### Abstract

Unitarity is a difficult concept to implement in canonical quantum gravity because of state non-normalizability and the problem of time. In this work, we take a realist approach based on pilot-wave theory to address this issue in the Ashtekar formulation of the Wheeler-de Witt equation. We use the postulate of a definite configuration in the theory to define a global time for the gravitational-fermionic system recently discussed in (Phys. Rev. D 106.10 (2022): 106012), by parameterizing a variation of a Weyl-spinor that depends on the Kodama state. The total Hamiltonian constraint yields a time-dependent Schrodinger equation, without semi-classical approximations, which we use to derive a local continuity equation over the configuration space. We implement the reality conditions at the level of the guidance equation, and obtain a real spin-connection, extrinsic curvature and triad along the system trajectory. The non-normalizable Kodama state is naturally factored out of the full quantum state in the conserved current density, opening the possibility for quantum-mechanical unitarity. We also give a pilot-wave generalisation of the notion of unitarity applicable to non-normalizable states, and show the existence of equilibrium density for our system. Lastly, we find unitary states in mini-superspace by finding an approximate solution to the Hamiltonian constraint.

###### Associate Professor of Physics

Researches quantum information, computation, and foundations.