Helpful new paper by Derek Wise

[marcus]

http://arxiv.org/abs/gr-qc/0611154
MacDowell-Mansouri gravity and Cartan geometry
Derek K. Wise
34 pages, 5 figures

"The geometric content of the MacDowell-Mansouri formulation of general relativity is best understood in terms of Cartan geometry. In particular, Cartan geometry gives clear geometric meaning to the MacDowell-Mansouri trick of combining the Levi-Civita connection and coframe field, or soldering form, into a single physical field. The Cartan perspective allows us to view physical spacetime as tangentially approximated by an arbitrary homogeneous 'model spacetime', including not only the flat Minkowski model, as is implicitly used in standard general relativity, but also de Sitter, anti de Sitter, or other models. A 'Cartan connection' gives a prescription for parallel transport from one 'tangent model spacetime' to another, along any path, giving a natural interpretation of the MacDowell-Mansouri connection as 'rolling' the model spacetime along physical spacetime. I explain Cartan geometry, and 'Cartan gauge theory', in which the gauge field is replaced by a Cartan connection. In particular, I discuss MacDowell-Mansouri gravity, as well as its recent reformulation in terms of BF theory, in the context of Cartan geometry."

things to notice:
care was taken to be intuitive and explain conceptually so as to generate more understanding----a certain bundle was described in terms of a hamster

the paper topics are what JB had that thread about earlier. Cartan geometry, rolling without slipping. JB had that thread about the forthcoming Baratin Freidel paper.

Derek Wise is doing his PhD thesis at UCR with Baez as advisor, but I think he also spent some time at Perimeter this year.

it is really nicely written.

the acknowledgments read like a list of people working on a certain approach that we've gotten some news about:

"..Acknowledgments
I thank John Baez, Aristide Baratin, Jim Dolan, Laurent Freidel, Bill Goldman, Jeff Morton,
Artem Starodubtsev, Danny Stevenson, and Josh Willis for helpful discussions..."

 

[marcus]

Garrett Lisi got a citation! Look at [19].

also the hamster-ball seems like a good idea---a kind of mobile squirrel cage able to travel around on the manifold

in the last sentence of the conclusions section at the end, DW talks about "the switch to Cartan-BF theory".

Garrett's work is cited at the bottom of page 4, where it talks about the MacDowell-Mansouri technique being used with a dfferent action based on BF theory and applied to various things including to particle physics.
===============

it seems obvious that since the universe is expanding you shouldn't use Minkowski flat as a tangent.
the tangent space should be deSitter
otherwise it won't glue right, the graft won't adhere

so Cartan-beef should be better than ordinary. what's to say? I think DW is on the right track and this is a seriously good paper---though it seems to just take the beginning steps---hope some others comment on it.

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

Derek Wise came by this morning and posted about this paper on the "deSitter group" thread
https://www.physicsforums.com/showpost.php?p=1176386&postcount=89
https://www.physicsforums.com/showthread.php?p=1176386#post1176386

 

[marcus]

Derek Wise strongly suggests we should read R.W.Sharpe's 1997 book

https://www.amazon.com/dp/0387947329/?tag=pfamazon01-20

I've got a lot of sales resistance to buying books but I'm sorely tempted in this case.

Amazon's price is just under $50
and there is a "like new" offer for slightly under $25 from some warehouse.

At the Amazon page they let you see the 3 page table of contents, and the several page index, and for style a few pages of the first chapter.

Sharpe is at University of Toronto
the short title of the book is "Differential Geometry"
it is a firstyear grad level textbook in the Springer series
Cartan diffy-geom is the next step beyond Riemann. It is better. the curriculum is going to change or has changed, and all the grad students are going to be learning Cartan D.G.-----it is not that it is new (Cartan made it up in 1920s). What is new is the realization.

Riemann-style D.G. is no longer cool. Cartan is cool.
But einstein Gen Rel was built on the old Riemann-style manifold, with the flat vectorspace kind of tangent space (euclidean or lorentzian, but flat either way) so now everything has to be rebuilt.

We are in a buzzword situation (MacDowell-Mansouri, Cartan) and when mathematicians get this way it is serious---you don't dismiss it as a fad.

You don't want the tangent space of your manifold to be flat anymore. it should be able to be expanding like deSitter. it should roll without twisting or slipping on the underlay manifold.

