Is Information Theory the Key to Quantum Gravity? A Rant from an Angry Physicist

[Haelfix]

Marcus, Cartan formulations of General relativity is formally equivalent to say the usual Palatini formulation (which is often used to study QG). Making your local patch Desitter instead of Minkowski is fine (in any formulation), assuming your vacuum is indeed Desitter (eg say a FRW asymptotic final state), but I don't see how a change of variables or language exactly buys you new physics, these things are rigorously isomorphic mathematically (see Spivak).

When we do field theory close to the QG scale, using the usual Poincare invariance, the cosmological constant term is basically explicit and manifest. If you change formulations, and the isometry group is now ds or Ads, you have to be a little more careful as the CC is somewhat hidden in the extra algebra, but be sure it has to be identical. At least semi classically.

Now quantization might be a different beast. For some reason (probably man made confusion) it seems half the time quantization doesn't commute with formalisms.

 

[marcus]

Making your local patch Desitter instead of Minkowski is fine...

Haelfix you seem to be talking about a local coordinate patch.
That is not what I was talking about.

If you are not familiar with the basics of Cartan geometry
(which does not use a tangent space, at least in the usual sense)
then why don't you have a look at Derek Wise's paper?

the link to it is back in my post, and also in Chris Hillman's post where he refers to the "hamsterball". Indeed it is the hamsterball that replaces the tangentspace, in Cartan geometry (so his comment is apposite here!)

 

[marcus]

Marcus, Cartan formulations of General relativity is formally equivalent to say the usual Palatini formulation (which is often used to study QG)...

This suggests to me that you don't realize what is meant by "Cartan geometry" in Derek Wise's title. Elie Cartan called it "generalized geometry".

(we aren't talking about moving frames or differential forms or other such things that Cartan is famous for having helped to invent. these were very popular in the 20th century. we are talking about something else.)

Cartan's generalized geometry was virtually ignored for most of the 20th century.

The only textbook on Cartan geometry I know of is by R.W.Sharpe
(Springer 1997). It is helping Cartan geometry become recognized after many years of neglect.
The textbook is called
"Differential Geometry: Cartan's Generalization of Klein's Erlangen Program."

I don't see how a Cartan manifold can be formally isomorphic to an ordinary differentiable manifold---they belong to two different categories. "isomorphic" AS WHAT?

Perhaps I am missing something remarkable here, but it seems obvious that a theory of gravity formulated on a Cartan manifold would HAVE to contain new physics, compared to GR (a theory of gravity formulated on a conventional differentiable manifold).

It might be wrong physics, but it would surely be different

 

[Haelfix]

Cartan connections is something one learns in a graduate class in differential geometry, and be sure its formally identical to the usual way of studying manifolds when restricted to say the known cases people are interested in (say GR manifolds). In fact his student Ehresman simplified this line of thought into the usual connections on principle bundles point of view so often used nowdays. This is straight out of Spivak.

I looked at Derek Wise's paper and indeed that's exactly what it is, he even makes it explicit, when he constructs MM gravity from the Palatini formulation. Lo and behold, we have the same field content between the two. They are formally isomorphic.

You are free of course to construct something else other than gravity, and the Cartan connection can be of use in dealing with some topological gauge theories like BF theory. Indeed its this latter that motivates the paper, since MM gravity looks a little bit like the former when you generalize the Cartan connection as Wise points out (what he calls generalized Cartan connections).

Anyway, the whole point of using different formalisms is that sometimes what appears hard in one context is easier to solve or intuitively 'see' in another, but so long as we are talking about general relativity the physics must be the same.

 

[marcus]

Cartan connections is something one learns in a graduate class in differential geometry, and be sure its formally identical to the usual way of studying manifolds when restricted to say the known cases people are interested in (say GR manifolds). In fact his student Ehresman simplified this line of thought into the usual connections on principle bundles point of view so often used nowdays. This is straight out of Spivak.

I looked at Derek Wise's paper and indeed that's exactly what it is, he even makes it explicit, when he constructs MM gravity from the Palatini formulation. Lo and behold, we have the same field content between the two. They are formally isomorphic...

