Wigner distributions for finite state systems without redundant phase point operators

We set up Wigner distributions for $N$ state quantum systems following a Dirac inspired approach. In contrast to much of the work on this case, requiring a $2N\times 2N$ phase space, particularly when $N$ is even, our approach is uniformly based on an $N\times N$ phase space grid and thereby avoids the necessity of having to invoke a `quadrupled' phase space and hence the attendant redundance. Both $N$ odd and even cases are analysed in detail and it is found that there are striking differences between the two. While the $N$ odd case permits full implementation of the marginals property, the even case does so only in a restricted sense. This has the consequence that in the even case one is led to several equally good definitions of the Wigner distributions as opposed to the odd case where the choice turns out to be unique.Comment: Latex, 14 page

Temperature reducing coating for metals subject to flame exposure Patent

Anodizing method for providing metal surfaces with temperature reducing coatings against flame

Classical Light Beams and Geometric Phases

We present a study of geometric phases in classical wave and polarisation optics using the basic mathematical framework of quantum mechanics. Important physical situations taken from scalar wave optics, pure polarisation optics, and the behaviour of polarisation in the eikonal or ray limit of Maxwell's equations in a transparent medium are considered. The case of a beam of light whose propagation direction and polarisation state are both subject to change is dealt with, attention being paid to the validity of Maxwell's equations at all stages. Global topological aspects of the space of all propagation directions are discussed using elementary group theoretical ideas, and the effects on geometric phases are elucidated.Comment: 23 pages, 1 figur

The Sampling Theorem and Coherent State Systems in Quantum Mechanics

The well known Poisson Summation Formula is analysed from the perspective of the coherent state systems associated with the Heisenberg--Weyl group. In particular, it is shown that the Poisson summation formula may be viewed abstractly as a relation between two sets of bases (Zak bases) arising as simultaneous eigenvectors of two commuting unitary operators in which geometric phase plays a key role. The Zak bases are shown to be interpretable as generalised coherent state systems of the Heisenberg--Weyl group and this, in turn, prompts analysis of the sampling theorem (an important and useful consequence of the Poisson Summation Formula) and its extension from a coherent state point of view leading to interesting results on properties of von Neumann and finer lattices based on standard and generalised coherent state systems.Comment: 20 pages, Late

Hamilton's theory of turns revisited

We present a new approach to Hamilton's theory of turns for the groups SO(3) and SU(2) which renders their properties, in particular their composition law, nearly trivial and immediately evident upon inspection. We show that the entire construction can be based on binary rotations rather than mirror reflections.Comment: 7 pages, 4 figure

Hamilton's Turns for the Lorentz Group

Hamilton in the course of his studies on quaternions came up with an elegant geometric picture for the group SU(2). In this picture the group elements are represented by ``turns'', which are equivalence classes of directed great circle arcs on the unit sphere $S^2$, in such a manner that the rule for composition of group elements takes the form of the familiar parallelogram law for the Euclidean translation group. It is only recently that this construction has been generalized to the simplest noncompact group $SU(1,1) = Sp(2, R) = SL(2,R)$, the double cover of SO(2,1). The present work develops a theory of turns for $SL(2,C)$, the double and universal cover of SO(3,1) and $SO(3,C)$, rendering a geometric representation in the spirit of Hamilton available for all low dimensional semisimple Lie groups of interest in physics. The geometric construction is illustrated through application to polar decomposition, and to the composition of Lorentz boosts and the resulting Wigner or Thomas rotation.Comment: 13 pages, Late

Algebraic theory of affine curvature tensors

We use curvature decompositions to construct generating sets for the space of algebraic curvature tensors and for the space of tensors with the same symmetries as those of a torsion free, Ricci symmetric connection; the latter naturally appear in relative hypersurface theory.Comment: The paper is dedicated to the memory of the first author (N. Blazic) who passed away Monday 10 October 200

Intracardiac Calcification - An Interesting Chest X-ray Report.

We report a case of chest X-ray finding of mitral annular calcification. Mitral annular calcification is a degenerative process involving the fibrous annulus of the mitral valve. It is generally an incidental finding associated with aging and atherosclerosis. It may lead to significant mitral regurgitation and can rarely cause symptomatic mitral stenosis. In addition, mitral annulus calcification may be associated with atrial fibrillation and cardiac arrhythmia. Calcification of mitral annulus in the chest X-ray generally follows the C-shape of the mitral annulus. Confirmation is by echocardiography. Symptomatic cases require repair by surgery