Spin Networks for Non-Compact Groups

TL;DR

This paper extends the concept of spin networks to non-compact groups, crucial for Lorentzian gravity models, by constructing a suitable measure, Hilbert space, and defining spin network states for groups like SL(2,R) and SL(2,C).

Contribution

It introduces a framework for defining gauge invariant spin network states for non-compact groups, addressing the challenges of quotient spaces and measure construction.

Findings

Constructed a natural measure and Hilbert space for non-compact gauge groups.

Defined spin network states as eigenvectors of commuting operators.

Applied the framework to groups SL(2,R) and SL(2,C).

Abstract

Spin networks are natural generalization of Wilson loops functionals. They have been extensively studied in the case where the gauge group is compact and it has been shown that they naturally form a basis of gauge invariant observables. Physically the restriction to compact gauge group is enough for the study of Yang-mills theories, however it is well known that non-compact groups naturally arise as internal gauge groups for Lorentzian gravity models. In this context a proper construction of gauge invariant observables is needed. The purpose of this work is to define the notion of spin network states for non-compact groups. We first built, by a careful gauge fixing procedure, a natural measure and a Hilbert space structure on the space of gauge invariant graph connection. Spin networks are then defined as generalized eigenvectors of a complete set of hermitic commuting operators. We show how the delicate issue of taking the quotient of a space by non compact groups can be address in term of algebraic geometry. We finally construct the full Hilbert space containing all spin network states. Having in mind application to gravity we illustrate our results for the groups SL(2,R), SL(2,C).