Non-Linear Canonical Transformations in Classical and Quantum Mechanics

$p$-Mechanics is a consistent physical theory which describes both classical and quantum mechanics simultaneously through the representation theory of the Heisenberg group. In this paper we describe how non-linear canonical transformations affect $p$-mechanical observables and states. Using this we show how canonical transformations change a quantum mechanical system. We seek an operator on the set of $p$-mechanical observables which corresponds to the classical canonical transformation. In order to do this we derive a set of integral equations which when solved will give us the coherent state expansion of this operator. The motivation for these integral equations comes from the work of Moshinsky and a variety of collaborators. We consider a number of examples and discuss the use of these equations for non-bijective transformations.Comment: The paper has been improved in light of a referee's report. The paper will appear in the Journal of Mathematical Physics. 24 pages, no figure

Дизайн-освіта в університеті мистецтв і дизайну Burg Giebichenstein (м. Халлє, Німеччина) на прикладі дизайну одягу

In the article the peculiarities of the system fashion designers at design education in Germany. The especially attention is given to research in interdisciplinary connections professionally oriented discipline, and the general methodic to special subject.В статье рассматриваются особенности подготовки дизайнеров одежды в системе дизайн-образования Германии. Особенное внимание уделяется исследованию междисциплинарных связей в профессионально-ориентированных дисциплинах, а также общей методике преподавания специализированных предметов.В статті розглядаються особливості підготовки дизайнерів одягу в системі дизайн-освіти Німеччини. Особлива увага приділяється дослідженню міждисциплінарних зв’язків у професійно-орієнтованих дисциплінах, а також загальній методиці викладання фахових предметів

Small oscillations and the Heisenberg Lie algebra

The Adler Kostant Symes [A-K-S] scheme is used to describe mechanical systems for quadratic Hamiltonians of $\mathbb R^{2n}$ on coadjoint orbits of the Heisenberg Lie group. The coadjoint orbits are realized in a solvable Lie algebra $\mathfrak g$ that admits an ad-invariant metric. Its quadratic induces the Hamiltonian on the orbits, whose Hamiltonian system is equivalent to that one on $\mathbb R^{2n}$. This system is a Lax pair equation whose solution can be computed with help of the Adjoint representation. For a certain class of functions, the Poisson commutativity on the coadjoint orbits in $\mathfrak g$ is related to the commutativity of a family of derivations of the 2n+1-dimensional Heisenberg Lie algebra $\mathfrak h_n$. Therefore the complete integrability is related to the existence of an n-dimensional abelian subalgebra of certain derivations in $\mathfrak h_n$. For instance, the motion of n-uncoupled harmonic oscillators near an equilibrium position can be described with this setting.Comment: 17 pages, it contains a theory about small oscillations in terms of the AKS schem

Recommendations on determination of interfacial tension at the interface between two fluids by the spinning drop method

There are studied popular methods of B. Vonnegut, H. Princen, J. Slattery, S. Torza for measuring the interfacial tension at the interface between two immiscible fluids by the spinning drop method. There is provided the method for calculating the geometric parameters of the spinning drop, and its results to the emergence of a spinning drop at the centre of the severely cylindrical area that correspond to the range of the ratio of the cube of the drop`s length to its volume of 24–120, and the range of the ratio of the spinning drop`s length to its diameter 4.00–0.35. Based on the obtained calculation results there is offered the method for determining the interfacial tension by using approximation dependence of the interfacial tension from the given drop volume, its length, difference in fluid densities and the angular velocity of the drop rotation. There are assessed methodological errors of the offered and the known methods for determining interfacial tension according to the rotating drop method. A flow chart and the overall appearance of the device that implements the proposed method for measuring the interfacial tension is presented

The gravity-related decoherence master equation from hybrid dynamics

Canonical coupling between classical and quantum systems cannot result in reversible equations, rather it leads to irreversible master equations. Coupling of quantized non-relativistic matter to gravity is illustrated by a simplistic example. The heuristic derivation yields the theory of gravity-related decoherence proposed longtime ago by Penrose and the author.Comment: 9pp, extended version of invited talk at Fifth International Workshop DICE2010 (Castello Pasquini/Castiglioncello/Tuscany, Sept. 13-17, 2010

Spin dynamics with non-abelian Berry gauge fields as a semiclassical constrained hamiltonian system

The dynamics of observables which are matrices depending on \hbar and taking values in classical phase space is defined retaining the terms up to the first order in \hbar of the Moyal bracket. Within this semiclassical approach a first order lagrangian involving gauge fields is studied as a constrained hamiltonian system. This provides a systematic study of spin dynamics in the presence of non-abelian Berry gauge fields. We applied the method to various types of dynamical spin systems and clarified some persisting discussions. In particular employing the Berry gauge field which generates the Thomas precession, we calculated the force exerted on an electron in the external electric and magnetic fields. Moreover, a simple semiclassical formulation of the spin Hall effect is accomplished.Comment: References and some clarification added. Published versio

Higher spin quaternion waves in the Klein-Gordon theory

Electromagnetic interactions are discussed in the context of the Klein-Gordon fermion equation. The Mott scattering amplitude is derived in leading order perturbation theory and the result of the Dirac theory is reproduced except for an overall factor of sixteen. The discrepancy is not resolved as the study points into another direction. The vertex structures involved in the scattering calculations indicate the relevance of a modified Klein-Gordon equation, which takes into account the number of polarization states of the considered quantum field. In this equation the d'Alembertian is acting on quaternion-like plane waves, which can be generalized to representations of arbitrary spin. The method provides the same relation between mass and spin that has been found previously by Majorana, Gelfand, and Yaglom in infinite spin theories