A causal quantum theory in phase space

The causal theory for the coherent state representation of quantum mechanics is derived. The general conditions for the classical limit are given and it is shown that phase space classical mechanics can be obtained as a limit even for stationary states, in contrast to the de Broglie-Bohm quantum theory of motion.     

A propagator theory applied to wave mechanics in phase space

The fact that the classical Liouville equation can be analyzed as a dynamical equation in Hilbert-Koopman (HK.) space is used in order to develop a perturbative method for the wave mechanics in phase space: an explicit solution of the Liouville equation inqp representation is exhibited. The connection between the solution obtained and the dynamics of correlations is established by computing theqp-kp transformation function in HK space. To elucidate the method, an application is presented and the result compared to that available in the literature.     

Connes distance of 2D harmonic oscillators in quantum phase space

We study the Connes distance of quantum states of two-dimensional (2D) harmonic oscillators in phase space. Using the Hilbert-Schmidt operatorial formulation, we construct a boson Fock space and a quantum Hilbert space, and obtain the Dirac operator and a spectral triple corresponding to a four-dimensional (4D) quantum phase space. Based on the ball condition, we obtain some constraint relations about the optimal elements. We construct the corresponding optimal elements and then derive the Connes distance between two arbitrary Fock states of 2D quantum harmonic oscillators. We prove that these two-dimensional distances satisfy the Pythagoras theorem. These results are significant for the study of geometric structures of noncommutative spaces, and it can also help us to study the physical properties of quantum systems in some kinds of noncommutative spaces.     

Grassmann phase space methods for fermions. I. Mode theory

In both quantum optics and cold atom physics, the behaviour of bosonic photons and atoms is often treated using phase space methods, where mode annihilation and creation operators are represented by c-number phase space variables, with the density operator equivalent to a distribution function of these variables. The anti-commutation rules for fermion annihilation, creation operators suggest the possibility of using anti-commuting Grassmann variables to represent these operators. However, in spite of the seminal work by Cahill and Glauber and a few applications, the use of Grassmann phase space methods in quantum–atom optics to treat fermionic systems is rather rare, though fermion coherent states using Grassmann variables are widely used in particle physics.     

Dirac equation with a magnetic field in 3D non-commutative phase space

For a spin-1/2 particle moving in a background magnetic field in noncommutative phase space, the Dirac equation is solved when the particle is allowed to move off the plane that the magnetic field is perpendicular to. It is shown that the motion of the charged particle along the magnetic field has the effect of increasing the magnetic field. In the classical limit, matrix elements of the velocity operator related to the probability give a clear physical picture. Along an effective magnetic field, the mechanical momentum is conserved and the motion perpendicular to the effective magnetic field follows a round orbit. If using the velocity operator defined by the coordinate operators, the motion becomes complicated.     

Electron-impact ionization and dissociative ionization of sulfur in the gas phase

We describe the methods and the results of investigation of the yield of positive ions formed as a result of electron-impact ionization of sulfur. The ionization energy for the basic molecule and the energies corresponding to the emergence of fragment ions are obtained from the ionization efficiency curves. The dynamics of formation of molecular sulfur ions in the temperature range 320–700 K is investigated. The energy dependences of efficiency S _n of the ion formation for n = 1–6 are analyzed, and their appearance energies are determined. The total cross section of sulfur ionization by a monochromatic electron beam is also investigated. Using the linear approximation method, we marked out features on the ionization function curve, which correspond to the ionization and excitation energies for multiply charged ions. The total cross section of the formation of negative sulfur ions is measured in the energy range 0–9 eV.     

On projective group properties of the 6D pseudo-Riemannian space

We study the six-dimensional pseudo-Riemannian spaces with two time-like coordinates that admit non-homothetic infinitesimal projective transformations. The metrics are manifestly obtained and the projective group properties are determined. We also find a generic definition of projective motion in the 6-dimensional rigid h-space. Presented at the International Colloquium “Integrable Systems and Quantum Symmetries”, Prague, 16–18 June 2005.     

The Van Vleck formula,Maslov theory,and phase space geometry

The Van Vleck formula is an approximate, semiclassical expression for the quantum propagator. It is the starting point for the derivation of the Gutzwiller trace formula, and through this, a variety of other expansions representing eigenvalues, wave functions, and matrix elements in terms of classical periodic orbits. These are currently among the best and most promising theoretical tools for understanding the asymptotic behavior of quantum systems whose classical analogs are chaotic. Nevertheless, there are currently several questions remaining about the meaning and validity of the Van Vleck formula, such as those involving its behavior for long times. This article surveys an important aspect of the Van Vleck formula, namely, the relationship between it and phase space geometry, as revealed by Maslov's theory of wave asymptotics. The geometrical constructions involved are developed with a minimum of mathematical formalism.     

