Bosonic and fermionic statistics in nonperturbative quantum gravity

TL;DR

This work investigates whether gravity in a nonperturbative quantum gravity setting must obey bosonic statistics. By implementing general covariance through invariance under active diffeomorphisms in loop quantum gravity, it shows that the kinematical space for geometry on a graph supports bosonic, fermionic, and mixed statistics depending on the graph and spin configuration, with explicit demonstrations on dipole, pentagram, and complete graphs. The main contribution is the explicit appearance of fermionic and mixed sectors arising from automorphism-invariance, challenging the assumption that gravitational quanta are strictly bosonic in the nonperturbative regime. This has implications for the formulation of group field theories and the statistical mechanics of quantum geometry, suggesting that gravity may exhibit richer statistical behavior than previously assumed, subject to further constraints from dynamics via the Hamiltonian constraint.

Abstract

The relation between spin and statistics in quantum field theory relies on Poincaré invariance, a symmetry that is lost in the presence of a gravitational field, and replaced in general relativity by the principle of general covariance. In a nonperturbative approach to quantum gravity, beyond the picture of gravitational perturbations propagating on a flat background, it is an open question whether the gravitational field must still satisfy a bosonic statistics. By implementing the principle of general covariance through the requirement of invariance under active diffeomorphisms in loop quantum gravity, we find that the space of kinematical states of the gravitational field includes not only bosonic states, but also subspaces of fermionic and mixed statistics.