2D(1+1) Quantum Gravity. Gravitational Quantum Stationary Hamilton-Jacobi Equation

Abstract

In the present article, we construct a 2D formulation of quantum gravity in the framework of a deterministic theory. In this context, a Quantum stationary Hamilton-Jacobi equation is derived from the Klein- Gordon equation written in the presence of a gravitational field. We show that this equation reduces to the Quantum stationary Hamilton- Jacobi equation when the gravitational field is not present in the 2D time-space. As a second step, we introduce the quantum gravitational Lagrangian for the quantum motion of a particle moving in the presence of a gravitational field. We, deduce the relationship between the gravitational quantum conjugate momentum and the velocity of the particle.