The Issue of Time Evolution in Quantum Gravity

Abstract

We discuss the relation between the concept of time and the dynamic structure of quantum gravity. We briefly review the problems of time associated with the standard procedures of gravity quantization. By explicitly utilizing York’s analysis of the geometrodynamic degrees of freedom, and imposing the constraints as expectation value equations, we describe a new procedure of gravity quantization. In particular, this “minimally constrained canonical” quantization procedure leads to a linear Schrödinger equation augmented by the super-Hamiltonian and supermomentum constraints imposed on expectation values. This approach supplies a description of time evolution in quantum geometrodynamics free from the standard problems of time associated with canonical approaches. Furthermore, the theory is applicable to the full theory of general relativity, without the need to impose symmetries or reduce dimensionality. Using this method we arrive at an analytic expression for the quantum evolution of the Kasner cosmology, as well as a numerically generated solution of the quantized Taub model. The key new feature of this approach is that we impose the constraints weakly — instead of the quantum-mechanical wave function satisfying the constraints exactly, it need only satisfy them on average. Thus the constraints only constrain the observables.

Time in Quantum Geometrodynamics

“…As a result, they cannot replace the full set of equations for geometrodynamic evolution. However, the resulting complete system of equations (dynamic equations on conformal superspace and constraint equations) is equivalent to this of the standard geometrodynamics on the superspace of 3-geometries [7]. For the purpose of quantization, we make a transition to the corresponding Schrödinger equation based entirely on dynamics and ignoring the system symmetries…”

“…[6], [7]). The important point is that these problems disappear as soon as the proper object of quantization (true geometrodynamic variables) is chosen.…”

“…The ADM square root quantization procedure is also based entirely on constraints, but in this procedure the set of canonical variables is split in two subsets, embedding variables (four of them altogether; one slicing parameter Ω and three coordinatization parameters α) and true dynamic variables β (two of them) [5], [6], [7]. The set of constraints is solved with respect to the momenta conjugate to the embedding variables.…”

“…It also seems to avoid the technical problems of time, such as the Hilbert space problem, and the spectral analysis problem, as it produces the Schrödinger equation for the state evolution, and the Hamiltonian does not include the square root operation. The procedure has been described previously elsewhere [6], [7], but we do not believe that it is widely known and provide a brief overview of it below.…”

Constraints in Quantum Geometrodynamics

“…The ADM square root quantization procedure is also based entirely on constraints, but in this procedure the set of canonical variables is split into two subsets: the embedding variables (four of them altogether; one slicing parameter Ω and three coordinatization parameters α) and the true dynamical variables β (two of them) [5,6,7]. The constraints are then solved with respect to the momenta conjugate to the embedding variables.…”