"No prior geometry" and the group field theory vehicle

Atyy recently pointed us to a provocative quote from MTW: Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior geometry" and for a geometric, coordinate-independent formulation of physics. Einstein described both demands by a single phrase, "general covariance." The "no prior geometry" demand actually fathered general relativity, but by doing so anonymously, disguised as "general covariance", it also fathered half a century of confusion.

page 431 in Gravitation by Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler (1973) Freeman.

In case someone wants to backtrack here's where it came up:

Here's another thought-provoking recent comment by Atyy from a different thread:

I'd like to comment on these ideas and perhaps others would as well.

Re: "No prior geometry" and the group field theory vehicle

I think it is a big advantage if we normally use online sources that everyone in the discussion can freely consult. It can be especially confusing when people quote out of context, and don't give a link providing the rest of the passage. Context can sometimes make all the difference.

I can't give a link to an online version of MTW "Gravitation" but I do have this article from the Institute of Physics 1993 Reports on Progress in Physics: http://www.pitt.edu/~jdnorton/papers/decades.pdf

For what I think is an extremely clear discussion of the "hole argument" see pages 801 and 802 of Norton's RPP article.
For a discussion of Einstein's 1915 statement of ontology (space and time not physically real) see page 804.
Norton is clearly an expert in these questions. It's not just textbook-style coverage--an IOP review like this has a different aim, more along the lines of "definitive monograph" that takes the time to lay out difficult issues carefully.

I might not necessarily agree with everything J.D. Norton says (for sure!) but I have to acknowledge his presumed expertise and balanced judgment. Also he highlights controversy and presents both sides where he finds disagreement. (There are certainly statements in the article which, if isolated and quoted out of context, I would have to suppress my urge to quarrel with. Presenting both sides is part of a scholar's job.)

John Stachel is another expert of like caliber. We might find something online by him.

Anyway hopefully this J.D. Norton article will be of use.

Re: "No prior geometry" and the group field theory vehicle

Nrqed, I'm very glad to see a comment! GFT (group field theory) is one possible way of doing background independent QG.

You work on a group manifold. Like a cartesian product of many copies of SU(2) or some other Lie group G---call the direct sum of N copies by the name G^{N}

Picking a point in the group manifold is like picking an N-tuple of group elements. A field defined on a group manifold---or simply some real or complex-valued function defined on a group manifold---specifies a spacetime geometry in a way explained in Rovelli's April 2010 paper.

This goes back to some papers by Reisenberger and Rovelli in the mid 1990s, and the idea especially in 3D goes back to others even before, but I mention the April 2010 paper because it is a good up-to-date source.

GFT is a flexible general framework that can be applied to formulate various QG. It is nothing specific by itself. It can be used as a technique to formulate simplicial QG and several other alternatives to LQG.

Atyy said something about LQG turning out eventually to be a "limit" of some unknown GFT. That is not quite right because LQG already is a GFT, and not just a limit, it IS that. You don't have to wait for something to happen, or take a limit.
Theories have various different equivalent formulations. The April paper describes various equivalent or convergent formulations of LQG, one being the GFT version.

There is a gimmick. For each N you can have the cartesian product of N copies of the group: G^{N}
and so you can have in a naive sense one GFT for each N. But then you can put them all together and let N go to infinity! You still get a countable basis for the Hilbert space.
So the Hilbertspace is still "separable"---the technical term as you may know.
So when you do a GFT treatment with triangulations you are not just dealing with one triangulation, but possibly all or very many. If you do a GFT treatment of LQG you are jnot just dealing with one fixed graph and a fixed number of spin networks with, eg. N links. You are able to take a big projective limit or direct sum limit and include all N.
Sometimes the idea that N might be fixed finite bothered me when I first encounter the group manifold idea, but it's OK.

Re: "No prior geometry" and the group field theory vehicle

You may be right, I don't know which way it will go, but I don't think the various formulations Rovelli mentions have been shown to be equivalent or convergent. And in my statement I had in mind canonical LQG.

Among the open problems listed by Rovelli:

13. Find a simple group feld theory [41] whose expansion gives (52).
14. Find the relation between this formalism and the way dynamics can be treated in the canonical theory. ... Can we construct the Hamiltonian operator in canonical LQG such that this is realized?

Furthermore, because I expect GFTs to have to be renormalized, I would say the present work looks at a "low energy" limit of GFT, along the lines of "Finally, if one regards group field theory as fundamental, rather than just a convenient computational tool to arrive at the spin foam vertex expansion, then one is led to take the coupling constant λ as a physical parameter which can run with the renormalization group flow. http://arxiv.org/abs/0909.4221"