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Spin Networks for Non-Compact Groups

Laurent Freidel, Etera R. Livine

TL;DR

The paper develops a framework to define and analyze spin-network states when the gauge group is non-compact, with a focus on Lorentzian gravity where groups like $SL(2,\mathbb{C})$ and $SL(2,\mathbb{R})$ arise. It builds a gauge-fixed, invariant measure on the quotient spaces $A_h=G^h/Ad(G)$ via a two-step gauge fixing and algebraic-geometry tools, enabling a Hilbert-space structure for cylindrical gauge-invariant functionals and a Laplacian-based spin-network basis. Explicit constructions are given for rank-one groups, including one-loop and two-petal cases, and the spectrum is connected to edge Casimirs and vertex intertwiners, recovering familiar SU(2) spin networks in appropriate limits. The work culminates in a non-compact Hilbert-space picture with a suggested Fock-space-like structure for the space of connections, revealing a path toward a rigorous quantum-theoretic treatment of non-compact gauge theories and their geometric operators in Lorentzian gravity contexts.

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).

Spin Networks for Non-Compact Groups

TL;DR

The paper develops a framework to define and analyze spin-network states when the gauge group is non-compact, with a focus on Lorentzian gravity where groups like and arise. It builds a gauge-fixed, invariant measure on the quotient spaces via a two-step gauge fixing and algebraic-geometry tools, enabling a Hilbert-space structure for cylindrical gauge-invariant functionals and a Laplacian-based spin-network basis. Explicit constructions are given for rank-one groups, including one-loop and two-petal cases, and the spectrum is connected to edge Casimirs and vertex intertwiners, recovering familiar SU(2) spin networks in appropriate limits. The work culminates in a non-compact Hilbert-space picture with a suggested Fock-space-like structure for the space of connections, revealing a path toward a rigorous quantum-theoretic treatment of non-compact gauge theories and their geometric operators in Lorentzian gravity contexts.

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).

Paper Structure

This paper contains 31 sections, 12 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Connection space and Cylindrical functions
  3. Gauge Fixing Cylindrical Functions
  4. Constructing Flowers
  5. Examples
  6. Reductive groups, Quotient space and algebraic geometry
  7. Reductive Lie group and Cartan subalgebra
  8. Orbit space invariant theory and algebraic geometry
  9. construction of the measure on $A_h$
  10. Construction of the measure on $A_2$
  11. Case of a unique Cartan subgroup $H$
  12. The case of many Cartan subgroups
  13. The measure on $A_h$
  14. Measure on an arbitrary graph
  15. Spin Networks States
  16. ...and 16 more sections

Key Result

Proposition 1

$G_{2}$ is a dense subset of $G\times G$, its complement is of Haar measure zero and $G_{2}/Ad(G)$ is a geometric quotient when $G$ is of rank one. Therefore, $A_{2}=G_{2}/Ad(G)$ is an Haussdorf manifold of dimension $dimG$ which separates rational functions and it is the base manifold of a homogene

Theorems & Definitions (17)

  • Proposition 1
  • Definition 1
  • Proposition 2
  • Theorem 1
  • Proposition 3
  • Definition 2
  • Definition 3
  • Proposition 4
  • Theorem 2
  • Definition 4
  • ...and 7 more