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

• Schreiberdk
In summary: But when you look at the whole group, the angles are all the same. So the rotations are a group action, and they act on a single space.
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

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.

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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

Schreiberdk said:
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 GN 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 GN 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 GN = 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 = GN.

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.)

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Schreiberdk said:
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).

## 1. What is Group Field Theory (GFT)?

Group Field Theory is a mathematical framework that combines elements of quantum field theory and group theory to study the dynamics of space-time at a fundamental level. It is a higher-dimensional generalization of matrix models and loop quantum gravity, and is often used to describe the quantum nature of space-time in theories of quantum gravity.

## 2. What are the main ideas behind GFT?

The main idea behind GFT is to describe space-time as a network of interacting building blocks, called spin networks, which are made up of elementary building blocks called atoms of space. These atoms of space are associated with individual group elements, and the interactions between them are described by group-theoretic operations.

## 3. What is the formalism used in GFT?

The formalism used in GFT is based on a field theoretic approach, where space-time is described as a field on a higher-dimensional manifold. The field is then quantized using techniques from quantum field theory, and the dynamics of the field is described by a set of equations called the GFT equations.

## 4. What are the origins of GFT?

The origins of GFT can be traced back to the efforts to reconcile general relativity and quantum mechanics, and the search for a theory of quantum gravity. It has been heavily influenced by loop quantum gravity and spin foam models, and has also drawn inspiration from matrix models and non-commutative geometry.

## 5. What are some potential applications of GFT?

GFT has potential applications in various fields, including quantum gravity, cosmology, and condensed matter physics. It can also be used to study the emergence of space-time from a more fundamental quantum structure, and to investigate the dynamics of black holes and other extreme phenomena in the universe.

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