Group Field Theory approach to Quantum Gravity

[marcus]

Here is a set of 100-some slides for a talk Daniele Oriti gave in September 2005 called
The Group Field Theory Approach to Quantum Gravity
http://www.phy.olemiss.edu/GR/qg05/talks/oriti.pdf

The talk was given at this conference in Sardinia:
http://www.phy.olemiss.edu/GR/qg05/
This is the 8-page paper representing the talk in the conference proceedings:
http://arxiv.org/abs/gr-qc/0512048
Quantum gravity as a group field theory: a sketch
Daniele Oriti
8 pages, 9 figures; to appear in the Proceedings of the Fourth Meeting on Constrained Dynamics and Quantum Gravity, Cala Gonone, Italy, September 12-16, 2005

"We give a very brief introduction to the group field theory approach to quantum gravity, a generalisation of matrix models for 2-dimensional quantum gravity to higher dimension, that has emerged recently from research in spin foam models."

So far here at PF we haven't discussed GFT approach to QG much.
Besides Oriti's one of the main examples of GFT is Laurent Freidel's research. There are various other materials about GFT readily available.

For instance there is a Perimeter video of Oriti giving Lecture #21 of the Smolin Lecture series. There the visuals are whatever Oriti writes on the blackboard. These slides cover some of the same material and are graphically very clear. The formats complement.

There is also an article by Oriti about this, for publication in a collection of QG essays.
have to go, I will fetch the link when I get back
should try to say what GFT is and why Oriti and some others favor it.

 

[marcus]

this is a paper Oriti had recently in Physical Review D
http://arxiv.org/abs/gr-qc/0512069
Generalised group field theories and quantum gravity transition amplitudes
Daniele Oriti
6 pages, 2 figures
Phys.Rev. D73 (2006) 061502
"We construct a generalised formalism for group field theories, in which the domain of the field is extended to include additional proper time variables, as well as their conjugate mass variables. This formalism allows for different types of quantum gravity transition amplitudes in perturbative expansion, and we show how both causal spin foam models and the usual a-causal ones can be derived from it, within a sum over triangulations of all topologies. We also highlight the relation of the so-derived causal transition amplitudes with simplicial gravity actions."

this has the same, or near the same, title as the talk he gave at October Loops '05 conference.

the message seems to be that GFT SUBSUMES usual spinfoam models and Loll-type triangulation gravity (CDT).

the central position of GFT---how you can specialize so as to derive other earlier kinds of Quantum Gravity---was something that Oriti was explaining in the Perimeter video (Smolin Lecture #21)
=================

Here are the slides of Oriti's Loops '05 talk.
They should match up fairly well with the Physics Review D paper.
http://loops05.aei.mpg.de/index_files/abstract_oriti.html
Parametrised group field theories and quantum gravity transition amplitudes
"We describe a generalisation of the group field theory formalism to include extra variables with the interpretation of proper time, and an action that contains derivative kinetic terms; we show how to construct the Feynman expansion of the field theory and how to obtain both causal and a-causal spin foam amplitudes from this generalised formalism."

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

Here is a chapter Oriti wrote for somebody else's QG book (not the one he is editing but also due out this year)
http://arxiv.org/abs/gr-qc/0512103
Quantum Gravity as a quantum field theory of simplicial geometry
Daniele Oriti
23 pages, 12 figures; to be published in 'Mathematical and Physical Aspects of Quantum Gravity', B. Fauser, J. Tolksdorf and E. Zeidler eds, Birkhaeuser, Basel (2006)

"This is an introduction to the group field theory approach to quantum gravity, with emphasis on motivations and basic formalism, more than on recent results; we elaborate on the various ingredients, both conceptual and formal, of the approach, giving some examples, and we discuss some perspectives of future developments."

==============
I believe he also wrote a chapter for his own book, but if i ever had it I have mislaid the link. He seems to be very focused on GFT. thinks it is the way to go.
It's new. And it is the approach that Laurent Freidel mostly uses, I believe.
It is probably a good idea to know something about it.

 

[Mike2]

The question I have is how is Group Field Theory any different than Algebraic Quantum Field Theory since group theory is part of abstract algebra?

