Noncommutativity in quantum cosmology and the Hierarchy problem

We study the quantum cosmology of an empty (4+1)-dimensional Kaluza-Klein cosmology with a negative cosmological constant and a FRW type metric with two scale factors, one for 4-D universe and the other for one compact extra dimension. By assuming the noncommutativity in the corresponding mini-superspace we suggest a solution for the Hierarchy problem, at the level of Wheeler-DeWitt equation. PACS: 98.80.Qc e-mail: [email protected] e-mail: [email protected] 1 Recently, the study of various physical theories from noncommutative point of view, such as string theory [1], quantum field theory (see for instance [2]), quantum mechanics [3], and classical mechanics [4], has been of particular interest. In particular, a new interest has been developed to study the noncommutative quantum cosmology [5], [6], [7]. In these studies, the influence of noncommutativity at early universe was explored by the formulation of a version of noncommutative quantum cosmology in which a deformation of mini-superspace is required instead of space-time deformation [5]. We know that in the study of homogeneous universes, the metric depends only on the time parameter. Thus, one can find a model with a finite dimensional configuration space, the so-called mini-superspace, whose variables are the three-metric components. The quantization of these models can be performed by using of the rules of quantum mechanics. The minisuperspace construction is a procedure to define quantum cosmological models in the search for describing the quantum features of the early universe. The influence of noncommutativity on the mini-superspace has already been considered for the cosmological models with KantowskiSachs and FRW metrics [5], [6]. Here, we consider the effect of noncommutativity on the configuration space in the model which was previously studied by one of the authors [8]. This model shows an empty (4+1) dimensional Kaluza-Klein universe with a negative cosmological constant and a FRW type metric having two scale factors. Following the idea in regarding the Wheeler-Dewitt equation more primitive than the classical Einstien equations 1 we study the noncommutativity in the quantum cosmology of this model and try to propose a solution for the Hierarchy problem, in the context of quantum cosmology. The experimental upper bound on the value of cosmological constant is extremely small. On the other hand, it is usually assumed that an effective cosmological constant describes the energy density of the vacuum < ρvac >. In fact, the vacuum energy density < ρvac > is a quantum field theory contribution to the effective cosmological constant Λeff = λ + κ < ρvac >, where λ is a small bare cosmological constant. The calculations show that, these contributions affect enormously the value of effective cosmological constant as < ρvac >∼ M P , where MP is the Planck mass which defines the ultraviolet cutoff scale of the quantum gravity. This is the well-known cosmological constant problem. However, there is another fundamental energy scale in nature, namely the electroweak scale MEW whose experimental investigation is of particular interest and the corresponding interaction has been probed successfully. Over the past two decades there was a great interest to explain the smallness of MEW/MP , which is known as the Hierarchy problem. Recently, a great amount of interest has been concentrated on solving this problem based on the existence of large extra dimensions [9]. Some attempts have also been done to solve this problem based on the noncommutativity in the space-time coordinates [10]. To the authors knowledge, this problem has not yet been paid attention by considering the noncommutativity in the mini-superspace coordinates of a quantum cosmology. In this paper, we introduce a mechanism based on the existence of noncommutativity in the mini-superspace of a quantum cosmology corresponding to an empty (4 + 1)-dimensional Kaluza-Klein cosmology. It is to be noted that the present mechanism does not claim to solve This idea was followed by Hawking and Page investigating the wormholes [11]