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Noncommutative Geometry and Cosmology
G. D. Barbosa∗ and N. Pinto-Neto†
Centro Brasileiro de Pesquisas F´ ısicas, CBPF, Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil We study some consequences of noncommutativity to homogeneous cosmologies by introducing a deformation of the commutation relation between the minisuperspace variables. The investigation is carried out for theKantowski-Sachs model by means of a comparative study of the universe evolution in four different scenarios: the classical commutative, classical noncommutative, quantum commutative, and quantum noncommutative. The comparison is rendered transparent by the use of the Bohmian formalism of quantum trajectories. As a result of our analysis, we found that noncommutativity can modify significantly the universeevolution, but cannot alter its singular behavior in the classical context. Quantum effects, on the other hand, can originate non-singular periodic universes in both commutative and noncommutative cases. The quantum noncommutative model is shown to present interesting properties, as the capability to give rise to non-trivial dynamics in situations where its commutative counterpart is necessarilystatic.
PACS numbers: 98.80.Qc,04.60.Kz,11.10.Nx,11.10.Lm

arXiv:hep-th/0407111v2 27 Oct 2004

I.

INTRODUCTION

Recently, there has been a great amount of work devoted to noncommutative theories (see, e.g., [1, 2]). The boom of interest in noncommutativity of the canonical type was triggered by works establishing its connection with string and M-theory [3], although previousinvestigations in the context of semiclassical gravity [4] have pointed out the relevance of noncommutative field theories. In addition to its relevance to string theory, the study of noncommutative theories is justified in its own by the opportunity it gives us to deal with interesting properties, such as the IR/UV mixing and nonlocality [5], Lorentz violation [6], and new physics at very short scale distances[1, 2, 7]. In the latest two years, several investigations have been carried out to clarify the possible role of noncommutativity in the cosmological scenario in a great variety of contexts. Among them, we quote the Newtonian cosmology [8], cosmological perturbation theory and noncommutative inflationary cosmology [9], noncommutative gravity [10], and quantum cosmology [11]. The latter, inparticular, provides an interesting arena for speculation on the possible connection between noncommutativity and quantum gravity. The common claim that noncommutativity leads to fuzzyness renders obscure the application of noncommutative ideas to the description of a quantum universe, which, according to the Copenhagen interpretation of quantum theory, has no objective reality. Indeed, the inadequacy ofthe Copenhagen interpretation for quantum cosmology has been stressed long time ago by several physicists, as Everett [12], Feynman [13], and Bell [14]. Recently, ’t Hooft [15] has argued that a reconsideration of hidden-variables theories is in turn necessary to account for the difficulties that appear in the unification of General Relativity with Quantum Theory. Perhaps the great riddle of quantumgravity is the comprehension of the behavior of spacetime (if this concept has a meaning) at the Planck scale. At very early times, when the universe was small and hot, even when its characteristic length scale was larger than the Planckian one, noncommutativity may have played a relevant role in its evolution. The aim of this work is to exploit this possibility by carrying out a comparative studyof the universe evolution in four different scenarios: classical commutative, classical noncommutative, quantum commutative, and quantum noncommutative. As our object of analysis, we chose the Kantowski-Sachs

∗ Electronic † Electronic

address: gbarbosa@cbpf.br address: nelsonpn@cbpf.br

2 universe (see, e.g., [16, 17, 18, 19, 20, 21]). A noncommutative version of the Kantowski-Sachs...
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