 

[marcus]

The question I have is how is Group Field Theory any different than Algebraic Quantum Field Theory since group theory is part of abstract algebra?

you are judging by the WORDS I think. that is just a surface thing, what people decided to call it. Names are often not a good clue to what something is about.

the theory called GFT does not seem to me to be primarily algebraic but rather it is (I would say) geometric in flavor
but you have to judge for yourself by reading

it looks to me as if the group G is merely the GROUND on which one defines the field. One takes a cartesian product of several copies of the group G and uses that, instead of spacetime, as a basis to build on.

and when you get around to integrating, you can do the integral over the group, instead of over a spacetime continuum or something else. groups often have a nice ready-made measure that can be used for integrating.

Please check for yourself, I don't want to mislead you. But I want to be sure that you understand that the GROUP IS NOT SPACETIME. (because in a sense there isn't any)
the cartesian product of copies of the group is used to define stuff on and integrate on, but it is NOT A SUBSTITUTE FOR SPACETIME. (because in a sense the theory doesn't have any)

AFAICS a mathematical object representing ready-made spacetime does not exist in the theory. But what does get constructed is Feynman diagrams. And averages of Feynman diagrams. I guess you can say that it is THESE things---these diagrams--- that ultimately represent spacetime, to the extent that there is something with that role in the theory.

this mother is background independent as all get-out because it does not even have a ready-made spacetime manifold in the picture. It doesn't even have a BARE one, without a metric on it.

spacetime and matter are both handled by the diagrams and the diagrams emerge or get built somehow. Space time is not something you give yourself at the beginning (as in some older theories) so that you can then build things on it and run model trains and stuff. It is something the point of the theory is to erect. Diagrams are an OBSERVABLE of the theory. Space/matter relations are observables that must be constructed. One of these frustrating "do-it-yourself" kits. Nature playing hard to get

=====how Algebraic QFT is different======
In algebraic QFT one gives onesself immediately for starters MINKOWSKI SPACETIME and you have all the open subsets (of this given Minkimowski spacetime) and one postulates a functor to the category of unital CEE-STAR ALGEBRAS. A C*-algebra is bigtime functional analysis algebra like the algebra of operators on a Hilbert space. It was already infinite dimensional before we were born.

the word "algebra" as in C*-algebra only superficially resembles the word "algebra" where you say that a finite dimensional Lie group is a group and therefore sounds like it should be told about in an "algebra" course.
Lie groups tend to be studied in differential geometry course or physics. A groups is NOT AN ALGEBRA in the sense of C*-algebra. a group has one binary operation, an algebra has several ops. corresponding to addition and multiplication and accessory stuff like multiplication by elements of a field.

When Oriti says "group" in GFT the group is usually finite-dimensional and there is no spacetime and there is no C*-algebra. there is no obvious or explicit connection with Algebraic QFT. Mathematicians can always MAKE UP connections between things (mathematics is about the made-up connections between things so they are good at it---they do it instinctively almost without noticing)

the spirit is different and I think
the verbal associations via the word "algebra" are confusing.

=================
what worries me about all this is suppose you want to replace the (say) finite dimensional Lie group by a QUANTUM group-----which is what (say) a finite dimensional Lie group looks like when you have had enough to drink. I think Freidel has ventured into that territory. the idea of it strikes me with forboding.

 

[Mike2]

AFAICS a mathematical object representing ready-made spacetime does not exist in the theory. But what does get constructed is Feynman diagrams. And averages of Feynman diagrams. I guess you can say that it is THESE things---these diagrams--- that ultimately represent spacetime, to the extent that there is something with that role in the theory.

this mother is background independent as all get-out because it does not even have a ready-made spacetime manifold in the picture. It doesn't even have a BARE one, without a metric on it.

spacetime and matter are both handled by the diagrams and the diagrams emerge or get built somehow. Space time is not something you give yourself at the beginning (as in some older theories) so that you can then build things on it and run model trains and stuff. It is something the point of the theory is to erect. Diagrams are an OBSERVABLE of the theory. Space/matter relations are observables that must be constructed. One of these frustrating "do-it-yourself" kits. Nature playing hard to get

If there is no spacetime to work from, then I have to wonder what ontology can be given. What intuitive "first principles" can be given as a "physical foundation" for the theory?

 

[marcus]

If there is no spacetime to work from, then I have to wonder what ontology can be given.

It's Feynman diagrams all the way down

Seriously, good thinking. You SHOULD be wondering "what ontology can be given"

Also already when you see the title of Oriti's book quoted as "approaches to a new understanding of space time and matter"
you or anybody should be wondering what ontology.

