Quantum Holonomy Theory

TL;DR

Quantum holonomy theory offers a non-perturbative framework for quantum gravity coupled with fermions, deriving classical constraints and fermionic dynamics from a novel algebraic and geometric approach.

Contribution

It introduces a new non-perturbative quantum gravity model based on holonomy-diffeomorphisms and Dirac operators, linking quantum gravity with fermionic fields.

Findings

Classical Hamiltonian constraint emerges in semi-classical limit

Operator constraint algebra reproduces classical relations

States connect Dirac operator expectation to fermionic Hamiltonian

Abstract

We present quantum holonomy theory, which is a non-perturbative theory of quantum gravity coupled to fermionic degrees of freedom. The theory is based on a C*-algebra that involves holonomy-diffeomorphisms on a 3-dimensional manifold and which encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Employing a Dirac type operator on the configuration space of Ashtekar connections we obtain a semi-classical state and a kinematical Hilbert space via its GNS construction. We use the Dirac type operator, which provides a metric structure over the space of Ashtekar connections, to define a scalar curvature operator, from which we obtain a candidate for a Hamilton operator. We show that the classical Hamilton constraint of general relativity emerges from this in a semi-classical limit and we then compute the operator constraint algebra. Also, we find states in the kinematical Hilbert space on which the expectation value of the Dirac type operator gives the Dirac Hamiltonian in a semi-classical limit and thus provides a connection to fermionic quantum field theory. Finally, an almost-commutative algebra emerges from the holonomy-diffeomorphism algebra in the same limit.