TwistedC⋆-algebra formulation of quantum cosmology with application to the Bianchi I model

Abstract

A twisted C ⋆ -algebra of the extended (noncommutative) Heisenberg-Weyl group has been constructed which takes into account the Uncertainty Principle for coordinates in the Planck length regime. This general construction is then used to generate an appropriate Hilbert space and observables for the noncommutative theory which, when applied to the Bianchi I Cosmology, leads to a new set of equations that describe the quantum evolution of the universe. We find that this formulation matches theories based on a reticular Heisenberg-Weyl algebra in the bouncing and expanding regions of a collapsing Bianchi universe. There is, however, an additional effect introduced by the dynamics generated by the noncommutativity. This is an oscillation in the spectrum of the volume operator of the universe, within the bouncing region of the commutative theories. We show that this effect is generic and produced by the noncommutative momentum exchange between the degrees of freedom in the cosmology. We give asymptotic and numerical solutions which show the above mentioned effects of the noncommutativity.

Gaussian fidelity distorted by external fields

“…This map, in the context of quantum field theory, is due to the connection between the noncommutative quantum mechanics and the standard quantum mechanics [17,18]. In fact, the extended noncommutative algebra, which is a case of the modified Heisenberg-Weyl algebra, is represented by the commutation relations given by…”

“…By defining ω = qB 0 /2m as being the frequency of coupling, and defining the new parameters [17,18] …”

“…Let one considers the case of a 2D harmonic oscillator of mass m, frequency ω 0 and charge q in a magnetic field B 0 perpendicular to the 2D plane. The Hamiltonian in this case is written as [17][18][19],…”

“…Following the Wigner-Weyl formalism of the quantum mechanics, the stargenfunctions for the Hamiltonian of a harmonic oscillator [17,18] in a magnetic field are obtained from the stargenvalue equation [17,18],…”

Einstein Equations for a Noncommutative Spacetime of Lie-Algebraic Type

“…By having an expression for the connection in a noncommutative Lie-algebraic spacetime we are able to render the Riemann, Ricci and Einstein tensors; these derive ntities are defined analogously to their classical counterparts and allow us to derive their classical counterparts plus quantum corrections up to first order in the parameter related to Planck's length. Similar questions have been posed for quantum cosmology seen as a minisuperspace of quantum gravity, by making use of a noncommutative algebra in the context of the Bianchi-I model [RVJM14].…”