Catarina Bastos

Weyl-Wigner Formulation of Noncommutative Quantum Mechanics

We address the phase space formulation of a noncommutative extension of quantum mechanics in arbitrary dimension, displaying both spatial and momentum noncommutativity. By resorting to a covariant generalization of the Weyl-Wigner transform and to the Seiberg-Witten map we construct an isomorphism between the operator and the phase space representations of the extended Heisenberg algebra. This map provides a systematic approach to derive the entire structure of noncommutative quantum mechanics in phase space. We construct the extended starproduct, Moyal bracket and propose a general definition of noncommutative states. We study the dynamical and eigenvalue equations of the theory and prove that the entire formalism is independent of the particular choice of Seiberg-Witten map. Our approach unifies and generalizes all the previous proposals for the phase space formulation of noncommutative quantum mechanics. For concreteness we rederive these proposals by restricting our formalism to some 2-dimensional spaces.

Black Holes and Phase Space Noncommutativity

We use the solutions of the noncommutative Wheeler-De Witt equation arising from a Kantowski-Sachs cosmological model to compute thermodynamic properties of the Schwarzschild black hole. We show that the noncommutativity in the momentum sector introduces a quadratic term in the potential function of the black hole minisuperspace model. This potential has a local minimum and thus the partition function can be computed by resorting to a saddle point evaluation in the neighbourhood of the minimum. The thermodynamics of the black hole is derived and the corrections to the usual Hawking temperature and entropy exhibit a dependence on the momentum noncommutative parameter, �. Moreover, we study the t = r = 0 singularity in the noncommutative regime and show that in this case the wave function of the system vanishes in the neighbourhood of t = r = 0.

Berry phase in the gravitational quantum well and the Seiberg-Witten map

Abstract We explicitly compute the geometrical Berry phase for the noncommutative gravitational quantum well for different SW maps. We find that they lead to different partial contributions to the Berry phase. For the most general map we obtain that Δ γ ( S ) ∼ η 3 , in a segment S of the path in the configuration space where η is the fundamental momentum scale for the noncommutative gravitational quantum well. For the full closed path, we find, through an explicit computation, that γ ( C ) = 0 . This result is consistent with the fact that physical properties are independent of the SW map and shows that these maps do not introduce degeneracies or level crossing in the noncommutative extensions of the gravitational quantum well.

Singularity problem and phase-space noncanonical noncommutativity

The Wheeler-DeWitt equation arising from a Kantowski-Sachs model is considered for a Schwarzschild black hole under the assumption that the scale factors and the associated momenta satisfy a noncanonical noncommutative extension of the Heisenberg-Weyl algebra. An integral of motion is used to factorize the wave function into an oscillatory part and a function of a configuration space variable. The latter is shown to be normalizable using asymptotic arguments. It is then shown that on the hypersurfaces of constant value of the argument of the wave functions oscillatory piece, the probability vanishes in the vicinity of the black hole singularity.

Entropic Gravity, Phase-Space Noncommutativity and the Equivalence Principle

We generalize E Verlindes entropic gravity reasoning to a phase-space noncommutativity setup. This allows us to impose a bound on the product of the noncommutative parameters based on the equivalence principle. The key feature of our analysis is an effective Plancks constant that naturally arises when accounting for the noncommutative features of the phase-space.

Noncanonical phase-space noncommutativity and the Kantowski-Sachs singularity for black holes

We consider a cosmological model based upon a non-canonical noncommutative extension of the Heisenberg-Weyl algebra to address the thermodynamical stability and the singularity problem of both the Schwarzschild and the Kantowski-Sachs black holes. The interior of the black hole is modelled by a noncommutative extension of the Wheeler-DeWitt equation. We compute the temperature and entropy of a Kantowski-Sachs black hole and compare our results with the Hawking values. Again, the noncommutativity in the momenta sector allows us to have a minimum in the potential, which is relevant in order to apply the Feynman-Hibbs procedure. For Kantowski-Sachs black holes, the same model is shown to generate a non-unitary dynamics, predicting vanishing total probability in the neighborhood of the singularity. This result effectively regularizes the Kantowski-Sachs singularity and generalizes a similar result, previously obtained for the case of Schwarzschild black hole.

Noncommutative black holes

One considers phase-space noncommutativity in the context of a Kantowski-Sachs cosmological model to study the interior of a Schwarzschild black hole. It is shown that the potential function of the corresponding quantum cosmology problem has a local minimum. One deduces the thermodynamics and show that the Hawking temperature and entropy exhibit an explicit dependence on the momentum noncommutativity parameter, η. Furthermore, the t = r = 0 singularity is analysed in the noncommutative regime and it is shown that the wave function vanishes in this limit.

Wigner Measures in Noncommutative Quantum Mechanics

We study the properties of quasi-distributions or Wigner measures in the context of noncommutative quantum mechanics. In particular, we obtain necessary and sufficient conditions for a phase-space function to be a noncommutative Wigner measure, for a Gaussian to be a noncommutative Wigner measure, and derive certain properties of the marginal distributions which are not shared by ordinary Wigner measures. Moreover, we derive the Robertson-Schrödinger uncertainty principle. Finally, we show explicitly how the set of noncommutative Wigner measures relates to the sets of Liouville and (commutative) Wigner measures.

Phase-space noncommutative formulation of Ozawa’s uncertainty principle

Ozawas measurement-disturbance relation is generalized to a phase-space noncommutative extension of quantum mechanics. It is shown that the measurement-disturbance relations have additional terms for backaction evading quadrature amplifiers and for noiseless quadrature transducers. Several distinctive features appear as a consequence of the noncommutative extension: measurement interactions which are noiseless, and observables which are undisturbed by a measurement, or of independent intervention in ordinary quantum mechanics, may acquire noise, become disturbed by the measurement, or no longer be an independent intervention in noncommutative quantum mechanics. It is also found that there can be states which violate Ozawas universal noise-disturbance trade-off relation, but verify its noncommutative deformation.

Interacting universes and the cosmological constant

Abstract In this Letter it is studied the effects that an interaction scheme among universes can have in the values of their cosmological constants. In the case of two interacting universes, the value of the cosmological constant of one of the universes becomes very close to zero at the expense of an increasing value of the cosmological constant of the partner universe. In the more general case of a chain of N interacting universes with periodic boundary conditions, the spectrum of the Hamiltonian splits into a large number of levels, each of them associated with a particular value of the cosmological constant, that can be occupied by single universes revealing a collective behavior that plainly shows that the multiverse is much more than the mere sum of its parts.