Some of us may have to buy Sharpe's book
====================

Here is a nice quote from Henri Poincaré (1902) that Sharpe puts at the start of chapter 1 introduction:

Is Euclidean geometry true? [The question] has no meaning...One geometry cannot be more true than another; it can only be more convenient.

 

[marcus]

Wiki has articles on Cartans: father Élie and son Henri.
http://en.wikipedia.org/wiki/Elie_Cartan
"Élie Joseph Cartan (9 April 1869 – 6 May 1951) was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. He also made significant contributions to mathematical physics, differential geometry, and group theory..."

Élie is the French for Elijah. I think it was the father Elijah who invented the "generalized" or new version of the continuum.

Derek Wise refers to Sharpe's book repeatedly. It is his reference [24], see for example pages 3, 7. 9. 13. 24, for starters.

=========
the son, Henri, was a Bourbaki and very into Categories.
http://en.wikipedia.org/wiki/Henri_Cartan
=========

But here we are talking about the reinvention of the continuum---so it is Elijah, Cartan senior.

Derek Wise gives this perspective on page 3:

==quote from DW==
One point of the present paper is to show that MacDowell–Mansouri theory is no trick after all, but rather a theory with a rich geometric structure, which may offer insights into the geometry of gravity itself. In fact, the secret to understanding the geometry behind their work had been around in some form for over 50 years by the time MacDowell and Mansouri introduced their theory.

The geometric foundations had been laid in the 1920s by Élie Cartan, but were for a long time largely forgotten. The relevant geometry is a generalization of Felix Klein’s celebrated Erlangen Programm to include inhomogeneous spaces, called ‘Cartan geometries’, or in Cartan’s own terms, espaces généralisés [9, 10]. The MacDowell–Mansouri gauge field is a special case of a ‘Cartan connection’, which encodes geometric information relating the geometry of spacetime to the geometry of a homogeneous ‘model spacetime’ such as de Sitter space. Cartan connections have been largely replaced in the literature by what is now the usual notion of ‘connection on a principal bundle’ [11], introduced by Cartan’s student Charles Ehresmann [12].

The MacDowell–Mansouri formalism has recently seen renewed interest among researchers in gravitational physics, especially over the past 5 years. Over a slightly longer period, there has been a resurgence in the mathematical literature of work related to Cartan geometry, no doubt due in part to the availability of the first modern introduction to the subject [24].

Yet it is not clear that there has been much communication between researchers on the two sides—physical and mathematical—of what is essentially the same topic.
==endquote from DW==

 

[marcus]

given that it is a very helpful paper with a lot of signposts it is natural to want to know a bit more about DW, and the ucr math department is very open and straightforward, from the looks of it.

Derek has a CV page that tells the title of his thesis (PhD expected 2007) and some papers he is working on with Laurent Freidel and John Baez and others.
This is pretty interesting because it tells you not just past research but where he is going. From his picture he looks like he might be around 30, not older than that anyway. (pictures are not always up-to-date)

I think Wise is an Irish name. I know a guy named Wise who has a crafty Leprechaun look in his eyes and whose people came from some county in Ireland. I forget which. He was almost a professional baseball player and musician but then he became a lawyer. His degree from Harvard was in music, of all things. He has a good baritone. So you can't always tell.

Oh here is the link to Derek Wise page:
http://math.ucr.edu/~derek/

Well the thing to focus on is the titles of the work in progress, I think.
Thesis title: Cartan Geometry, Topological Quantum Field Theory, and Gravity

Papers in Preparation
1. On the representation theory of the Poincaré 2-group, with John Baez, Aristide Baratin, Laurent Freidel, and Jeffrey Morton.
2. Cartan geometry and geometric structures in BF theory and gravity.
3. Higher gauge theory: chain field theory, volumetric field theory, and p-form electromagnetism.PRESENTATIONS
Abstracts and slides at http://math.ucr.edu/~derek/talks.html

