The Chern-Simons invariant as the natural time variable for classical and quantum cosmology

Abstract

We propose that the Chern-Simons invariant of the Ashtekar-Sen connection (or its imaginary part in the Lorentzian case) is the natural internal time coordinate for classical and quantum cosmology. The reasons for this are: 1) It is a function on the gauge and di eomorphism invariant con guration space, whose gradient is orthogonal to the two p h ysical degrees of freedom, in the metric de ned by the Ashtekar formulation of general relativity.2) The imaginary part of the Chern-Simons form reduces in the limit of small cosmological constant, , and solutions close to DeSitter spacetime, to the York extrinsic time coordinate. 3) Small matter-eld excitations of the Chern-Simons state satisfy, b y virtue of the quantum constraints, a functional Schroedinger equation in which the matter elds evolve o n a D eSitter background in the Chern-Simons time. We then propose this is the natural vacuum state of the theory for 6 = 0. 4) This time coordinate is periodic on the Euclidean con guration space, due to the large gauge transformations, which means that physical expectation values for all states in non-perturbative quantum gravity will satisfy the KMScondition, and may then be interpreted as thermal states. We also nd evidence that the theory with 6 = 0 is unstable as the cosmological constant term contributes third time derivatives to the Hamiltonian constraint. Finally, forms for the physical hamiltonian and inner product are suggested and a new action principle for general relativity, as a geodesic principle on the connection superspace, is found. smolin@phys.psu.edu y soo@phys.psu.edu 1

On the role of conformal three-geometries in the dynamics of general relativity

“…Recently Smolin and Soo argued that since the proper arena of the dynamics is the phase space rather than the spacetime and in a canonical quantum theory the carrying space of the wave functions is the configuration space, we should find a natural time variable in the configuration space and not in the spacetime. For such a natural time variable in the configuration space they suggested the imaginary part of the Chern-Simons functional built from the complex Ashtekar connection [13]. Although the quantum dynamics must be formulated in the configuration space (or on the phase space endowed with an appropriate polarization), we think that in the classical theory time in the phase space or configuration space (i.e.…”

Quantum Gravity and Inflation

“…These reduce, in the limit of vanishing slow role parameter,V /V , to the Chern-Simons invariant of the Ashtekar connection. This is good, as the latter is known to be the Hamilton-Jacobi function for deSitter spacetime [9,13,12,14]. By exponentiating the actions of these solutions, one obtains a semiclassical state that reduces in the same limit to the Kodama state.…”

“…Furthermore, for constant cosmological constant, there is an exact solution to the quantum constraints that define the full quantum general relativity, discovered by Kodama [9], which has both an exact Planck scale description and a semiclassical interpretation in terms of deSitter spacetime. While there are open issues of interpretation concerning this state [10,11,19], it is also true that it can be used as the basis of both non-perturbative and semiclassical caclulations [12,13,14]. Furthermore, exact results in the loop representation have made possible an understanding of the temperature and entropy of deSitter spacetime [12,14] in terms of the kinematics of the quantum gravitational field.…”

My Journey Into the Physics of David Finkelstein

“…Under a large gauge transformation characterized by winding number n, Y CS → Y CS + 4π 2 n [15], and the Chern-Simons state transforms as CS (A) → e inθ CS (A) [6]. The inclusion of matter perturbations reproduce standard quantum field theory on de Sitter background [16], while linearizing the quantum theory one recovers long-wavelength gravitons on de Sitter [11]. In this sense the Chern-Simons state is a genuine ground state of the theory.…”