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

TL;DR

The paper identifies the Chern-Simons invariant of the Ashtekar-Sen connection as a natural internal time variable for both classical and quantum cosmology, deriving a geodesic formulation of general relativity on the connection configuration space. In the Euclidean sector, τ_CS acts as a timelike coordinate with a conformal structure that isolates the physical degrees of freedom, while in the small-Λ limit its imaginary part reduces to a York-type extrinsic time, enabling a functional Schrödinger evolution for matter on a DeSitter background with CS time as the clock. A canonical transformation yields a Hamiltonian that evolves the gravitational DOF in CS time, and the Kodama CS state is argued to be a natural vacuum for Λ ≠ 0, with perturbations around it behaving like quantum fields on DeSitter space. The Euclidean theory exhibits a periodic CS time, implying that nonperturbative quantum gravity is intrinsically thermal through a KMS condition, with the semiclassical DeSitter case reproducing the Hawking-like temperature T_cosmological = (1/(2π)) sqrt(Λ/(3G)). The work also outlines an inner product on the CS time slice and analyzes perturbations near the conformally self-dual sector, concluding with open questions about reality conditions and global structure of the diffeomorphism-invariant configuration space.

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, $Λ$, 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 $Λ\neq 0$. 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.