

Richard, we note the way you choose to represent youself personally on this open physics forum.
I would not add one way or the other if personal truth and intellectual honesty are requirements here.
Again I want to say that I support your efforts to communicate your vision. This seems to be a safe place to do that as you do not claim any professionalism and you have only to gain from the educated and tolerant people willing to help. As we all know, finding assistance for creative thinkers is at best maddening.
I will conclude with more Talmud as you seem to be communicating with someone who has studied such and is willing to post it on a physics forum... "Three went in to the Garden only one came out unharmed". This means to say that three were smart enough to get into the garden or receive information, but only one was fortified enough to endure the light.
Suzanne

marcus

Dec18-04, 11:53 AM

\left(\begin{array}{cc}N\\k\end{array}\right) = \frac{N!}{k!(N-k)!}

for this, you have to know what N! the factorial of N is,
and you should know the convention that 0!, the zero factorial, equals one.

the N-choose-k numbers are those appearing in the "Pascal triangle"

\left(\begin{array}{cc}3\\0\end{array}\right) = 1

\left(\begin{array}{cc}3\\1\end{array}\right) = 3

\left(\begin{array}{cc}3\\2\end{array}\right) = 3

\left(\begin{array}{cc}3\\3\end{array}\right) = 1

\left(\begin{array}{cc}4\\0\end{array}\right) = 1

\left(\begin{array}{cc}4\\1\end{array}\right) = 4

\left(\begin{array}{cc}4\\2\end{array}\right) = 6

\left(\begin{array}{cc}4\\3\end{array}\right) = 4

\left(\begin{array}{cc}4\\4\end{array}\right) = 1

\left(\begin{array}{cc}5\\0\end{array}\right) = 1

\left(\begin{array}{cc}5\\1\end{array}\right) = 5

\left(\begin{array}{cc}5\\2\end{array}\right) = 10

\left(\begin{array}{cc}5\\3\end{array}\right) = 10

\left(\begin{array}{cc}5\\4\end{array}\right) = 5

\left(\begin{array}{cc}5\\5\end{array}\right) = 1

the number of triangle faces of a tetrahedron is the number
of THREEPOINTED simplices belonging to a FOURPOINTED
is four choose three, namely 4.

and to take another example the number of edges of a tetrahedron, the number of
TWOPOINTED simplices belonging to a FOURPOINTED
is four choose two, namely 6.
==============

now we go to 4 dimensions. the basic simplex in 4D, the socalled "4-simplex" is a FIVEPOINTED simplex

how many of ITSELF does the fivepoint simplex have?
five choose five, namely 1.

how many tetrahedrons does it have, as its threedimensional "sides"?
well a tetrahedron is a FOURPOINTED so to specify one you have to choose 4 points from the 5
so it is five choose four, namely 5

and how many triangles?
five choose three, namely 10

and how many edges?
five choose two, namely 10

and how many vertices?
we already said,
five choose one, namely 5

just as a game, to complete the sequence, let us ask
a further question
and how many NOTHINGS does the basic fivepoint simplex have?
five choose zero, namely 1

nightcleaner

Dec18-04, 01:29 PM

Hi Marcus and Shoshana

Thanks for the wisdom, S., always appreciated, even when not well taken. And Marcus, always appreciated even when not understood. I am still trying. I have to go make coffee and do chores, but will return to Pascal and the N! formulation this afternoon offline.

Last night I tried to fit the N!'s into the formula to come out with the triangle, and it worked some of the time, but not for every trial. Probably I don't remember some of the details of factorials and will have to go to Wickipedia for a refresher. I thought about the zero factorial, and also about how to make a factorial of a negative.

3!=1+2=3
4!=3!+3=6
5!=4!+4=10

so,
2!=1!+1=1
1!=0!+0=0

0!=-1!-1 so -1! should equal 1?

Hmmmm.

No that can't be right. Will have to have coffee.