Spivak "Comprehensive Introduction..." is an interesting book.
https://www.amazon.com/dp/0914098713/?tag=pfamazon01-20
great cover art.
dropped out of academia and started his own press called "Publish or Perish"

I guess you would be talking about volume two of the five-volume set.

still not sure we are communicating, however.

Elie Cartan invented a lot of stuff (some I referred to earlier) including (with others like his student Ehresmann) the usual idea of connection. But what I said was I'm not talking about that other stuff. Cartan geometry the phrase in Derek Wise title is different from just doing conventional diffy geom using apparatus invented by Cartan. Cartan called it "generalized geometry", Wise and others call it Cartan geometry.

If you want to learn about it, and see if it has "new physics" potential, then I don't think you can rely on volume 2 of Spivak. I think you may need Sharpe's textbook. Garrett Lisi, who sometimes posts here, has Sharpe. Also Derek Wise referred to Sharpe. Also by carefully reading Wise' paper you can probably find out a lot.

I haven't read Spivak volume 2, but from what I see about it, I wouldn't assume it would give an intro to Cartan ("generalized") geometry. Could be wrong of course, perhaps I'll have a look at the math department library

=========
PS, Haelfix here is an exerpt from a reviewer's summary of volume 2
"...Kozul's concept of the connection and this is introduced in Chapter 6. First, note that the connection here is one of the versions of the introduced by Kozul as a map of pairs of vector fields to a vector field. Another useful version, not studied in volume II, is to consider the connection as a Hessian which maps any smooth function to a bilinear form on the tangent space. Second, note that Chapter 6 is usually the starting point for most treatments of curvature in differential geometry (e.g Do Carmo's "Riemannian Geometry"). Without the motivating material from the previous chapters, it would be difficult to understand the need for(or the point of) Kozul's connection.

Cartan's theory of curvature via a study of moving frames is detailed in Chapter 7. The author is careful to intuitively motivate Cartan's deviation from Euclidean concept as represented in the structure equations. Cartan's curvature tensor is shown to agree with Riemann's tensor, the "Test Case" is revisited, and the well-known fact that the curvature determines the Riemannian metric is established.

Building on the orthonormal frames from the previous chapter, Spivak now considers Ehresmann's theory of connections in principal bundles in Chapter 8. The main results here introduce the Ehresmann connection on the frame bundle, and gives the Kozul connection as a Lie derivative, thought of as the Cartan connection obtained from the Ehresmann connection..."

 

[ccdantas]

Here is something about "Cartan geometry" written by John Baez over at n-Cat Cafe

http://golem.ph.utexas.edu/category/2006/09/klein_2geometry_v.html#c004832"

 

[john baez]

Derek Wise

If I am not mistaken, he is a rather pleasant chap with a British accent.

No - while he's quite pleasant, Derek Wise doesn't have a British accent. He's from Colorado; I've attached a photo of him.

He's my grad student in the U. C. Riverside math department. He just passed his thesis defense last Wednesday! He hasn't come up to U. C. Davis yet; he should show up in the late summer. He'll be in the math department, but he plans to talk a bunch with Steve Carlip. Look him up if you don't bump into him.

He needs to polish his thesis a bit more before submitting it and putting it on the arXiv. It'll be called Topological Gauge Theory, Cartan Geometry and Gravity. It will subsume our paper with Alissa Crans on http://arxiv.org/abs/gr-qc/0603085" .

The idea is, quite briefly, that Klein geometry is to topological gauge theories like 3d gravity as Cartan geometry is to the MacDowell-Mansouri formulation of 4d gravity and its relatives. I explained a few of the ideas at more length in http://math.ucr.edu/home/baez/week232.html" of This Week's Finds, but I'll try to give a more thorough explanation of what we're up to when his thesis hits the arXiv!

A number of physics grad students here (except for a handful that I've met) don't believe in a lot of the stuff they write though. Very disappointing to say the least; perhaps, hopefully, he'll be one of the handful exceptions.

