The Narcowich-Wigner spectrum of a pure state
We completely characterize the Narcowich-Wigner spectrum of Wigner functions associated with pure states.
math-ph↗
arXiv subjects
Nuno Costa Dias
Publications and source records attributed to Nuno Costa Dias.
Phase-Space Noncommutative Quantum Cosmology
We present a noncommutative extension of Quantum Cosmology and study the Kantowski-Sachs (KS) cosmological model requiring that the two scale factors of the KS metric, the coordinates of the system, and their conjugate canonical momenta do not commute. Through the ADM formalism, we obtain the Wheeler-DeWitt (WDW) equation for the noncommutative system. The Seiberg-Witten map is used to transform the noncommutative equation into a commutative one, i.e. into an equation with commutative variables, which depend on the noncommutative parameters, $θ$ and $η$. Numerical solutions are found both for the classical and the quantum formulations of the system. These solutions are used to characterize the dynamics and the state of the universe. From the classical solutions we obtain the behavior of quantities such as the volume expansion, the shear and the characteristic volume. However the analysis of these quantities does not lead to any restriction on the value of the noncommutative parameters, $θ$ and $η$. On the other hand, for the quantum system, one can obtain, via the numerical solution of the WDW equation, the wave function of the universe both for commutative as well as for the noncommutative models. Interestingly, we find that the existence of suitable solutions of the WDW equation imposes bounds on the values of the noncommutative parameters. Moreover, the noncommutativity in the momenta leads to damping of the wave function implying that this noncommutativity can be of relevance for the selection of possible initial states of the early universe.
gr-qc↗
Deformation Quantization of Confined Systems
The Weyl-Wigner formulation of quantum confined systems poses several interesting problems. The energy stargenvalue equation, as well as the dynamical equation does not display the expected solutions. In this paper we review some previous results in the subject and add some new contributions. We reformulate the confined energy eigenvalue equation by adding to the Hamiltonian a new (distributional) boundary potential. The new Hamiltonian is proved to be globally defined and self-adjoint. Moreover, it yields the correct Weyl-Wigner formulation of the confined system.
quant-ph↗
Environment-Induced Decoherence in Noncommutative Quantum Mechanics
We address the question of the appearence of ordinary quantum mechanics in the context of noncommutative quantum mechanics. We obtain the noncommutative extension of the Hu-Paz-Zhang master equation for a Brownian particle linearly coupled to a bath of harmonic oscillators. We consider the particular case of an Ohmic regime.
hep-th↗
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.
hep-th↗
Features of Moyal Trajectories
We study the Moyal evolution of the canonical position and momentum variables. We compare it with the classical evolution and show that, contrary to what is commonly found in the literature, the two dynamics do not coincide. We prove that this divergence is quite general by studying Hamiltonians of the form $p^2 /2m + V(q)$. Several alternative formulations of Moyal dynamics are then suggested. We introduce the concept of starfunction and use it to reformulate the Moyal equations in terms of a system of ordinary differential equations on the noncommutative Moyal plane. We then use this formulation to study the semiclassical expansion of Moyal trajectories, which is cast in terms of a (order by order in $\hbar$) recursive hierarchy of i) first order partial differential equations as well as ii) systems of first order ordinary differential equations. The latter formulation is derived independently for analytic Hamiltonians as well as for the more general case of smooth local integrable ones. We present various examples illustrating these results.
quant-ph↗
Comment on "On infinite walls in deformation quantization"
We discuss a recent method proposed by Kryukov and Walton to address boundary-value problems in the context of deformation quantization. We compare their method with our own approach and establish a connection between the two formalisms.
quant-ph↗
Stargenfunctions, generally parametrized systems and a causal formulation of phase space quantum mechanics
We address the deformation quantization of generally parametrized systems displaying a natural time variable. The purpose of this exercise is twofold: first, to illustrate through a pedagogical example the potential of quantum phase space methods in the context of constrained systems and particularly of generally covariant systems. Second, to show that a causal representation for quantum phase space quasidistributions can be easily achieved through general parametrization. This result is succinctly discussed.
math-ph↗
Deformation Quantization and Wigner Functions
We review the Weyl-Wigner formulation of quantum mechanics in phase space. We discuss the concept of Narcowich-Wigner spectrum and use it to state necessary and sufficient conditions for a phase space function to be a Wigner distribution. Based on this formalism we analize the modifications introduced by the presence of boundaries. Finally, we discuss the concept of environment-induced decoherence in the context of the Weyl-Wigner approach.
hep-th↗
Admissible states in quantum phase space
We address the question of which phase space functionals might represent a quantum state. We derive necessary and sufficient conditions for both pure and mixed phase space quantum states. From the pure state quantum condition we obtain a formula for the momentum correlations of arbitrary order and derive explicit expressions for the wavefunctions in terms of time dependent and independent Wigner functions. We show that the pure state quantum condition is preserved by the Moyal (but not by the classical Liouville) time evolution and is consistent with a generic stargenvalue equation. As a by-product Baker's converse construction is generalized both to an arbitrary stargenvalue equation, associated to a generic phase space symbol, as well as to the time dependent case. These results are properly extended to the mixed state quantum condition, which is proved to imply the Heisenberg uncertainty relations. Globally, this formalism yields the complete characterization of the kinematical structure of Wigner quantum mechanics. The previous results are then succinctly generalized for various quasi-distributions. Finally, the formalism is illustrated through the simple examples of the harmonic oscillator and the free Gaussian wave packet. As a by-product, we obtain in the former example an integral representation of the Hermite polynomials.