By the way, just finished reading "The man who loved only numbers" about Paul Erdos, a famous mathematician whom I had never heard of. I guess from the reading that he is only famous among mathematicians. An inspiring story for impoverished seekers of truth. However the title is misleading. The way i read it, Erdoes loved many people, and loved to talk, but numbers were the only thing that made any sense to him. Apparently he often began conversations, even with old friends whom he had not seen for long periods of time, with something like "Hello. Let n be an even integer........" He was a brilliant, kind, and funny man who was a beloved pain in the *** to most everyone he knew, especially to those who understood him.

btw, Marcus i think Rees is the real stuff and the others are the imitators, but I could be wrong. Sir Martin Rees, Lord High Astronomer to the British Crown or some such fooferall. He is at Cambridge, I think, and was one of Brian Greene's professors.

Be well. Coffee calls.

nc

marcus

Dec18-04, 01:42 PM

0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120

to get the next you always multiply the number you are at.

so, start at 0!= 1 and to get the next, namely 1!, multiply by one
then to get the next, namely 2!, multiply what you have already by two
then to get the next, namely 3!, multiply what you have already by three

Act III of "Lady Windermere's Fan"
We are all in the gutter, but some of us are looking at the stars.
brevity and generality

nightcleaner

Dec18-04, 01:46 PM

Oh yeah, do I feel dumb. I just went to Wikipedia and still have not gotten coffee. Ugh. So I'll try it again with multiples instead of adds. Cheese. BRB

Ahhh, the bitter and the sweet.

0! = 1
1! = 1 =0!x1
2! = 2 =1!x2
3! = 6 =2!x3
4! = 24 =3!x4
5! = 120 =4!x5

So we could imagine 0!= 1 = -1!x0. So -1! is the number which, when multiplied by zero, equals one. Now there is an insistant explosion for you.

5! = 120 =4!x5
4! = 24 =3!x4
3! = 6 =2!x3
2! = 2 =1!x2
1! = 1 =0!x1
0! = 1 =-1!x0
-1!= n =-2!x-1

comments?

We could multiply both sides of the last equation by -1, then 1!=-n=2!. Not so good. Could be a problem with multiplication. Maybe -1! x -1 does not equal 1!

-1! = n = -2! x -1
-1 x (-1!) = -n = -1 x (-2! x -1)
-1 x (-1!) = -n = -1 x (-2!) x -1 = -2!

I suppose we could say that n is infinity, so 0 x infinity is unity. Unity is no infinities? Sort of seems to make sense. One is to infinity as zero is to one?

I am getting cross-eyed. Maybe there just are no factorials of negative numbers, as Wikipedia said. Anyway I promised to work on Pascal and the dimensional stuff for Marcus. I'll go offline and do that now instead of trying to invert infinities.

nc

Lacy33

Dec18-04, 01:49 PM

Hi Marcus and Shoshana
Will have to have coffee.
c

This is very good Richard. You go have some more coffee and I will take the trash out.

marcus

Dec20-04, 08:21 PM

\left(\begin{array}{cc}N\\k\end{array}\right) = \frac{N!}{k!(N-k)!}

suppose we want to use this formula to find out what 4-choose-2 is

(this is the number of edges on a 3-simplex, i.e. a fourpointer, a tetrahedron,
or you can think it's the number of pairs of points you can pick from a batch of four)

so N=4 and k = 2 and we use the formula

\left(\begin{array}{cc}4\\2\end{array}\right) = \frac{4!}{2!(4-2)!} = = \frac{4!}{2!2!}

\left(\begin{array}{cc}4\\2\end{array}\right) = \frac{4!}{2!2!}= \frac{4*3*2*1}{2*1*2*1} = \frac{4*3}{2*1} = \frac{12}{2} = 6

so a tetrahedron has 6 edges. but you knew that already. one can visualize. it was just to check to see if the formula works!

but how about a 4-simplex living in 4D space, they have 5 points
lets see how many edges
it would be 5-choose-2

\left(\begin{array}{cc}5\\2\end{array}\right) = \frac{5!}{2!3!}= \frac{5*4*3*2*1}{2*1*3*2*1} = \frac{5*4}{2*1} = \frac{20}{2} = 10

