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Observables in quantum gravity

Alejandro Perez, Carlo Rovelli

TL;DR

This work introduces $W$ functions, or $n$-net functions, as gauge-invariant observables in quantum gravity that encode transitions between spin-network eigenstates and generalize $n$-point data for geometry. It develops both a canonical formulation, where $W(s,s')$ is the matrix element of the Hamiltonian-constraint projector, and a covariant, group-field-theory formulation, where $W(s)$ arises as gauge-invariant correlators and the perturbative expansion sums over spin foams (discretized 4-geometries). Under a positivity condition, the GNS reconstruction recovers the physical Hilbert space and an operator algebra from $W$, thereby linking loop quantum gravity to spin-foam dynamics. The paper also presents a solvable free model demonstrating positivity and explicit Hilbert-space construction, and discusses the role of the reality of $P$ and possible complex generalizations, outlining future directions for a fully background-independent quantum gravity framework.

Abstract

We study a family of physical observable quantities in quantum gravity. We denote them W functions, or n-net functions. They represent transition amplitudes between quantum states of the geometry, are analogous to the n-point functions in quantum field theory, but depend on spin networks with n connected components. In particular, they include the three-geometry to three-geometry transition amplitude. The W functions are scalar under four-dimensional diffeomorphism, and fully gauge invariant. They capture the physical content of the quantum gravitational theory. We show that W functions are the natural n-point functions of the field theoretical formulation of the gravitational spin foam models. They can be computed from a perturbation expansion, which can be interpreted as a sum-over-four-geometries. Therefore the W functions bridge between the canonical (loop) and the covariant (spinfoam) formulations of quantum gravity. Following Wightman, the physical Hilbert space of the theory can be reconstructed from the W functions, if a suitable positivity condition is satisfied. We compute explicitly the W functions in a "free" model in which the interaction giving the gravitational vertex is shut off, and we show that, in this simple case, we have positivity, the physical Hilbert space of the theory can be constructed explicitly and the theory admits a well defined interpretation in terms of diffeomorphism invariant transition amplitudes between quantized geometries.

Observables in quantum gravity

TL;DR

This work introduces functions, or -net functions, as gauge-invariant observables in quantum gravity that encode transitions between spin-network eigenstates and generalize -point data for geometry. It develops both a canonical formulation, where is the matrix element of the Hamiltonian-constraint projector, and a covariant, group-field-theory formulation, where arises as gauge-invariant correlators and the perturbative expansion sums over spin foams (discretized 4-geometries). Under a positivity condition, the GNS reconstruction recovers the physical Hilbert space and an operator algebra from , thereby linking loop quantum gravity to spin-foam dynamics. The paper also presents a solvable free model demonstrating positivity and explicit Hilbert-space construction, and discusses the role of the reality of and possible complex generalizations, outlining future directions for a fully background-independent quantum gravity framework.

Abstract

We study a family of physical observable quantities in quantum gravity. We denote them W functions, or n-net functions. They represent transition amplitudes between quantum states of the geometry, are analogous to the n-point functions in quantum field theory, but depend on spin networks with n connected components. In particular, they include the three-geometry to three-geometry transition amplitude. The W functions are scalar under four-dimensional diffeomorphism, and fully gauge invariant. They capture the physical content of the quantum gravitational theory. We show that W functions are the natural n-point functions of the field theoretical formulation of the gravitational spin foam models. They can be computed from a perturbation expansion, which can be interpreted as a sum-over-four-geometries. Therefore the W functions bridge between the canonical (loop) and the covariant (spinfoam) formulations of quantum gravity. Following Wightman, the physical Hilbert space of the theory can be reconstructed from the W functions, if a suitable positivity condition is satisfied. We compute explicitly the W functions in a "free" model in which the interaction giving the gravitational vertex is shut off, and we show that, in this simple case, we have positivity, the physical Hilbert space of the theory can be constructed explicitly and the theory admits a well defined interpretation in terms of diffeomorphism invariant transition amplitudes between quantized geometries.

Paper Structure

This paper contains 7 sections, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The 2-net function $W(s,s')$ in canonical quantum gravity
  3. Reality of $P$ and $W$ functions
  4. $W(s)$ in field theories over a group
  5. A "free" theory
  6. Complex $P$
  7. Conclusion

Figures (3)

  • Figure 1: The three-geometry to three-geometry transition amplitude $W(s,s')$ between two connected three-geometries.
  • Figure 2: The transition amplitude between a doubly connected and a connected three-geometry.
  • Figure 3: The reality condition on $P$.