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We introduce and study continuity properties of the Gabor product ♮ g and relate it to the well-known product formula for the short-time Fourier transform (STFT). We derive a phase space representation of the cubic nonlinear Schrödinger equation in terms of the Gabor product, and discuss how the Gabor product can be used in the study of nonlinear dynamics of mixed quantum states. [ABSTRACT FROM AUTHOR]

The notion of partial trace of a density operator is essential for the understanding of the entanglement and separability properties of quantum states. In this paper, we investigate these notions putting an emphasis on the geometrical properties of the covariance ellipsoids of the reduced states. We thereafter focus on Gaussian states and we give new and easily numerically implementable sufficient conditions for the separability of all Gaussian states. Unlike the positive partial transposition criterion, none of these conditions is however necessary. [ABSTRACT FROM AUTHOR]

We prove two results about Wigner distributions. Firstly, that the Wigner transform is the only sesquilinear map S(Rn)×S(Rn) S(R2n) which is bounded and covariant under phase-space translations and linear symplectomorphisms. Consequently, the Wigner distributions form the only set of quasidistributions which is invariant under linear symplectic transformations. Secondly, we prove that the maximal group of (linear or non-linear) coordinate transformations that preserves the set of (pure or mixed) Wigner distributions consists of the translations and the linear symplectic and antisymplectic transformations. [ABSTRACT FROM AUTHOR]

We show that the cross Wigner function can be written in the form where is the Fourier transform of ϕ and Ŝ is a metaplectic operator that projects onto a linear symplectomorphism S consisting of a rotation along an ellipse in phase space (or in the time-frequency space). This formulation can be extended to generic Weyl symbols and yields an interesting fractional generalization of the Weyl-Wigner formalism. It also provides a suitable approach to study the Bopp phase space representation of quantum mechanics, familiar from deformation quantization. Using the "metaplectic formulation" of the Wigner transform, we construct a complete set of intertwiners relating the Weyl and the Bopp pseudo-differential operators. This is an important result that allows us to prove the spectral and dynamical equivalence of the Schrödinger and the Bopp representations of quantum mechanics. [ABSTRACT FROM AUTHOR]

We consider a noncommutative description of graphene. This description consists of a Dirac equation for massless Dirac fermions plus noncommutative corrections, which are treated in the presence of an external magnetic field. We argue that, being a two-dimensional Dirac system, graphene is particularly interesting to test noncommutativity. We find that momentum noncommutativity affects the energy levels of graphene and we obtain a bound for the momentum noncommutative parameter. [ABSTRACT FROM AUTHOR]

We show that the deformation quantization of noncommutative quantum mechanics previously considered by Dias and Prata [“Weyl–Wigner formulation of noncommutative quantum mechanics,” J. Math. Phys. 49, 072101 (2008)] and Bastos, Dias, and Prata [“Wigner measures in non-commutative quantum mechanics,” e-print arXiv:math-ph/0907.4438v1; Commun. Math. Phys. (to appear)] can be expressed as a Weyl calculus on a double phase space. We study the properties of the star-product thus defined and prove a spectral theorem for the star-genvalue equation using an extension of the methods recently initiated by de Gosson and Luef [“A new approach to the *-genvalue equation,” Lett. Math. Phys. 85, 173–183 (2008)]. [ABSTRACT FROM AUTHOR]

We present a phase-space noncommutative version of quantum mechanics and apply this extension to Quantum Cosmology. We motivate this type of noncommutative algebra through the gravitational quantum well (GQW) where the noncommutativity between momenta is shown to be relevant. We also discuss some qualitative features of the GQW such as the Berry phase. In the context of quantum cosmology we consider a Kantowski-Sachs cosmological model and obtain the Wheeler-DeWitt (WDW) equation for the noncommutative system through the ADM formalism and a suitable Seiberg-Witten (SW) map. The WDW equation is explicitly dependent on the noncommutative parameters, θ and η. We obtain numerical solutions of the noncommutative WDW equation for different values of the noncommutative parameters. We conclude that the noncommutativity in the momenta sector leads to a damped wave function implying that this type of noncommmutativity can be relevant for a selection of possible initial states for the universe. [ABSTRACT FROM AUTHOR]

We address the phase-space formulation of a noncommutative extension of quantum mechanics in arbitrary dimension, displaying both spatial and momentum noncommutativities. By resorting to a covariant generalization of the Weyl–Wigner transform and to the Darboux 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 star product and 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 the Darboux 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 two-dimensional spaces. [ABSTRACT FROM AUTHOR]

We address the question of the appearance of ordinary quantum mechanics in the context of noncommutative quantum mechanics. We review our derivation of the noncommutative extension of the Hu–Paz–Zhang master equation for a Brownian particle linearly coupled to a bath of harmonic oscillators. As a new contribution, we consider the particular case of an Ohmic regime. [ABSTRACT FROM AUTHOR]

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 p2/2m+V(q). Several alternative formulations of Moyal dynamics are then suggested. We introduce the concept of star function 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 h) 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 locally integrable ones. We present various examples illustrating these results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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 wave functions 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. [Copyright &y& Elsevier]

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 nondiagonal 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. [Copyright &y& Elsevier]

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. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

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 "stargenvalue" 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. [ABSTRACT FROM AUTHOR]

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. [Copyright &y& Elsevier]

We consider the Weyl quantization on a flat non-standard symplectic vector space. We focus mainly on the properties of the Wigner functions defined therein. In particular we show that the sets of Wigner functions on distinct symplectic spaces are different but have non-empty intersections. This extends previous results to arbitrary dimension and arbitrary (constant) symplectic structure. As a by-product we introduce and prove several concepts and results on non-standard symplectic spaces which generalize those on the standard symplectic space, namely, the symplectic spectrum, Williamson’s theorem, and Narcowich-Wigner spectra. We also show how Wigner functions on non-standard symplectic spaces behave under the action of an arbitrary linear coordinate transformation. [ABSTRACT FROM AUTHOR]

Abstract: For an arbitrary pseudo-differential operator with Weyl symbol , we consider the pseudo-differential operators associated with the Weyl symbols , where for all and s is a linear symplectomorphism of . We call the operators symplectic dimensional extensions of A. In this paper we study the relation between A and in detail, in particular their regularity, invertibility and spectral properties. We obtain an explicit formula allowing to express the eigenfunctions of in terms of those of A. We use this formalism to construct new classes of pseudo-differential operators, which are extensions of the Shubin classes of globally hypoelliptic operators. We show that the operators in the new classes share the invertibility and spectral properties of the operators in but not the global hypoellipticity property. Finally, we study a few examples of operators that belong to the new classes and which are important in mathematical physics. [Copyright &y& Elsevier]

Abstract: The usual Weyl calculus is intimately associated with the choice of the standard symplectic structure on . In this paper we will show that the replacement of this structure by an arbitrary symplectic structure leads to a pseudo-differential calculus of operators acting on functions or distributions defined, not on but rather on . These operators are intertwined with the standard Weyl pseudo-differential operators using an infinite family of partial isometries of indexed by . This allows us to obtain spectral and regularity results for our operators using Shubinʼs symbol classes and Feichtingerʼs modulation spaces. [Copyright &y& Elsevier]