Introduction to Group Field Theory: Main Ideas, Formalism, and Origins

[Schreiberdk]

Hi there PF

I just want to ask, what are the main ideas behind Group Field Theory? How does it work, and what is the formalism? And why is it called GROUP Field Theory? :)

\Schreiber

 

[marcus]

ordinary fields are functions defined on the ordinary x-y-z coordinates of flat space.

what values the functions take does not matter for this discussion. they might be real number valued, or complex number vaued, or vector valued etc.

what matters is the domain of definition.

suppose instead of being defined on the ordinary xyz coords of flat space, or txyz coords of flat 4D the fields are functions defined on a Lie group like SU(2).

Or some other Lie group, call it G.

Or on a cartesian product of N copies of G. Like GxGxG...xG.

Then it is a group type of field theory.

 

[Schreiberdk]

That really makes quite good sense, except: Does a group form a "space" in the same way ordinary coordinate-systems does (sort of the same way a Hilbert space works)?

\Schreiber

 

[marcus]

That really makes quite good sense, except: Does a group form a "space" in the same way ordinary coordinate-systems does (sort of the same way a Hilbert space works)?

\Schreiber

Other people may answer that question differently. Here is how I think of it.

In GFT you think of a finite simplification of space that is just a graph of nodes and links.

a finite set of locations joined by paths.

and a GEOMETRY is a set of Lie-group labels on the links that say what happens to you when you move along the link. do you get rotated? if so how? What transformation happens to you as you run from this place to that place?

so a point in GxGxG...xG is an N-tuple of group elements which label the graph and determine its shape (a "position" in the space of all shapes that graph could have, or a "configuration" of the graph)

so a FUNCTION defined on G^N is like a WAVEFUNCTION for a particle. except that instead of being defined on the real line, on the possible positions of the particle, it is defined on the space of possible geometries of this simplified N-world, this graph with N links.

Or let me not say wavefunction, let me call it a quantum state of geometry.

I am talking very loosely.

anyway the functions defined on G^N make a HILBERTSPACE. There are some technicalities like making sure they are square integrable, and factoring out some redundancy. (Two N-tuples can describe the same experience of running around in the graph if they are offset copies of each other in a certain sense.)

Then you have to take the limit as N --> ∞

But instead of taking the limit, let us just think of this finite world of the graph, and its geometry. A truncation of the infinite degrees of freedom that realworld geometry has. (or maybe it doesn't really). So keep N finite.

Then G^N = GxGx...xG is like the world of geometries, the geometric possibilities.
So instead of doing field theory on space time = txyz
we do field theory on the world of possible configurations = G^N.

This is a sloppy impressionistic introduction. Someone else can perhaps improve.

There is a Loops 2011 talk by Dan Oriti on GFT. The video should be available.
http://www.iem.csic.es/loops11/

A more direct link:
http://loops11.iem.csic.es/loops11/...velopments&catid=35:plenary-lectures&Itemid=1

(This was a template I got from Unusualname: &view=article&id=76%3Acarlo-rovelli-the-covariant-version-of-loop-quantum-gravity-definition-of-the-theory-results-open-problems&catid=35%3Aplenary-lectures&Itemid=1
which can be used to construct direct links.)

 

[tom.stoer]

Does a group form a "space" in the same way ordinary coordinate-systems does?

A Lie group is a topological manifold, so locally you can introduce coordinate patches as for any other topological space. Think about the Euler rotations angles as coordinates for SO(3).