1. A Tour of Homogeneous Spacetimes, 16 November 2006.
Quantum Gravity Seminar, University of California, Riverside.
2. Loop Braids and Quandles, 24 October 2006.
University of Ottawa, Ottawa, Ontario, Canada.
3. Volumetric Field Theory, 21 October 2006.
Category Theory Octoberfest, University of Ottawa, Ottawa, Ontario, Canada.
4. Higher Electromagnetism and Volumetric Field Theory, 18 October 2006.
Toplogy Seminar, University of California, Riverside.
5. Exotic Statistics and Particle Types in 3- and 4d BF Theory, 13 July 2006.
Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada
Derek K. Wise 3
6. Group-valued momentum and exotic statistics in BF theory, 4 July 2006.
University of Western Ontario, London, Ontario, Canada.
7. Chain Field Theory, 29 June 2006.
International Category Theory Conference, CT 2006, White Point Beach, Nova Scotia, Canada.
8. Discrete p-form electromagnetism as a Chain Field Theory, 11 October 2005
LOOPS ’05 conference on background-free quantum gravity, Albert Einstein Institut, Golm, Germany.
9. Lagrangians for General Relativity: Palatini and MacDowell-Mansouri formulations, 7 February
2006.
Quantum Gravity Seminar, University of California, Riverside.
10. The Geometry of MacDowell-Mansouri Gravity, 9 February 2006.
Quantum Gravity Seminar, University of California, Riverside.
...
...

DW says he spent a month at Perimeter this past summer. He has some slides for a talk he gave there in July 2006.
I think they might be helpful---have some additional intuition about what is in the paper by Baez, Wise, Crans.

 

[marcus]

...One geometry cannot be more true than another; it can only be more convenient.

As everyone has probably noticed, Baez and Wise are in the MATH department at riverside. Math rules. Ultimately the major advances in physics---that get closer to the true, real fundamental degrees of freedom that nature actually is----these advances come from mathematics.

And that quote from Henri Poincaré is slightly ironical or I take it in a contrarian sense, because the sign that something is the true real ontology of nature is (besides passing empirical tests better than the others) that a trained mathematician finds it "convenient"

that is what true is.

if Cartan geometry is more convenient than Riemann-type, then it is more real

heh heh

 

[ccdantas]

Hi Marcus,

Thanks for the posts on Derek's new paper. I am reading it, and it is always a pleasure to learn from papers so clearly written such as this one.

Best wishes,
Christine

 

[marcus]

... I am reading it, and it is always a pleasure to learn from papers so clearly written such as this one.

Hi Christine,

well I went and ordered Sharpe's text
amazon had a "half-price" deal for $24.95 or something

if this is the new kind of continuum, more "convenient" than what Riemann invented in around 1850, then I thought I should try to learn diffy-geom from that point of view. Sure hope I can understand it.

I'm so glad you are enjoying the DW paper! It is in fact very clearly (and patiently) written, as if he wanted people to understand.

Actually hamster-balls may be real pet toys that you can buy, not something made up.

 

[garrett]

Hey Marcus,

I spent some time with Sharpe's book a year ago, and found much of it to be kind of rough going. You may be better off just reading Derek's paper, in which he's done a great job of making the subject understandable. For more mathematical detail I really like the introductory part of this paper Derek references:

http://arxiv.org/abs/math.DG/9412232

But Sharpe is good, and it is THE mathematical reference on the subject -- so don't feel that you've wasted your money.

 

[ccdantas]

Does anyone have impressions about this one?

https://www.amazon.com/gp/product/0821833758/?tag=pfamazon01-20

(And yes, his hamster drawings are cute and informative at the same time! And no, I have no intentions to buy hamsters, my dog has already build up an interesting topology over my grass, the hamsters would fall into the holes or be eaten by the dog -- no way... )

Best wishes,
Christine

 

[garrett]

Hi Christine,
I haven't read it, but looking at the table of contents... it looks more like it focuses on the differential geometry (and symmetries) arising from systems of differential equations. This was Cartan's original playground. But I wouldn't recommend the book if you're interested in understanding GR via frames, or for Cartan geometry.

 

[derekwise]

well I went and ordered Sharpe's text
amazon had a "half-price" deal for $24.95 or something

Wow, that's a great deal! I really should buy that book too, given the amount of advertising I seem to be doing for it. I've had it checked out from the UCR library for probably 2 years. :rofl: Unfortunately, the amazon sale seems to be over...

-DeReK

 

[ccdantas]

Thanks Garrett!

 

[garrett]

Hey Marcus, this should make your head explode:

I was just talking with Sabine and we noticed that the idea of having de Sitter space as a tangent manifold (which naturally arises in Cartan geometry) meshes with the idea of having momentum valued in de Sitter space, which gives DSR:
http://arxiv.org/abs/hep-th/0207279

Might be worth thinking about.

