José Antonio Zapata
Centro de Ciencias Matemáticas
UNAM Campus Morelia Morelia
CP 58190 Michoacán, México

Before being here:
Bs Physics UNAM, México
PhD Physics Penn State, USA
Postdoc RRI, India


Renormalization tailored to loop quantization

Let us consider that loop quantization as described in the section "Spin foam models in the continuum" is a formalism able to define quantum field theories on a given spacetime manifold. The definition of such theories involves limits characterizing certain projective measures, and these limits are the natural objects to write from the point of view of the projective techniques inherent to loop quantization.
The missing ingredient was a constructive method for such theories; truly nontrivial examples seamed to be out of reach. On the other hand, the lattice gauge theory developed by Wilson has many similarities with loop quantization, and in its core it has a constructive procedure: Wilson's renormalization procedure. From this point of view, loop quantized theories as defined in the section cited above should be seen as the continuum limit of completely renormalized lattice theories. In the article shown below this point of view is developed and loop quantization is presented as a quantization procedure which needed to be completed. An implementation of Wilson's renormalization is exhibited as the missing part needed to make the quantization procedure operational.

Loop quantization from a lattice gauge theory perspective
Jose A. Zapata, Class.Quant.Grav. 21 (2004) L115-L122

An interacting relativistic quantum field theory
constructed by loop quantization

We had a good idea of how to implement Wilson's renormalization procedure in the context of loop quantization; we only needed to materialize our intuition with a nontrivial example.
Mc Coy, Tracy and Wu proved ground-breaking results on the convergence of the double scaling limit of the n-point functions of the 2d Ising model. These results let us give a concrete example of an interacting quantum field theory defined and constructed by loop quantization techniques.

Loop quantization as a continuum limit
Elisa Manrique, Robert Oeckl, Axel Weber, Jose A. Zapata, Class.Quant.Grav. 23 (2006) 3393-3404