Quantization of the damped harmonic oscillator revisited

We return to the description of the damped harmonic oscillator with an assessment of previous works, in particular the Bateman-Caldirola-Kanai model and a new model proposed by one of the authors. We argue the latter has better high energy behavior and is connected to existing open-systems approaches. (C) 2011 Elsevier B.V. All rights reserved.

Coherent state quantization of paragrassmann algebras

By using a coherent state quantization of paragrassmann variables, operators are constructed in finite Hilbert spaces. We thus obtain in a straightforward way a matrix representation of the paragrassmann algebra. This algebra of finite matrices realizes a deformed Weyl-Heisenberg algebra. The study of mean values in coherent states of some of these operators leads to interesting conclusions.

The Stern-Gerlach Phenomenon According to Classical Electrodynamics

We present a description of the Stem-Gerlach type experiments using only the concepts of classical electrodynamics and the Newton`s equations of motion. The quantization of the projections of the spin (or the projections of the magnetic dipole) is not introduced in our calculations. The main characteristic of our approach is a quantitative analysis of the motion of the magnetic atoms at the entrance of the magnetic field region. This study reveals a mechanism which modifies continuously the orientation of the magnetic dipole of the atom in a very short time interval, at the entrance of the magnetic field region. The mechanism is based on the conservation of the total energy associated with a magnetic dipole which moves in a non uniform magnetic field generated by an electromagnet. A detailed quantitative comparison with the (1922) Stem-Gerlach experiment and the didactical (1967) experiment by J.R. Zacharias is presented. We conclude, contrary to the original Stern-Gerlach statement, that the classical explanations are not ruled out by the experimental data.

Dynamical noncommutativity

The model of dynamical noncommutativity is proposed. The system consists of two interrelated parts. The first of them describes the physical degrees of freedom with the coordinates q(1) and q(2), and the second corresponds to the noncommutativity eta which has a proper dynamics. After quantization, the commutator of two physical coordinates is proportional to the function of eta. The interesting feature of our model is the dependence of nonlocality on the energy of the system. The more the energy, the more the nonlocality. The leading contribution is due to the mode of noncommutativity; however, the physical degrees of freedom also contribute in nonlocality in higher orders in theta .

Spacetime noncommutativity with a bifermionic parameter

We consider a Moyal plane and propose to make the noncommutativity parameter Theta(mu nu) bifermionic, i.e. composed of two fermionic (Grassmann odd) parameters. The Moyal product then contains a finite number of derivatives, which avoid the difficulties of the standard approach. As an example, we construct a two-dimensional noncommutative field theory model based on the Moyal product with a bifermionic parameter and show that it has a locally conserved energy-momentum tensor. The model has no problem with the canonical quantization and appears to be renormalizable.

Pseudoclassical description of scalar particle in non-abelian background and path-integral representations

Path-integral representations for a scalar particle propagator in non-Abelian external backgrounds are derived. To this aim, we generalize the procedure proposed by Gitman and Schvartsman of path-integral construction to any representation of SU(N) given in terms of antisymmetric generators. And for arbitrary representations of SU(N), we present an alternative construction by means of fermionic coherent states. From the path-integral representations we derive pseudoclassical actions for a scalar particle placed in non-Abelian backgrounds. These actions are classically analyzed and then quantized to prove their consistency.

Path integral representations in noncommutative quantum mechanics and noncommutative version of Berezin-Marinov action

It is known that the actions of field theories on a noncommutative space-time can be written as some modified (we call them theta-modified) classical actions already on the commutative space-time (introducing a star product). Then the quantization of such modified actions reproduces both space-time noncommutativity and the usual quantum mechanical features of the corresponding field theory. In the present article, we discuss the problem of constructing theta-modified actions for relativistic QM. We construct such actions for relativistic spinless and spinning particles. The key idea is to extract theta-modified actions of the relativistic particles from path-integral representations of the corresponding noncommutative field theory propagators. We consider the Klein-Gordon and Dirac equations for the causal propagators in such theories. Then we construct for the propagators path-integral representations. Effective actions in such representations we treat as theta-modified actions of the relativistic particles. To confirm the interpretation, we canonically quantize these actions. Thus, we obtain the Klein-Gordon and Dirac equations in the noncommutative field theories. The theta-modified action of the relativistic spinning particle is just a generalization of the Berezin-Marinov pseudoclassical action for the noncommutative case.

Semiclassical and quantum motions on the non-commutative plane

We study the canonical and the coherent state quantizations of a particle moving in a magnetic field on the non-commutative plane. Using a theta-modified action, we perform the canonical quantization and analyze the gauge dependence of the theory. We compare coherent states quantizations obtained through Malkin-Man`ko states and circular squeezed states. The relation between these states and the ""classical"" trajectories is investigated, and we present numerical explorations of some semiclassical quantities. (C) 2009 Elsevier B.V. All rights reserved.

