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

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

Masked volume wise principal component analysis of small adrenocortical tumours in dynamic [11C]-metomidate positron emission tomography

<p>Abstract</p> <p>Background</p> <p>In previous clinical Positron Emission Tomography (PET) studies novel approaches for application of Principal Component Analysis (PCA) on dynamic PET images such as Masked Volume Wise PCA (MVW-PCA) have been introduced. MVW-PCA was shown to be a feasible multivariate analysis technique, which, without modeling assumptions, could extract and separate organs and tissues with different kinetic behaviors into different principal components (MVW-PCs) and improve the image quality.</p> <p>Methods</p> <p>In this study, MVW-PCA was applied to 14 dynamic 11C-metomidate-PET (MTO-PET) examinations of 7 patients with small adrenocortical tumours. MTO-PET was performed before and 3 days after starting per oral cortisone treatment. The whole dataset, reconstructed by filtered back projection (FBP) 0–45 minutes after the tracer injection, was used to study the tracer pharmacokinetics.</p> <p>Results</p> <p>Early, intermediate and late pharmacokinetic phases could be isolated in this manner. The MVW-PC1 images correlated well to the conventionally summed image data (15–45 minutes) but the image noise in the former was considerably lower. PET measurements performed by defining "hot spot" regions of interest (ROIs) comprising 4 contiguous pixels with the highest radioactivity concentration showed a trend towards higher SUVs when the ROIs were outlined in the MVW-PC1 component than in the summed images. Time activity curves derived from "50% cut-off" ROIs based on an isocontour function whereby the pixels with SUVs between 50 to 100% of the highest radioactivity concentration were delineated, showed a significant decrease of the SUVs in normal adrenal glands and in adrenocortical adenomas after cortisone treatment.</p> <p>Conclusion</p> <p>In addition to the clear decrease in image noise and the improved contrast between different structures with MVW-PCA, the results indicate that the definition of ROIs may be more accurate and precise in MVW-PC1 images than in conventional summed images. This might improve the precision of PET measurements, for instance in therapy monitoring as well as for delineation of the tumour in radiation therapy planning.</p

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

Quasi-Quantum Groups, Knots, Three-Manifolds, and Topological Field Theory

We show how to construct, starting from a quasi-Hopf algebra, or quasi-quantum group, invariants of knots and links. In some cases, these invariants give rise to invariants of the three-manifolds obtained by surgery along these links. This happens for a finite-dimensional quasi-quantum group, whose definition involves a finite group $G$, and a 3-cocycle \om, which was first studied by Dijkgraaf, Pasquier and Roche. We treat this example in more detail, and argue that in this case the invariants agree with the partition function of the topological field theory of Dijkgraaf and Witten depending on the same data G, \,\om.Comment: 30 page