hep-th↗
Formal solutions of stargenvalue equations
The formal solution of a general stargenvalue equation is presented, its properties studied and a geometrical interpretation given in terms of star-hypersurfaces in quantum phase space. Our approach deals with discrete and continuous spectra in a unified fashion and includes a systematic treatment of non-diagonal stargenfunctions. The formalism is used to obtain a complete formal solution of Wigner quantum mechanics in the Heisenberg picture and to write a general formula for the stargenfunctions of Hamiltonians quadratic in the phase space variables in arbitrary dimension. A variety of systems is then used to illustrate the former results.
quant-ph↗
Time dependent transformations in deformation quantization
We study the action of time dependent canonical and coordinate transformations in phase space quantum mechanics. We extend the covariant formulation of the theory by providing a formalism that is fully invariant under both standard and time dependent coordinate transformations. This result considerably enlarges the set of possible phase space representations of quantum mechanics and makes it possible to construct a causal representation for the distributional sector of Wigner quantum mechanics.
hep-th↗
Classicality Criteria
We present two possible criteria quantifying the degree of classicality of an arbitrary (finite dimensional) dynamical system. The inputs for these criteria are the classical dynamical structure of the system together with the quantum and the classical data providing the two alternative descriptions of its initial time configuration. It is proved that a general quantum system satisfying the criteria up to some extend displays a time evolution consistent with the classical predictions up to some degree and thus it is argued that the criteria provide a suitable measure of classicality. The features of the formalism are illustrated through two simple examples.
quant-ph↗
Bohmian trajectories and quantum phase space distributions
We prove that most quasi-distributions can be written in a form similar to that of the de Broglie-Bohm distribution, except that ordinary products are replaced by some suitable non-commutative star product. In doing so, we show that the Hamilton-Jacobi trajectories and the concept of "classical pure state" are common features to all phase space formulations of quantum mechanics. Furthermore, these results provide an explicit quantization prescription for classical distributions.
quant-ph↗
Wigner Functions with Boundaries
We consider the general Wigner function for a particle confined to a finite interval and subject to Dirichlet boundary conditions. We derive the boundary corrections to the "star-genvalue" equation and to the time evolution equation. These corrections can be cast in the form of a boundary potential contributing to the total Hamiltonian which together with a subsidiary boundary condition is responsible for the discretization of the energy levels. We show that a completely analogous formulation (in terms of boundary potentials) is also possible in standard operator quantum mechanics and that the Wigner and the operator formulations are also in one-to-one correspondence in the confined case. In particular, we extend Baker's converse construction to bounded systems. Finally, we elaborate on the applications of the formalism to the subject of Wigner trajectories, namely in the context of collision processes and quantum systems displaying chaotic behavior in the classical limit.
quant-ph↗
Causal Interpretation and Quantum Phase Space
We show that the de Broglie-Bohm interpretation can be easily implemented in quantum phase space through the method of quasi-distributions. This method establishes a connection with the formalism of the Wigner function. As a by-product, we obtain the rules for evaluating the expectation values and probabilities associated with a general observable in the de Broglie-Bohm formulation. Finally, we discuss some aspects of the dynamics.
quant-ph↗
Generalized Weyl-Wigner map and Vey quantum mechanics
The Weyl-Wigner map yields the entire structure of Moyal quantum mechanics directly from the standard operator formulation. The covariant generalization of Moyal theory, also known as Vey quantum mechanics, was presented in the literature many years ago. However, a derivation of the formalism directly from standard operator quantum mechanics, clarifying the relation between the two formulations is still missing. In this paper we present a covariant generalization of the Weyl order prescription and of the Weyl-Wigner map and use them to derive Vey quantum mechanics directly from the standard operator formulation. The procedure displays some interesting features: it yields all the key ingredients and provides a more straightforward interpretation of the Vey theory including a direct implementation of unitary operator transformations as phase space coordinate transformations in the Vey idiom. These features are illustrated through a simple example.
quant-ph↗
Half Quantization
A general dynamical system composed by two coupled sectors is considered. The initial time configuration of one of these sectors is described by a set of classical data while the other is described by standard quantum data. These dynamical systems will be named half quantum. The aim of this paper is to derive the dynamical evolution of a general half quantum system from its full quantum formulation. The standard approach would be to use quantum mechanics to make predictions for the time evolution of the half quantum initial data. The main problem is how can quantum mechanics be applied to a dynamical system whose initial time configuration is not described by a set of fully quantum data. A solution to this problem is presented and used, as a guideline to obtain a general formulation of coupled classical-quantum dynamics. Finally, a quantization prescription mapping a given classical theory to the correspondent half quantum one is presented.
quant-ph↗