Loop space representation of quantum fermions and gravity

Abstract

We study the quantum fermions + gravity system, that is, the gravitational counterpart of QED. We start from the standard Einstein-Weyl theory, reformulated in terms of Ashtekar variables; and we construct its non-perturbative quantum theory by extending the loop representation of general relativity. To this aim, we construct the fermion equivalent of the loop variables, and we define the quantum theory as a representation of their Poisson algebra. Not surprisingly, fermions can be accounted for in the loop representation simply by including open curves into loop space, as expected from lattice Yang-Mills theory. We explicitly construct the diffeomorphism and Hamiltonian operators. The first can be fully solved as in pure gravity. The second is constructed by using a background independent regularization technique. The theory retains the clean geometrical features of pure quantum gravity. In particular, the Hamiltonian constraint admits the same simple geometrical interpretation as its pure gravity counterpart: it is the operator that shifts curves along themselves (shift operator). Quite surprisingly, we believe, this simple action codes the full dynamics of the interacting fermion-gravity theory. To unravel the dynamics of the theory we study the evolution of the fermion-gravity system in the physical time defined by an additional coupled (clock-) scalar field. We explicitly construct the Hamiltonian operator that evolves the system in this physical time. We show that this Hamiltonian is finite, diffeomorphism invariant, and has a simple geometrical action confined to the intersections and the end points of the “loops”. The quantum theory of fermions + gravity evolving in the clock time is finally given by the combinatorial and geometrical action of this Hamiltonian on a set of graphs with a finite number of end points. This geometrical action defines the topological Feynman rules of the theory.