Heisenberg Quantum Representation of the Bianchi I Cosmological Model

Abstract

A Bianchi-I type cosmological model is considered. A Hamiltonian function for the Bianchi geometry is developed from differential geometric methods and general relativity. The Hamiltonian function allows the model to be analyzed via dynamical methods of classical mechanics. The system is then quantized by way of the the usual methods of transition from classical mechanics to quantum mechanics.

The motivation to express the system in the Heisenberg representation stems from the Heisenberg equations of motion being more closely parallel to the classical equations. This correspondence is largely due to dynamical time-dependent quantum mechanical operators consisting of derivative or matrix operators fashioned from classical observables that are considered to be time dependent in the classical theory. In the Heisenberg picture the time-dependent operators corresponding to classical observables act on a state vector in Hilbert space which is not time-dependent.

In contrast, the Schrodinger representation expresses evolution by means of application of non-dynamic non-time-dependent operators which act on a dynamic evolving time-dependent wave function. The correspondence is somewhat faulty because in the Schrodinger picture time-independent operators replace time-dependent classical observables.

It is important to consider Heisenberg evolution because the concepts of time in relativistic space-time geometric systems become difficult to analyze in a consistent fashion because of Lorentz transformations and a vanishing Hamiltonian function.

An interpretation of a global time function with respect to local measurements of time should be consistently and logically related, and made to coincide with the fact that measurements of such observables as local times must be performed within the system itself. This is a unique problem in quantum cosmology because in usual quantum mechanical systems the observer is defined to be external to the system being analysed.