The Chern-Simons Invariant as the Natural Time Variable for Classical and Quantum Cosmology

TL;DR

This paper introduces the Chern-Simons invariant as a natural internal time variable in classical and quantum cosmology, linking it to gauge invariance, the York time, and thermal states in quantum gravity.

Contribution

It proposes the Chern-Simons invariant as a novel internal time coordinate, with properties connecting classical and quantum cosmology, gauge invariance, and thermality in non-perturbative quantum gravity.

Findings

Chern-Simons invariant is gauge and diffeomorphism invariant.

It reduces to York time near DeSitter spacetime.

Quantum states evolve with Chern-Simons time, satisfying a Schrödinger equation.

Abstract

We propose that the Chern-Simons invariant of the Ashtekar-Sen connection is the natural internal time coordinate for classical and quantum cosmology. The reasons for this are a number of interesting properties of this functional, which we describe here. 1)It is a function on the gauge and diffeomorphism invariant configuration space, whose gradient is orthogonal to the two physical degrees of freedom, in the metric defined by the Ashtekar formulation of general relativity. 2)The imaginary part of the Chern-Simons form reduces in the limit of small cosmological constant, $\Lambda$, and solutions close to DeSitter spacetime, to the York extrinsic time coordinate. 3)Small matter-field excitations of the Chern-Simons state satisfy, by virtue of the quantum constraints, a functional Schroedinger equation in which the matter fields evolve on a DeSitter background in the Chern-Simons time. We then n propose this is the natural vacuum state of the theory for $\Lambda \neq 0$. 4)This time coordinate is periodic on the configuration space of Euclideanized spacetimes, due to the large gauge transformations, which means that physical expectation values for all states in non-perturbative quantum gravity will satisfy the $KMS$ condition, and may then be interpreted as thermal states. 5)Forms for the physical hamiltonians and inner product which support the proposal are suggested, and a new action principle for general relativity, as a geodesic principle on the connection superspace, is found.