Coherent states of a particle in a magnetic field and the Stieltjes moment problem

A solution to a version of the Stieltjes moment. problem is presented. Using this solution, we construct a family of coherent states of a charged particle in a uniform magnetic field. We prove that these states form an overcomplete set that is normalized and resolves the unity. By the help of these coherent states we construct the Fock-Bergmann representation related to the particle quantization. This quantization procedure takes into account a circle topology of the classical motion. (C) 2009 Elsevier B.V. All rights reserved.

Field on Poincare Group and Quantum Description of Orientable Objects

We propose an approach to the quantum-mechanical description of relativistic orientable objects. It generalizes Wigner`s ideas concerning the treatment of nonrelativistic orientable objects (in particular, a nonrelativistic rotator) with the help of two reference frames (space-fixed and body-fixed). A technical realization of this generalization (for instance, in 3+1 dimensions) amounts to introducing wave functions that depend on elements of the Poincar, group G. A complete set of transformations that test the symmetries of an orientable object and of the embedding space belongs to the group I =GxG. All such transformations can be studied by considering a generalized regular representation of G in the space of scalar functions on the group, f(x,z), that depend on the Minkowski space points xaG/Spin(3,1) as well as on the orientation variables given by the elements z of a matrix ZaSpin(3,1). In particular, the field f(x,z) is a generating function of the usual spin-tensor multi-component fields. In the theory under consideration, there are four different types of spinors, and an orientable object is characterized by ten quantum numbers. We study the corresponding relativistic wave equations and their symmetry properties.

A new proposal for the picture changing operators in the minimal pure spinor formalism

Using a new proposal for the ""picture lowering"" operators, we compute the tree level scattering amplitude in the minimal pure spinor formalism by performing the integration over the pure spinor space as a multidimensional Cauchy-type integral. The amplitude will be written in terms of the projective pure spinor variables, which turns out to be useful to relate rigorously the minimal and non-minimal versions of the pure spinor formalism. The natural language for relating these formalisms is the. Cech-Dolbeault isomorphism. Moreover, the Dolbeault cocycle corresponding to the tree-level scattering amplitude must be evaluated in SO(10)/SU(5) instead of the whole pure spinor space, which means that the origin is removed from this space. Also, the. Cech-Dolbeault language plays a key role for proving the invariance of the scattering amplitude under BRST, Lorentz and supersymmetry transformations, as well as the decoupling of unphysical states. We also relate the Green`s function for the massless scalar field in ten dimensions to the tree-level scattering amplitude and comment about the scattering amplitude at higher orders. In contrast with the traditional picture lowering operators, with our new proposal the tree level scattering amplitude is independent of the constant spinors introduced to define them and the BRST exact terms decouple without integrating over these constant spinors.

Was the Stern-Gerlach Phenomenon Classically Described?

A criticism of a recent article published in this journal, claiming to have reached a classical description of the Stern-Gerlach phenomenon, is presented here. The author of the article, among other mistakes, wrongly writes the total energy of each silver atom and, moreover, presents a nonsensical equation, from which his results and the conclusion of his article are derived.

Star products made (somewhat) easier

We develop an approach to the deformation quantization on the real plane with an arbitrary Poisson structure which is based on Weyl symmetrically ordered operator products. By using a polydifferential representation for the deformed coordinates, xj we are able to formulate a simple and effective iterative procedure which allowed us to calculate the fourth-order star product (and may be extended to the fifth order at the expense of tedious but otherwise straightforward calculations). Modulo some cohomology issues which we do not consider here, the method gives an explicit and physics-friendly description of the star products.

Landau analog levels for dipoles in non-commutative space and phase space - Landau analog levels for dipoles

We investigate the analog of Landau quantization, for a neutral polarized particle in the presence of homogeneous electric and magnetic external fields, in the context of non-commutative quantum mechanics. This particle, possessing electric and magnetic dipole moments, interacts with the fields via the Aharonov-Casher and He-McKellar-Wilkens effects. For this model we obtain the Landau energy spectrum and the radial eigenfunctions of the non-commutative space coordinates and non-commutative phase space coordinates. Also we show that the case of non-commutative phase space can be treated as a special case of the usual non-commutative space coordinates.

Influence of interlayer tunneling on the quantized Hall phases in intentionally disordered multilayer structures

Stability of the quantized Hall phases is studied in weakly coupled multilayers as a function of the interlayer correlations controlled by the interlayer tunneling and by the random variation of the well thicknesses. A strong enough interlayer disorder destroys the symmetry responsible for the quantization of the Hall conductivity, resulting in the breakdown of the quantum Hall effect. A clear difference between the dimensionalities of the metallic and insulating quantum Hall phases is demonstrated. The sharpness of the quantized Hall steps obtained in the coupled multilayers with different degrees of randomization was found consistent with the calculated interlayer tunneling energies. The observed width of the transition between the quantized Hall states in random multilayers is explained in terms of the local fluctuations of the electron density.