Bosonic and fermionic statistics in nonperturbative quantum gravity

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.

💡 Research Summary

The paper investigates whether the gravitational field, when quantized non‑perturbatively, must obey purely bosonic statistics, as is usually assumed for a metric tensor field, or whether more general statistics can arise once the principle of general covariance is enforced. Working within the framework of Loop Quantum Gravity (LQG), the authors implement general covariance by demanding invariance under active diffeomorphisms, which in the discrete setting of LQG correspond to automorphisms of the underlying spin‑network graph.

In the canonical formulation of LQG the kinematical Hilbert space K is built from square‑integrable functions of SU(2) holonomies on a fixed graph Γ. After imposing the Gauss constraint (SU(2) gauge invariance) one obtains the space (\bar K). The diffeomorphism constraint is then realized as invariance under graph automorphisms; physically distinct states are equivalence classes under the action of the automorphism group Aut(Γ). A generic kinematical state |Ψ⟩∈K can be written as a group‑averaged superposition of a non‑invariant seed state |(\bar Ψ)⟩, i.e.
\