Field theories over an homogenousmomentum space:

symmetries and non-commutative space-times

Florian Girellitogether with E. Livine

University of Sydney

Work in progress: all comments are welcome!

Group field theories for spinfoams

A group field theory (GFT) is a (often non-local) scalar field theory over a (product of) Lie group

A GFT is a generalization of matrix models.

A GFT is a tool to generate spinfoams: each Feynman diagram is a spinfoam, ie a 2-complex decorated by representations of G

Example: GFT for 3d BF

Field theory on group

Example: GFT for Barrett-Crane

Field theory on coset space

We have a field defined either on

A group G

A coset G/H

Key idea: Interpret the group or the coset as momentum space

If this is a field theory study it as a field theory!

Symmetry analysis Canonical quantization Renormalisation analysis ….

Field theory with a group as momentum space

For group field theories defined on a group G, the symmetries are encoded in different types of quantum groups:

Cf Talk in Beijing at LOOPS 09

The general bicrossproduct construction by Majid (to appear soon)

Group field theory on AN(p) ie field theory on kappa Minkowski or kappa’ Minkowski.

Other examples could be constructed for example from the factorization(work in progress)

The Drinfeld double: Group field on SU(2) is a well known example. (Freidel, Livine, Noui, Mourad, Krasnov,…)

In both cases, the constructed dual non-commutative space-time is of the Lie algebra type.

Field theory with a coset as momentum space

G/H is momentum space. I choose the specific example , ie the (3d) momentum space is the hyperboloid

The generators are given by the boosts

Momentum is a coordinate on the hyperboloid.

The addition of momenta is given by the group product

The addition is non-commutative and non-associative in general!

Same idea as in Special Relativity

: relativistic speed : 3d speed

This addition is associative only when vectors are colinenar!

Thomas precession

Scalar field as a function on G/H

Dirac delta function

We introduce a convolution product inherited from the coset structure:

To cook a field action defined in momentum space, we needA scalar field, The Dirac delta function for momentum conservation How they and their product transform under Lorentz and translations, A measure invariant under Lorentz group, A propagator and interaction term constructed from convolution product.

The action of symmetries is defined as:

Lorentz transformations Translations

Adjoint action

Need space-time ….

Convolution product is transforming as

We introduce a propagator invariant under action of H,

We can introduce an action, invariant under H

Default notation:

We introduce the plane-wave and a star product between plane-waves to encode the non-trivial momentum addition

Snyder space-time

Using the measure on G/H, we define the Fourier transform

The star product is the dual of the convolution product. It is non-associative!

Consider the position operator

This is describing Snyder space-time!!

H transformations in space-time are induced on positions using the Fourier transform

Calculate explicitely the bracket

In particular, we have a linear realization

Performing the Fourier transform, we obtain an action defined on Snyder space-time, invariant under H as well as translations, thanks to the (modified) momentum conservation.

Consider the realization of the translations in momentum space

Things to be done

Differential calculus for Snyder space-time?

What kind of quantum group do we have?

Study the physics of Snyder space-time: non-associativity is tough…

Look at symmetries for the GFT behind the Quantum Gravity models(Snyder symmetry is in the Barrett-Crane model?!)

Things done

I have identified the symmetries for a scalar field theory defined on a coset space, interpreted as momentum space.

I have identified the dual space-time as Snyder space-time.

I have identified a star product to encode Snyder non-commutativity.

Summary and outlook

PHENOMENOLOGY

NON-COMMUTATIVE GEOMETRYLOOP QUANTUM GRAVITYSPINFOAM

GROUP FIELD

THEORY

Fuzzy spheres,

Flat semi-classical space-time:Kappa-Minkowski, Snyder…

Common (mathematical) framework betweenQuantum Gravity and Non-Commutative Geometry!