David Gross peered into that abyss recently when he was quoted as saying physics needs some new ideas including a new idea of space and time including that there might not be any. (any space and time, not new ideas---sure there'll always be new ideas). We should pay attention better to what these great men say, shouldn't we? here I can't remember what exactly he said!

 

[marcus]

If there is no spacetime to work from, then I have to wonder what ontology can be given...

I hunted down that David Gross insight by finding where Carlo Rovelli quotes it in http://arxiv.org/gr-qc/0604045

=====Rovelli "unfinished revolution"====
...In doing so, they confuse the details of the Einstein’s equations (which might well be modified at high energy), with the new understanding of space and time brought by GR.

This is coded in the background independence of the fundamental theory and expresses Einstein’s discovery that spacetime is not a fixed background, as it was assumed in special relativistic physics, but rather a dynamical field.

Nowadays this fact is finally being recognized even by those who have long refused to admit that GR forces a revolution in the way to think about space and time, such as some of the leading voices in string theory. In a recent interview [1], for instance, Nobel laureate David Gross says: “ [...] this revolution will likely change the way we think about space and time, maybe even eliminate them completely as a basis for our description of reality”.

This is of course something that has been known since the 1930’s [2] by anybody who has taken seriously the problem of the implications of GR and QM.

The problem of the conceptual novelty of GR, which the string approach has tried to throw out of the door, comes back by the window.

The scientists trying to resist quantum theory or background independence remind me of Tycho Brahe, who tried hard to conciliate Copernicus advances with the “irrefutable evidence” that the Earth is immovable at the center of the universe. To let the background spacetime go is perhaps as difficult as letting go of the unmovable background Earth. The world may not be the way it appears in the tiny garden of our daily experience.
===endquote===

References cited in this exerpt:

[1] Gross D, 2006, in Viewpoints on string theory, NOVA science programming on air and online, http://www.pbs.org/wgbh/nova/elegant/view-gross.html.

[2] Bronstein MP, 1936, “Quantentheories schwacher Gravitationsfelder”, Physikalische Zeitschrift der Sowietunion 9 140.
====================

So evidently Rovelli listened and heard Gross say something that we didnt hear. If you eliminate space and time and there is just fields, then maybe space and time have no ontology because they are just illusions that emerge at the scale of everyday life, and maybe it really is (as the ancient Tortoise said)
FEYNMAN DIAGRAMS ALL THE WAY DOWN

How can one imagine spatial relations without space?

how can one keep track of what particle is adjacent to which other particles, without the convenient device of putting them all in a single space-tank. well, it looks like one has to start keeping lists of adjacency. Yuk. Lists are the spinach of the mind. Well but how else, if you don't have the space-tank.
Space-tank is just a CONVENIENT APPLIANCE for keeping simulaneous account of a whole bunch spatial relationships (between beside, above before after below inside outside) all you have to do is locate it in the tank and you can go away and forget it. the tank keeps track.

one senses the enormous mental economy and convenience, and one clings to the tank.

BUT WHAT IF IT WAS ALWAYS A BOOK-KEEPING DEVICE and was never really there? the ancient Tortoise knows but he is reticent.

 

[Mike2]

Space-tank is just a CONVENIENT APPLIANCE for keeping simulaneous account of a whole bunch spatial relationships (between beside, above before after below inside outside) all you have to do is locate it in the tank and you can go away and forget it. the tank keeps track.

I wonder if it may be that category/topos theory might be the basis of this Group Field Theory. As I recall, Kea has taken the position that category/topos theory is the basis of the Algebraic approach to QFT which eliminates the need for the background eliments of spacetime and relies more on global things for the topology involved. In this way category theory provides a background independence to the situation. And as I recall, groups (especially Lie groups) have an associated algebra so that a Group Field Theory would have an associated Algebraic QFT as well.

If so, and you might wish to check with Kea, then this works for me. I would suggest that the most basic thing is the logic, independent of what elements (of a background spacetime) to which it is applied. By logic I mean the union and intersection of sets by which topologies are made of. Previously, these topologies were described by unions and intersections in terms of the elements of sets. But if I'm not mistaken, category/topos theory somehow forms their topologies without reference to the elements. So it seems that it might serve to provide background independence. The ontology (or fundamental principle) would be that logic is supreme, and not the elements (of spacetime) previously used as at least one example to describe it.

I'm a little new to these ideas. I could very well have misinterpreted things here. I would appreciate any correction or confirmation anyone could authuritatively give. Thank you.