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Fock representations from U(1) holonomy algebras

Madhavan Varadarajan

Abstract

We revisit the quantization of U(1) holonomy algebras using the abelian C* algebra based techniques which form the mathematical underpinnings of current efforts to construct loop quantum gravity. In particular, we clarify the role of ``smeared loops'' and of Poincare invariance in the construction of Fock representations of these algebras. This enables us to critically re-examine early pioneering efforts to construct Fock space representations of linearised gravity and free Maxwell theory from holonomy algebras through an application of the (then current) techniques of loop quantum gravity.

Fock representations from U(1) holonomy algebras

Abstract

We revisit the quantization of U(1) holonomy algebras using the abelian C* algebra based techniques which form the mathematical underpinnings of current efforts to construct loop quantum gravity. In particular, we clarify the role of ``smeared loops'' and of Poincare invariance in the construction of Fock representations of these algebras. This enables us to critically re-examine early pioneering efforts to construct Fock space representations of linearised gravity and free Maxwell theory from holonomy algebras through an application of the (then current) techniques of loop quantum gravity.

Paper Structure

This paper contains 13 sections, 100 equations.

Table of Contents

  1. 1. Introduction
  2. 2. Review of the construction and representation theory of $\overline{\cal{HA}}$.
  3. 3. ${\overline{\cal{HA}}}_r$ and the isomorphism $I_r$
  4. 4. Fock representation from ${\overline{\cal{HA}}}_r$
  5. 5. Induced representations through $I_r$.
  6. 6. Discussion
  7. Appendix