QSD VI : Quantum Poincaré Algebra and a Quantum Positivity of Energy Theorem for Canonical Quantum Gravity
T. Thiemann
TL;DR
This work addresses how to realize Poincaré symmetry at spatial infinity in canonical quantum gravity by quantizing the little group generators of the asymptotic Poincaré group. It develops two operator orderings for the ADM energy: a densely defined form on the full Hilbert space and a restricted, asymptotically flat, tangle-protected form that yields a nonnegative, discrete spectrum, thereby establishing a quantum positivity of energy theorem. By introducing a quantum Gauss constraint treatment and a boundary-focused energy expression, the authors show energy can be localized to boundary vertices and flows along spin-network edges, offering a quantum interpretation of gravitational energy in loop quantum gravity. The little-group algebra is shown to be anomaly-free and faithfully represents the classical algebra, ensuring that unitary irreducible representations of the Poincaré group can be built from the quantum theory, with the spin-network basis providing a non-linear Fock-like structure for quantum gravity.
Abstract
We quantize the generators of the little subgroup of the asymptotic Poincaré 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.