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A Background Independent Algebra in Quantum Gravity

Edward Witten

TL;DR

The paper introduces a background-independent operator product algebra for observers along their worldlines, aiming to make quantum gravity observables independent of a fixed spacetime background. It develops the static patch realization with a maximum-entropy state and proves a tracial property for dressed observables, linking entropy to relative entropy with a universal no-boundary state. A universal no-boundary state is proposed to define entropy across different spacetimes, with a detailed look at de Sitter vacua and the generalized entropy S_gen, up to a universal additive constant. The work further analyzes the role of unnormalizable no-boundary states, the possible von Neumann algebra types (I vs II), and extensions to asymptotic observers via large-N AdS/CFT, where a background-independent algebra emerges from deformation quantization. Together, these ideas offer a framework for observer-centric, background-independent quantum gravity and connect gravitational entropy to relative entropy through a universal no-boundary construct.

Abstract

We propose an algebra of operators along an observer's worldline as a background-independent algebra in quantum gravity. In that context, it is natural to think of the Hartle-Hawking no boundary state as a universal state of maximum entropy, and to define entropy in terms of the relative entropy with this state. In the case that the only spacetimes considered correspond to de Sitter vacua with different values of the cosmological constant, this definition leads to sensible results.

A Background Independent Algebra in Quantum Gravity

TL;DR

The paper introduces a background-independent operator product algebra for observers along their worldlines, aiming to make quantum gravity observables independent of a fixed spacetime background. It develops the static patch realization with a maximum-entropy state and proves a tracial property for dressed observables, linking entropy to relative entropy with a universal no-boundary state. A universal no-boundary state is proposed to define entropy across different spacetimes, with a detailed look at de Sitter vacua and the generalized entropy S_gen, up to a universal additive constant. The work further analyzes the role of unnormalizable no-boundary states, the possible von Neumann algebra types (I vs II), and extensions to asymptotic observers via large-N AdS/CFT, where a background-independent algebra emerges from deformation quantization. Together, these ideas offer a framework for observer-centric, background-independent quantum gravity and connect gravitational entropy to relative entropy through a universal no-boundary construct.

Abstract

We propose an algebra of operators along an observer's worldline as a background-independent algebra in quantum gravity. In that context, it is natural to think of the Hartle-Hawking no boundary state as a universal state of maximum entropy, and to define entropy in terms of the relative entropy with this state. In the case that the only spacetimes considered correspond to de Sitter vacua with different values of the cosmological constant, this definition leads to sensible results.
Paper Structure (13 sections, 86 equations, 2 figures)

This paper contains 13 sections, 86 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A Background-Independent Operator Product Algebra
  3. The Static Patch
  4. The Maximum Entropy State
  5. A Proof of The Tracial Property
  6. Some Further Properties
  7. Entropy And The No Boundary State
  8. $\Psi_{\rm dS}$ and $\Psi_{\rm HH}$
  9. A Universal No Boundary State?
  10. More On The No Boundary State
  11. Unnormalizable States and Unbounded Entropies
  12. The "Type" Of The Von Neumann Algebra
  13. Spacetimes That Do Have Asymptotic Observers

Figures (2)

  • Figure 1: A Penrose diagram for de Sitter space. Time flows upward; the far future is at the top of the diagram and the far past is at the bottom. Coordinates have been chosen so that the observer's worldline is the left edge of the diagram. The region causally accessible to the observer is the static patch, which is shaded green. It is bounded by the past and future horizons of the observer, as shown.
  • Figure 2: A two-sphere $S^D$ containing an "equator" $W\cong S^{D-1}$ orthogonal to a great circle $\gamma_E$. Drawn is the case $D=2$, so $W$ is another great circle. $W$ and $\gamma_E$ intersect at two points and accordingly the continuation of $\gamma_E$ to Lorentz signature has two components.