[Paper Review] Observables in quantum gravity

Alejandro Pérez, Carlo Rovelli|arXiv (Cornell University)|Apr 12, 2001

Noncommutative and Quantum Gravity Theories19 references3 citations

TL;DR

This paper proposes W functions—gauge-invariant, diffeomorphism-invariant observables in quantum gravity that generalize n-point functions from quantum field theory. They bridge canonical (loop quantum gravity) and covariant (spinfoam) formulations by encoding physical transition amplitudes between quantized geometries, and under a positivity condition, allow full reconstruction of the physical Hilbert space via the GNS construction.

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.

Motivation & Objective

• To identify a complete set of physical observables in non-perturbative quantum gravity that are gauge-invariant and diffeomorphism-invariant.
• To establish a connection between the canonical (loop quantum gravity) and covariant (spinfoam) formulations of quantum gravity through a common observable framework.
• To demonstrate that W functions—generalized n-point functions—can reconstruct the physical Hilbert space via the GNS construction under a positivity condition.
• To show that perturbative expansions of group field theories yield W functions that correspond to sum-over-4-geometries in spinfoam models.
• To clarify the physical interpretation of transition amplitudes between discrete quantum geometries, analogous to particle amplitudes in QFT.

Proposed method

• Define W functions as functionals W(s) over an algebra A of abstract spin networks, representing transition amplitudes between eigenstates of three-geometry with fixed quanta of geometry.
• Establish that W functions are fully gauge invariant and scalar under four-dimensional diffeomorphisms, ensuring physical observability.
• Use the GNS construction to reconstruct the physical Hilbert space from W functions, provided a positivity condition holds.
• Link W functions to spin foam models by showing they arise as n-point functions in group field theories, which provide a perturbative definition of dynamics.
• Demonstrate that in a free model (with no gravitational vertex), W functions satisfy positivity and allow explicit construction of the physical Hilbert space.
• Draw analogy to Wightman distributions in QFT, showing that the core idea of reconstructing QFT from n-point functions extends to generally covariant quantum gravity.

Experimental results

Research questions

• RQ1 Can a complete set of physical observables be defined in quantum gravity that are gauge-invariant and diffeomorphism-invariant?
• RQ2 How can the canonical and covariant formulations of quantum gravity be connected through a common observable structure?
• RQ3 Under what conditions can the physical Hilbert space of quantum gravity be reconstructed from n-point functions?
• RQ4 What is the physical interpretation of transition amplitudes between discrete quantum geometries labeled by spin networks?
• RQ5 How do W functions in group field theory relate to spinfoam amplitudes and sum-over-4-geometries?

Key findings

• W functions are well-defined, diffeomorphism-invariant observables that represent transition amplitudes between quantum states of geometry labeled by spin networks.
• The W functions generalize the three-geometry to three-geometry amplitude to arbitrary n-point functions, analogous to Wightman distributions in QFT.
• In a free model (with no gravitational vertex), the W functions satisfy the required positivity condition, allowing explicit reconstruction of the physical Hilbert space via the GNS construction.
• Perturbative expansions of group field theories yield W functions that correspond to sum-over-4-geometries, providing a direct link between field theory and spinfoam models.
• The W functions serve as the natural n-point functions in the field-theoretical formulation of spinfoam models, unifying canonical and covariant approaches.
• The framework allows a background-independent reconstruction of quantum gravity dynamics from observable quantities, extending Wightman's program to generally covariant theories.