Loop Quantum Gravity with Scalar Field as a Physical Time Variable

Abstract: Loop quantum gravity [1–3] is based on a canonical quantization of general relativity in the Ashtekar formulation, in which the basic variables are the connection Aa and the densitized triad Ea i of the (inverse) spatial metric qab. In this formulation, general relativity is encoded in the Gauss, diffeomorphism, and Hamiltonian constraints, arising from gauge invariance under local rotations of the triad, and under diffeomorphisms tangent and orthogonal to the spatial surfaces of the 3+1 decomposition of spacetime. In the quantum theory, implementation of the Gauss and diffeomorphism constraints is straightforward. Their solution space is spanned by the so-called spin network states, which give a kinematical description of a quantized, discrete spatial geometry. However, describing the dynamics of these states through quantizing the Hamiltonian constraint and looking for its solutions has proven to be technically extremely challenging. An alternative approach to the problem of dynamics in loop quantum gravity is provided by the so-called method of deparametrization, in which we consider gravity coupled to a matter field, and use the matter field as a physical, relational time variable, with respect to which the evolution of the quantum state of the gravitational field is described.