\left(\begin{array}{cc}5\\0\end{array}\right) = 1

\left(\begin{array}{cc}5\\1\end{array}\right) = 5

\left(\begin{array}{cc}5\\2\end{array}\right) = 10

\left(\begin{array}{cc}5\\3\end{array}\right) = 10

...

nightcleaner

Dec20-04, 09:13 PM

0 1

1 1 1 1
0 1
2 2 2 1 2 1
0 1 2

3 3 3 3 1 3 3 1
0 1 2 3
4 4 4 4 4 1 4 6 4 1
0 1 2 3 4
n pick k pascal

1 1 1 1
1 2 3 4

2 2 2 2 Riemann tensor
1 2 3 4

3 3 3 3
1 2 3 4

4 4 4 4
1 2 3 4

It now appears to me that the six elements outside the orange triangle could be the redundant elements of the Riemann Tensor?

I got this form of the Riemann tensor from page 41 of Michio Kaku's Hyperspace.

Then I wonder what it would mean, if anything, to extend the Pascal and n-pick-k triangles to cover the other six Riemann terms? Could we talk meaningfully about n-pick-k where k is larger than n? If I have three objects and I pick four of them, do I have one imaginary object? Or do I have one of the three objects twice? Or do I have to go into fractional objects? As if I have three apples to divide among four picknickers.

And what, if anything, lies beyond the edges of the Pascal triangle?

I admit i don't know how to read the math in the Riemann tensor pages of Wikipedia. However a search of Wikipedia for Pascal's triangle turned up the following page, which may be interesting.

http://www.4dsolutions.net/ocn/urner.html

This is from a paper published in 1998 and shows a relationship between the Pascal triangle and the cubeoctahedron or isomatix, a favorite topic for geomancers such as myself.

So if the Riemann tensor is folded over the edge of a 4d Pascal simplex.....

nightcleaner

Dec22-04, 08:27 AM

http://www.4dsolutions.net/ocn/urner.html

I repeat this here. It is a most interesting approach to 4d visualization. I would like to discuss this with anyone interested. I intend to study this and follow the links given, and report what I find here.

The basic idea is starting with the Pascal Triangle. As the link above says, there is a form of the Pascal triangle called the pegboard. It is used as in the game pachinko, where a triangulated pegboard is used to vector balls which are inserted at the apex. The balls fall through the pegs, bouncing off sucessive layers of pegs in a random walk manner, and end up in collection tubes at the bottom of the board. When balls are inserted at the apex, they fall through the triangulated peg board and end up in one of the tubes at the bottom. The distribution into the tubes follows a Gaussian pattern, the familiar Bell curve. You really have to go look at this link, it says it much better and gives pictures.

Anyway, if we require a three dimensional version of pachinko, we can use the Kepler stack. Insert a very small ball into the interstices between the balls in the Kepler stack, and it will bounce down through the spaces between the balls in the Kepler stack in a three dimensional random walk. We could collect the small balls in tubes at the bottom of the stack and they would show a two dimensional Gaussian distribution.

So, the Kepler stack is shown to act as a three dimensional machine to filter random events into a two dimensional analog of the Bell curve. The link gives a three dimensional Pascal triangle which I will try to transcribe below, since I have not had any luck trying to copy it from the link given above to this forum.

1 1 1 1 1
1 1 2 2 3 3 4 4
1 2 1 3 6 3 6 12 6
1 3 3 1 4 12 12 4
1 4 6 4 1

It was a lot of finnicky word getting the code to come out right, but I think I got it. You see each layer fits on top of the one to the right, making a pyramid. The sides of the pyramid are the Pascal triangles.

I have chores today and work tonight, so will be gone until probably tomorrow. I am not sure how the authors got 12 in the center of the fifth layer, but will study it some more at first opportunity. Comments?

be well,

nc

ps oh yeah, the twelve comes from adding the three numbers from the layer above, so 6+3+3=12. Whats the next layer?