QSD VI : Quantum Poincar\'e Algebra and a Quantum Positivity of Energy Theorem for Canonical Quantum Gravity

TL;DR

This paper quantizes the Poincaré generators in canonical quantum gravity, establishing a densely defined, self-adjoint ADM energy operator and proving a quantum positivity of energy theorem, challenging previous stability speculations.

Contribution

It introduces a quantum version of the Poincaré algebra in canonical quantum gravity and proves a positivity of energy theorem within this framework.

Findings

The ADM energy operator is densely defined, symmetric, and essentially self-adjoint.

A quantum positivity of energy theorem analogous to the classical one is established.

The quantum symmetry algebra faithfully reproduces the classical algebra.

Abstract

We quantize the generators of the little subgroup of the asymptotic Poincar\'e group of Lorentzian four-dimensional canonical quantum gravity in the continuum. In particular, the resulting ADM energy operator is densely defined on an appropriate Hilbert space, symmetric and essentially self-adjoint. Moreover, we prove a quantum analogue of the classical positivity of energy theorem due to Schoen and Yau. The proof uses a certain technical restriction on the space of states at spatial infinity which is suggested to us given the asymptotically flat structure available. The theorem demonstrates that several of the speculations regarding the stability of the theory, recently spelled out by Smolin, are false once a quantum version of the pre-assumptions underlying the classical positivity of energy theorem is imposed in the quantum theory as well. The quantum symmetry algebra corresponding to the generators of the little group faithfully represents the classical algebra.