Cosmology: The Noncommutative Quantum and Classical Cosmology

During the early days of quantum mechanics and quantum field theory, continuous space-time and Lorentz symmetry was considered inappropriate to describe the small scale structure of the universe. It was also argued that one should introduce a fundamental length scale, limiting the precision of position measurements. The introduction of fundamental length is suggested to cure the ultraviolet divergencies occurring in quantum field theory. H. Snyder was the first to formulate these ideasmathematically (1), introducing noncommutative coordinates brings an uncertainty in the position. The success of the renormalisation made people forget about these ideas for some time. But when the quantization of gravity was considered thoroughly, it became clear that the usual concepts of space-time are inadequate and that spacetime has to be quantised or noncommutative, in some way. Quantum cosmology, is a simplified approach to the study of the very early universe, which means that the gravitational and matter variables have been reduced to a finite number of degrees of freedom (these models were extensively studied by means of Hamiltonian methods in the 1970’s, (for reviews see (2; 3)); for homogenous cosmological models the metric depends only on time, this permits to integrate the space dependence and obtain a model with a finite dimensional configuration space, minisuperspace, whose variables are the 3-metric components. One way to extract useful dynamical information is through a WKB type method. The semiclassical or WKB approximations are usually discussed in text books on nonrelativistic quantum mechanics in the context of stationary states, i.e., determination of the energy eigenvalues and eigenfunctions (4). This approximation can also be used to obtain approximate and in some cases exact solutions of the dynamical problem, i.e., full Schroedinger equation, so the utility of the semiclassical approximation in obtaining exact solutions of the Schroedinger equation has not yet fully explored. The same seems to be the case for the relativistic quantum mechanics. The importance of the semiclassical approximation in the relativistic case is probably best appreciated in quantum cosmology (5), specifically, in the analysis of the Wheeler-DeWitt equation, which is essentially a Klein-Gordon equation on the minisuperspace (6). In the last few years there have been several attempts to study the possible effects of noncommutativity in the classical cosmological scenario (7–9). The proposal of authors in Ref 9 introduces the effects 18