 

[marcus]

Hey Marcus, this should make your head explode:

I was just talking with Sabine and we noticed that the idea of having de Sitter space as a tangent manifold (which naturally arises in Cartan geometry) meshes with the idea of having momentum valued in de Sitter space, which gives DSR:
http://arxiv.org/abs/hep-th/0207279

Might be worth thinking about.

Good.
To me that means it leads to a falsifiable prediction (for GLAST in 2007/2008) similar to what Smolin has been suggesting might occur.
I hope this is confirmed by enough of the ringleaders that there will be a sense that something is irrevocably bet on GLAST's GRB observations, and that it will be made more precise over the next few months.

Win or lose it is so important to bring some of these issues to empirical trial.

Thanks for the Jerzy K-G paper! I will spell out the link you gave:
http://arxiv.org/abs/hep-th/0207279
De Sitter space as an arena for Doubly Special Relativity
Jerzy Kowalski-Glikman
7 pages
Phys.Lett. B547 (2002) 291-296

"We show that Doubly Special Relativity (DSR) can be viewed as a theory with energy-momentum space being the four dimensional de Sitter space. Different formulations (bases) of the DSR theory considered so far can be therefore understood as different coordinate systems on this space. The emerging geometrical picture makes it possible to understand the universality of the non-commutative structure of space-time of Doubly Special Relativity. Moreover, it suggests how to construct the most natural DSR bases and makes it possible to address the long standing problem of total momentum of many particle systems from a different perspective. "

I should put a link to Bee's blog about DSR.

http://backreaction.blogspot.com/2006/12/deformed-special-relativity.html

 

[marcus]

Derek's July 2006 slides

Derek's July 2006 slides unlock stuff. They are at the right level, not too superficial, not too hard, not saying too much or with too much sophistication---for me. So I experience them as communicating very efficiently and unlocking stuff which boggled me a bit when i read it in the Baez, Wise, Crans paper.

the July slides do not overlap with this new paper on Cartan and M-M.

they connect with it but they deal with something else----exotic matter statistics coming from the GROUP OF MOTIONS.

you get various different group of motions from different setups---it is a general idea.

the definition of what is a MOTION (an element of this group) makes sense, like the way the definition of a homotopy makes sense. it has to be the way it is so you can do the things you want.

these slides have good pictures. I would recommend studying them a little and learning about the group of motions (which can be the braid group or the loop-braid, or the integers Z, or permutation group, whatever depending on the setup.)

It is easy, only 19 hand-drawn slides. I want to share this with someone

go to this menu
http://math.ucr.edu/%7Ederek/talks.html
and the 5th item from the top is "Exotic Statistics"...the 13 July talk

there may be other good stuff on this menu
================

in understanding the July slide talk it probably helped to have already tried with partial success to read the March 2006 Baez Wise Crans.
maybe one has to study from both.
http://arxiv.org/abs/gr-qc/0603085
Exotic Statistics for Strings in 4d BF Theory
John C. Baez, Derek K. Wise, Alissa S. Crans
41 pages, many figures.

"After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that string-like defects in 4d BF theory obey exotic statistics governed by the 'loop braid group'. This group has a set of generators that switch two strings just as one would normally switch point particles, but also a set of generators that switch two strings by passing one through the other. ..."
==================

Derek Wise lists some papers IN PREPARATION including

1. On the representation theory of the Poincaré 2-group, with John Baez, Aristide Baratin, Laurent Freidel, and Jeffrey Morton.
2. Cartan geometry and geometric structures in BF theory and gravity.

this is to think about and fit together with what we have seen, to get ideas of what to expect. About #2, Garrett said something came up in conversation with Bee Hossenfelder to the effect that a gravity using deSitter Cartan geometry could have (a curved momentum space) DSR and although Garrett didnt say it explicitly, this suggests a possibility of a gammaray dispersion QG effect that could be looked for in GRB data. DSR is more exposed to falsification than some other facets of QG. Anyway one can hope that things are creeping in that direction.

 

[Dcase]

On page 6 of the original DK Wise paper of this thread, he has a figure of two catenoids representing tangent de Sitter spaces.

