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On the superselection theory of the Weyl algebra for diffeomorphism invariant quantum gauge theories

Hanno Sahlmann, Thomas Thiemann

TL;DR

This work addresses the kinematical question of whether the Ashtekar–Lewandowski (AL) diffeomorphism-invariant representation of the holonomy–flux Weyl algebra is unique under mild extra assumptions. It introduces a C*-algebra closely related to the Weyl algebra and develops the general representation theory, showing irreducible, diffeomorphism-invariant representations decompose into measures on the generalized connection space and are constrained by the natural pull-back action of diffeomorphisms and gauge transformations. Under ir-reducibility, weak continuity of the Weyl-parameterized fluxes, and natural symmetry actions, the AL measure $\mu_0$ is forced and the representation space collapses to a single copy of ${\cal H}_0$, implying unitary equivalence to the AL representation; a streamlined proof is provided under stronger gauge-invariance assumptions. These results indicate that background independence and diffeomorphism invariance strongly constrain quantization freedom in loop quantum gravity and support the AL framework as a canonical choice for the diffeomorphism-invariant sector.

Abstract

Much of the work in loop quantum gravity and quantum geometry rests on a mathematically rigorous integration theory on spaces of distributional connections. Most notably, a diffeomorphism invariant representation of the algebra of basic observables of the theory, the Ashtekar-Lewandowski representation, has been constructed. This representation is singled out by its mathematical elegance, and up to now, no other diffeomorphism invariant representation has been constructed. This raises the question whether it is unique in a precise sense. In the present article we take steps towards answering this question. Our main result is that upon imposing relatively mild additional assumptions, the AL-representation is indeed unique. As an important tool which is also interesting in its own right, we introduce a C*-algebra which is very similar to the Weyl algebra used in the canonical quantization of free quantum field theories.

On the superselection theory of the Weyl algebra for diffeomorphism invariant quantum gauge theories

TL;DR

This work addresses the kinematical question of whether the Ashtekar–Lewandowski (AL) diffeomorphism-invariant representation of the holonomy–flux Weyl algebra is unique under mild extra assumptions. It introduces a C*-algebra closely related to the Weyl algebra and develops the general representation theory, showing irreducible, diffeomorphism-invariant representations decompose into measures on the generalized connection space and are constrained by the natural pull-back action of diffeomorphisms and gauge transformations. Under ir-reducibility, weak continuity of the Weyl-parameterized fluxes, and natural symmetry actions, the AL measure is forced and the representation space collapses to a single copy of , implying unitary equivalence to the AL representation; a streamlined proof is provided under stronger gauge-invariance assumptions. These results indicate that background independence and diffeomorphism invariance strongly constrain quantization freedom in loop quantum gravity and support the AL framework as a canonical choice for the diffeomorphism-invariant sector.

Abstract

Much of the work in loop quantum gravity and quantum geometry rests on a mathematically rigorous integration theory on spaces of distributional connections. Most notably, a diffeomorphism invariant representation of the algebra of basic observables of the theory, the Ashtekar-Lewandowski representation, has been constructed. This representation is singled out by its mathematical elegance, and up to now, no other diffeomorphism invariant representation has been constructed. This raises the question whether it is unique in a precise sense. In the present article we take steps towards answering this question. Our main result is that upon imposing relatively mild additional assumptions, the AL-representation is indeed unique. As an important tool which is also interesting in its own right, we introduce a C*-algebra which is very similar to the Weyl algebra used in the canonical quantization of free quantum field theories.

Paper Structure

This paper contains 6 sections, 7 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General Representation Theory of $\mathfrak{A}$
  4. Diffeomorphism Invariant Representations of $\mathfrak{A}$ and a Uniqueness Theorem
  5. A simpler proof under stronger assumptions
  6. Construction of more general $C^\infty-$vectors for the Fluxes

Key Result

Lemma 2.1

i) The vector fields $R^j_{x,[e]}$ satisfy the following commutation relations where $[\tau_j,\tau_k]=f_{jk}\;^l \tau_l$ defines the structure constantsSince $G$ is a compact, connected Lie group, we have $G/D\cong A\times S$ where $D$ is a central discrete subgroup and $A,S$ are Abelian and semisimple Lie groups respectively. Indices are dragged w.r.t. the Cartan-Killing metr

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.1
  • Definition 3.1
  • Lemma 3.1
  • Definition 4.1
  • Theorem 4.1
  • ...and 5 more