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Loop Quantum Gravity Vacuum with Nondegenerate Geometry

Tim Koslowski, Hanno Sahlmann

TL;DR

The Ashtekar–Lewandowski vacuum in loop quantum gravity is a degenerate, no-geometry state. Koslowski and Sahlmann develop nondegenerate vacuum representations for both the holonomy-flux (Lie) algebra and the Weyl (C*)-algebra, introducing a background geometry E^(0) that yields geometry eigenstates and condensate-like excitations. They show that geometric operators acquire background-dependent shifts and construct automorphism-invariant frameworks via direct sums over symmetry orbits and group averaging. This geometric-condensate perspective opens routes to effective field theory on quantum geometries, potential links to Loop Quantum Cosmology, and broader generalizations of LQG states and dynamics.

Abstract

In loop quantum gravity, states of the gravitational field turn out to be excitations over a vacuum state that is sharply peaked on a degenerate spatial geometry. While this vacuum is singled out as fundamental due to its invariance properties, it is also important to consider states that describe non-degenerate geometries. Such states have features of Bose condensate ground states. We discuss their construction for the Lie algebra as well as the Weyl algebra setting, and point out possible applications in effective field theory, Loop Quantum Cosmology, as well as further generalizations.

Loop Quantum Gravity Vacuum with Nondegenerate Geometry

TL;DR

The Ashtekar–Lewandowski vacuum in loop quantum gravity is a degenerate, no-geometry state. Koslowski and Sahlmann develop nondegenerate vacuum representations for both the holonomy-flux (Lie) algebra and the Weyl (C*)-algebra, introducing a background geometry E^(0) that yields geometry eigenstates and condensate-like excitations. They show that geometric operators acquire background-dependent shifts and construct automorphism-invariant frameworks via direct sums over symmetry orbits and group averaging. This geometric-condensate perspective opens routes to effective field theory on quantum geometries, potential links to Loop Quantum Cosmology, and broader generalizations of LQG states and dynamics.

Abstract

In loop quantum gravity, states of the gravitational field turn out to be excitations over a vacuum state that is sharply peaked on a degenerate spatial geometry. While this vacuum is singled out as fundamental due to its invariance properties, it is also important to consider states that describe non-degenerate geometries. Such states have features of Bose condensate ground states. We discuss their construction for the Lie algebra as well as the Weyl algebra setting, and point out possible applications in effective field theory, Loop Quantum Cosmology, as well as further generalizations.

Paper Structure

This paper contains 12 sections, 2 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Algebras underlying Loop Quantum Gravity
  3. Holonomy-flux algebra
  4. Holonomy-flux algebra
  5. $\boldsymbol{C^*}$-algebra (Weyl algebra)
  6. F/LOST representation theorem
  7. Nondegenerate vacua and representations
  8. Nondegenerate vacua and representations
  9. Representations of the holonomy-flux algebra with nondegenerate background geometry
  10. Nondegenerate representations of the Weyl algebra
  11. Physical interpretation and comparison
  12. Outlook

Key Result

Proposition 1

The standard regularization and quantization procedures for area and volume operators can be applied to the new representations and lead to well defined operators. We find with $V^{\rm vac}$, $A^{\rm vac}$ the geometric operators in the vacuum representation, and $V^{(0)}_R$, $A^{(0)}_S$ the classical values in the background geometry.

Theorems & Definitions (2)

  • Proposition 1: Sahlmann:2010hn
  • Proposition 2