Completeness of Coherent States Associated with Self-Similar Potentials and Ramanujan's Integral Extension of the Beta Function

A decomposition of identity is given as a complex integral over the coherent states associated with a class of shape-invariant self-similar potentials. There is a remarkable connection between these coherent states and Ramanujan's integral extension of the beta function.Comment: 9 pages of Late

Lagrangian and Hamiltonian Formalism on a Quantum Plane

We examine the problem of defining Lagrangian and Hamiltonian mechanics for a particle moving on a quantum plane $Q_{q,p}$. For Lagrangian mechanics, we first define a tangent quantum plane $TQ_{q,p}$ spanned by noncommuting particle coordinates and velocities. Using techniques similar to those of Wess and Zumino, we construct two different differential calculi on $TQ_{q,p}$. These two differential calculi can in principle give rise to two different particle dynamics, starting from a single Lagrangian. For Hamiltonian mechanics, we define a phase space $T^*Q_{q,p}$ spanned by noncommuting particle coordinates and momenta. The commutation relations for the momenta can be determined only after knowing their functional dependence on coordinates and velocities. Thus these commutation relations, as well as the differential calculus on $T^*Q_{q,p}$, depend on the initial choice of Lagrangian. We obtain the deformed Hamilton's equations of motion and the deformed Poisson brackets, and their definitions also depend on our initial choice of Lagrangian. We illustrate these ideas for two sample Lagrangians. The first system we examine corresponds to that of a nonrelativistic particle in a scalar potential. The other Lagrangian we consider is first order in time derivative

Approaching the Problem of Time with a Combined Semiclassical-Records-Histories Scheme

I approach the Problem of Time and other foundations of Quantum Cosmology using a combined histories, timeless and semiclassical approach. This approach is along the lines pursued by Halliwell. It involves the timeless probabilities for dynamical trajectories entering regions of configuration space, which are computed within the semiclassical regime. Moreover, the objects that Halliwell uses in this approach commute with the Hamiltonian constraint, H. This approach has not hitherto been considered for models that also possess nontrivial linear constraints, Lin. This paper carries this out for some concrete relational particle models (RPM's). If there is also commutation with Lin - the Kuchar observables condition - the constructed objects are Dirac observables. Moreover, this paper shows that the problem of Kuchar observables is explicitly resolved for 1- and 2-d RPM's. Then as a first route to Halliwell's approach for nontrivial linear constraints that is also a construction of Dirac observables, I consider theories for which Kuchar observables are formally known, giving the relational triangle as an example. As a second route, I apply an indirect method that generalizes both group-averaging and Barbour's best matching. For conceptual clarity, my study involves the simpler case of Halliwell 2003 sharp-edged window function. I leave the elsewise-improved softened case of Halliwell 2009 for a subsequent Paper II. Finally, I provide comments on Halliwell's approach and how well it fares as regards the various facets of the Problem of Time and as an implementation of QM propositions.Comment: An improved version of the text, and with various further references. 25 pages, 4 figure

Multi Parametric Deformed Heisenberg Algebras: A Route to Complexity

We introduce a generalization of the Heisenberg algebra which is written in terms of a functional of one generator of the algebra, $f(J_0)$, that can be any analytical function. When $f$ is linear with slope $\theta$, we show that the algebra in this case corresponds to $q$-oscillators for $q^2 = \tan \theta$. The case where $f$ is a polynomial of order $n$ in $J_0$ corresponds to a $n$-parameter deformed Heisenberg algebra. The representations of the algebra, when $f$ is any analytical function, are shown to be obtained through the study of the stability of the fixed points of $f$ and their composed functions. The case when $f$ is a quadratic polynomial in $J_0$, the simplest non-linear scheme which is able to create chaotic behavior, is analyzed in detail and special regions in the parameter space give representations that cannot be continuously deformed to representations of Heisenberg algebra.Comment: latex, 17 pages, 5 PS figures; to be published in J. Phys. A: Math and Gen (2001); a few sentences were added in order to clarify some point

Wolves in the Wolds: Late Capitalism, the English Eerie, and the Wyrd Case of ‘Old Stinker’ the Hull Werewolf

In this article, I depart from the earlier opinions of Emily Gerard, Sabine Baring-Gould, and others, who explained the disappearance of the werewolf in folklore as following the extinction of the wolf. I argue instead that British literature is distinctive in representing a history of werewolf sightings in places in Britain where there were once wolves. I draw on the idea of absence, manifestations of the English eerie, and the turbulence of England in the era of late capitalism to illuminate my analysis of the representation of contemporary werewolf sightingsPeer reviewe