Like me, he's a mathematician. We believe what we write, because we've proved it.

 

[Chronos]

As usual, john baez is right on track. The other universes are fortunate he does not reside there.

 

[Angryphysicist]

No - while he's quite pleasant, Derek Wise doesn't have a British accent. He's from Colorado; I've attached a photo of him.

He's my grad student in the U. C. Riverside math department. He just passed his thesis defense last Wednesday! He hasn't come up to U. C. Davis yet; he should show up in the late summer. He'll be in the math department, but he plans to talk a bunch with Steve Carlip. Look him up if you don't bump into him.

I'll definitely have to look him up when he gets here, I can say by the photo (and the fact that he's not in Davis yet ) that I have not met him yet.

From the title of his thesis, he sounds like a fascinating fellow. I'd be intrigued to read it.

 

[john baez]

From the title of his thesis, he sounds like a fascinating fellow. I'd be intrigued to read it.

Derek wants to polish his thesis a bit more before putting it on the arXiv. But, I'm giving a talk on it this Tuesday, and you (and everyone!) can see the transparencies now:

http://math.ucr.edu/home/baez/derek/" .

Looking back on this thread I see some discussion about whether the Cartan-geometric approach gives new physics or not. There are lots of things to do with Cartan geometry, but the most interesting one to Derek is MacDowell-Mansouri gravity. This is not primarily a new theory. It's just a reformulation of ordinary general relativity. In fact, it's completely equivalent when the coframe field is nondegenerate.

However, Cartan geometry explains how MacDowell-Mansouri gravity gets to have the DeSitter or anti-DeSitter group as gauge group! This seemed very mysterious before. Cartan geometry also explains how MacDowell-Mansouri gravity gets to be a perturbation of a topological quantum field theory. These things suggest some interesting new ways of studying gravity - see the talk and references for details.

(The business about the coframe field being nondegenerate sounds technical - but in fact Witten's new paper argues that this technicality is the door through which the Monster group sneaks into 3d quantum gravity!)

 

[marcus]

Derek wants to polish his thesis a bit more before putting it on the arXiv. But, I'm giving a talk on it this Tuesday, and you (and everyone!) can see the transparencies now:

http://math.ucr.edu/home/baez/derek/" .

YAY!
I can only admire your sense of timing. Cartanization of qg is ready to roll. If Derek spends too much time polishing he will miss the tide.

So you launch him now. Does both him and us a great favor. thanks

=========
Oh hell, you stop the story at slide 23, just when it is getting good!

 

[P=K2ρ4/3]

John Baez students

I have no idea---can't help guess. Derek will officially be in the math department at davis. Might not be around there regularly yet: his postdoc contract begins in the Fall. Here's a picture of John Baez students taken summer 2004
http://math.ucr.edu/home/baez/students.html

That's my new desktop background.

 

[duke_nemmerle]

has this blog shut or am I missing something? Either way, if it's down I'd love it if the angryphysicist could figure out a way to keep us up to date on his adventures.

EDIT: forgive me, I've been going to angryphysicist rather than angryphysics

EDIT AGAIN: Haha, I finally just went back to the very first post, it's http://angryphysicist.wordpress.com/about/

 

[Angryphysicist]

Latest adventures...

has this blog shut or am I missing something? Either way, if it's down I'd love it if the angryphysicist could figure out a way to keep us up to date on his adventures.

Sorry, I would be more active in the blog (and here too!) BUT I am out on the town with a few mates. You're only young once!

As for my latest adventures, it's writing a series on writing file systems and I have been thinking about trying to use the Einstein field equations to describe Higgs bosons as a sort of quantization of curved spacetime.

There is some intuitive feeling that the Field equations is intimately related to the Higgs field, but I need to think this through a little more thoroughly.

And then, I am attending the first ever D programming language conference in Seattle, Washington...and I would like to work on a demand to make D more low level and a way to kill the garbage collector for systems programming.

 

[Chronos]

Information theory is the right road to QG IMO, Angryphysicist. I am certain the universe will obey a QIT description once all the false gods are disrobed.