Tristan Needham in ‘Visual Complex Analysos’ [1997-2002] demonstrates the transformation of a helicoid into a catenoid. [Figure 3 in 12 Flows and harmonic functions, I Hamonic Duals, 1 Dual flows]

Using Needham’s ‘amplitwist’ concept, it appears that the catenoid [a category of hollow cylinder?] could be transformed into two different tori:

1 - conventional, uniform torus by closing the top and bottom with the joining leaving the center of the catenoid on the interior of this torus.
Then separating the center of this torus until an hollow Riemannian-like sphere is formed.
top down
\.../
/...\
bottom up to uniform torus

2 - skewed, cyclide torus by joining the top and bottom such that the center of the catenoid is part of the exterior of this torus.
Then separating the center of the old catenoid to form a double-pseudosphere.
top right
\.../
/...\
bottom right to cyclide torus

This may help explain why space-time is curved,
and why nucleic acid helices are found within various 3-balls [capsids and cells],
and why planets and probably fundamental particles have helical trajectories.

 

[marcus]

John Baez just announced some good news.
Derek is finishing is PhD thesis this spring and has gotten a postdoc job at the Math Department at UC Davis.
Baez gives this link to Derek's homepage
http://math.ucr.edu/~derek/

AFAIK UC Davis is a good place to be for Quantum Gravity. It has Steve Carlip
http://www.physics.ucdavis.edu/Text/carlip_steve.html
Carlip does a variety of kinds of research including Quantum Geometry (alias non-string QG)
His home page has a lot of links to resources about QG and GR
http://www.physics.ucdavis.edu/Text/Carlip.html

Carlip joined the UC Davis physics faculty in 1990 after a postdoc at Princeton Institute for Advanced Study.

UC Berkeley has nobody of comparable stature in non-string QG, as far as I know. The only person they have of comparable note is Bousso, a string theorist. Davis is only an hour north on the freeway from Berkeley. People in more or less any department will commute back and forth for seminars and colloquium talks. Somebody should get Berkeley MSRI to have a workshop on MacDowell-Mansouri gravity and SO(4,1) Cartan geometry. Wise could commute in from Davis.

I think if you are looking on the west coast then Riverside and Davis are the best places you could be for non-string QG.
Incidentally Steve Carlip just posted two BH entropy papers on arxiv this past week.
http://arxiv.org/abs/gr-qc/0702107
Black Hole Thermodynamics from Euclidean Horizon Constraints
S. Carlip
4 pages

http://arxiv.org/abs/gr-qc/0702094
Black Hole Entropy and the Problem of Universality
Steven Carlip
10 pages; talk at DICE 2006, Piombino, Italy

 

[marcus]

...Derek Wise lists some papers IN PREPARATION including

1. On the representation theory of the Poincaré 2-group, with John Baez, Aristide Baratin, Laurent Freidel, and Jeffrey Morton.
...
...

I'd like to hear more about this. One of the coauthors, Jeffrey Morton, is a Baez student and the news is he's finishing his PhD and has a postdoc at University of Western Ontario (UWO)
(Dan Christensen has a joint faculty position there in math, physics, and computing---Josh Willis went there from Penn State)

Right now, if non-string QG is going to pick up momentum in the US and Canada, I'd say the most interesting places outside Penn State and Perimeter are UC Riverside, UC Davis, UWO.

I wonder if they can pull it off, to understand the representations of the Poincaré 2-group. Seems like it would make some waves.

 

[john baez]

I'd like to hear more about this.

Okay.

Right now, if non-string QG is going to pick up momentum in the US and Canada, I'd say the most interesting places outside Penn State and Perimeter are UC Riverside, UC Davis, UWO.

As soon as Derek and Jeff finish, I won't have any students pestering me to work on quantum gravity, so I'll mainly do other stuff.

My student Mike Stay is going to quit studies at UCR and work for Google, but I hope to keep working with him on http://math.ucr.edu/home/baez/qg-winter2007/index.html#computation". I'll only have two students left at UCR: Alex Hoffnung, who is beginning work on string theory from a 2-categorical viewpoint, and John Huerta, who is starting work on grand unified theories.

I wonder if they can pull it off, to understand the representations of the Poincaré 2-group. Seems like it would make some waves.

I'd like to make some particles, not just waves.

We've classified the irreps of the Poincaré group, but we were hoping to get a 4d TQFT out of it, and this is harder - we probably won't go that far in this paper.

 

[garrett]

Ah, GUTs...
The Coleman-Mandula theorem killed the idea of joining the Poincare algebra non-trivially with (Yang-Mills) internal symmetries. But ditching Poincare for De Sitter revives the idea of finding a unified Cartan Geometry / Yang-Mills GUT theory that includes gravity. Fun stuff!