Deformation Quantization: Quantum Mechanics Lives and Works in Phase-Space

Wigner's quasi-probability distribution function in phase-space is a special (Weyl) representation of the density matrix. It has been useful in describing quantum transport in quantum optics; nuclear physics; decoherence (eg, quantum computing); quantum chaos; "Welcher Weg" discussions; semiclassical limits. It is also of importance in signal processing. Nevertheless, a remarkable aspect of its internal logic, pioneered by Moyal, has only emerged in the last quarter-century: It furnishes a third, alternative, formulation of Quantum Mechanics, independent of the conventional Hilbert Space, or Path Integral formulations. In this logically complete and self-standing formulation, one need not choose sides--coordinate or momentum space. It works in full phase-space, accommodating the uncertainty principle. This is an introductory overview of the formulation with simple illustrations.Comment: LaTeX, 22 pages, 2 figure

Comment on ``Conduction states in oxide perovskites: Three manifestations of Ti3+^{3 +} Jahn-Teller polarons in barium titanate''

In this comment to [S. Lenjer, O. F. Schirmer, H. Hesse, and Th. W. Kool, Phys. Rev. B {\bf 66}, 165106 (2002)] we discuss the electronic structure of oxygen vacancies in perovskites. First principles computations are in favour of rather deep levels in these vacancies, and Lenjer et al suggest that the electrons' interaction energy is negative, but data on electroconductivity are against.Comment: 2 pages, no figure

On the Spectrum of Field Quadratures for a Finite Number of Photons

The spectrum and eigenstates of any field quadrature operator restricted to a finite number $N$ of photons are studied, in terms of the Hermite polynomials. By (naturally) defining \textit{approximate} eigenstates, which represent highly localized wavefunctions with up to $N$ photons, one can arrive at an appropriate notion of limit for the spectrum of the quadrature as $N$ goes to infinity, in the sense that the limit coincides with the spectrum of the infinite-dimensional quadrature operator. In particular, this notion allows the spectra of truncated phase operators to tend to the complete unit circle, as one would expect. A regular structure for the zeros of the Christoffel-Darboux kernel is also shown.Comment: 16 pages, 11 figure

A dilemma in representing observables in quantum mechanics

There are self-adjoint operators which determine both spectral and semispectral measures. These measures have very different commutativity and covariance properties. This fact poses a serious question on the physical meaning of such a self-adjoint operator and its associated operator measures.Comment: 10 page

Obstruction Results in Quantization Theory

We define the quantization structures for Poisson algebras necessary to generalise Groenewold and Van Hove's result that there is no consistent quantization for the Poisson algebra of Euclidean phase space. Recently a similar obstruction was obtained for the sphere, though surprising enough there is no obstruction to the quantization of the torus. In this paper we want to analyze the circumstances under which such obstructions appear. In this context we review the known results for the Poisson algebras of Euclidean space, the sphere and the torus.Comment: 34 pages, Latex. To appear in J. Nonlinear Scienc

The Geometry of Quantum Mechanics

A recent notion in theoretical physics is that not all quantum theories arise from quantising a classical system. Also, a given quantum model may possess more than just one classical limit. These facts find strong evidence in string duality and M-theory, and it has been suggested that they should also have a counterpart in quantum mechanics. In view of these developments we propose "dequantisation", a mechanism to render a quantum theory classical. Specifically, we present a geometric procedure to "dequantise" a given quantum mechanics (regardless of its classical origin, if any) to possibly different classical limits, whose quantisation gives back the original quantum theory. The standard classical limit $\hbar\to 0$ arises as a particular case of our approach.Comment: 15 pages, LaTe

Aberrant computational mechanisms of social learning and decision-making in schizophrenia and borderline personality disorder

Psychiatric disorders are ubiquitously characterized by debilitating social impairments. These difficulties are thought to emerge from aberrant social inference. In order to elucidate the underlying computational mechanisms, patients diagnosed with major depressive disorder (N = 29), schizophrenia (N = 31), and borderline personality disorder (N = 31) as well as healthy controls (N = 34) performed a probabilistic reward learning task in which participants could learn from social and nonsocial information. Patients with schizophrenia and borderline personality disorder performed more poorly on the task than healthy controls and patients with major depressive disorder. Broken down by domain, borderline personality disorder patients performed better in the social compared to the non-social domain. In contrast, controls and MDD patients showed the opposite pattern and SCZ patients showed no difference between domains. In effect, borderline personality disorder patients gave up a possible overall performance advantage by concentrating their learning in the social at the expense of the non-social domain. We used computational modeling to assess learning and decision-making parameters estimated for each participant from their behavior. This enabled additional insights into the underlying learning and decision-making mechanisms. Patients with borderline personality disorder showed slower learning from social and non-social information and an exaggerated sensitivity to changes in environmental volatility, both in the non-social and the social domain, but more so in the latter. Regarding decision-making the modeling revealed that compared to controls and major depression patients, patients with borderline personality disorder and schizophrenia showed a stronger reliance on social relative to non-social information when making choices. Depressed patients did not differ significantly from controls in this respect. Overall, our results are consistent with the notion of a general interpersonal hypersensitivity in borderline personality disorder and schizophrenia based on a shared computational mechanism characterized by an over-reliance on beliefs about others in making decisions and by an exaggerated need to make sense of others during learning specifically in borderline personality disorder