Formulas for transverse ionization cooling in solenoidal focusing channels

Ionization cooling in solenoidal channels, such as that envisioned for the future muon colliders or neutrino factories, is studied. Assuming that the interaction with the ionization material is weak, the evolution of the transverse emittance and angular momentum can be determined analytically. Simple and practical formulas are derived for a general cooling configuration as well as for periodic channels. The prediction of these formulas agrees well with those obtained from simulation codes. The method developed here should be useful to other areas of beam physics involving solenoidal focusing.     

The reduced phase space of spherically symmetric Einstein-Maxwell theory including a cosmological constant

We extend here the canonical treatment of spherically symmetric (quantum) gravity to the most simple matter coupling, namely spherically symmetric Maxwell theory with or without a cosmological constant. The quantization is based on the reduced phase space which is coordinatized by the mass and the electric charge as well as their canonically conjugate momenta, whose geometrical interpretation is explored.The dimension of the reduced phase space depends on the topology chosen, quite similar to the case of pure (2+1) gravity.We investigate several conceptual and technical details that might be of interest for full (3+1) gravity. We use the new canonical variables introduced by Ashtekar, which simplifies the analysis tremendously.     

Mean field theory with correlations in space and time for the ferromagnetic phase transition

A mean field theory with correlations in space and time is proposed which implicitly takes into account the effect of thermal spin fluctuations. The theory describes the kinetic critical slowing down of the relaxation of the magnetization near the ferromagnetic phase transition in crystalline and amorphous ferromagnets. Furthermore, it provides an explanation for the large temperature range of apparent “critical” behaviour as observed experimentally for the paramagnetic susceptibility of amorphous ferromagnets.     

Linear response theory in nuclei

A new method is proposed for the treatment of a perturbing field as well as the perturbation due to the addition of a particle. The variational equations obtained from a Thouless-type trial function are linearized. The higher order equations are similar to the first order equations, only the inhomogenous terms are different. These equations have been derived for the Hartree-Fock-Bogolyubov (HFB) ground state of the N-particle system. An application of the method to the exactly soluble Lipkin model is also presented. The proposed method meets a high degree of success in the model case. We further argue that the present method is superior to the blocked HFB method and also numerically simpler, for the addition of a particle.     

Moduli space for conifolds as intersection of orthogonal D6 branes

We show that a system of parallel D3 branes near a conifold singularity can be mapped onto an intersecting configuration of orthogonal branes in type IIA string theory. Using this brane configuration, we analyze the Higgs moduli space of the associated field theory. The dimension of the Higgs moduli space is computed from a geometrical analysis of the conifold singularity. Our results provide evidence for an extended s-rule. In addition, a discrepancy between the prediction of the brane configuration and the result obtained from a geometrical analysis is noted. This discrepancy can be traced back to worldsheet instanton effects.     

Semiclassical Dirac theory of tunnel ionization

We present analytic tunnel ionization rates for hydrogenlike ions in ultrahigh intensity laser fields, as obtained from a semiclassical solution of the three-dimensional Dirac equation. This presents the first quantitative determination of tunneling in atomic ions in the relativistic regime. Our theory opens the possibility to study strong laser field processes with highly charged ions, where relativistic ionization plays a dominant role.     

Quantum theory in de-Sitter space

A general method for constructing fields in spaces with transitive group of transformations is presented. Quantum-theory of free fields with spin 0, 1/2, and the connection of spin and statistics in de-Sitter space of constant positive curvature are discussed.This work has been supported in part by the Oesterreichischer Forschungsrat.     

Path integration in phase space

We develop a method for defining and performing path integrals in phase space. This work generalizes to phase space the method formulated in recent years for path integration in configuration space. We give the relationship between the Wiener measure and the Liouville measure.Dedicated to Achille Papapetrou on the occasion of his retirement.This paper was written while the author was Visiting Professor in the Departments of Physics and Mathematics at the University of Texas.     

Shape invariance in phase space

Shape invariance is a powerful solvability condition, that allows for complete knowledge of the energy spectrum, and eigenfunctions of a system. After a short introduction into the deformation quantization formalism, this paper explores the implications of the supersymmetric quantum mechanics and shape invariance techniques to the phase space formalism. We show that shape invariance induces a new set of relations between the Wigner functions of the system, that allows for their direct calculation, once we know one of them. The simple harmonic oscillator and the Morse potential are solved as examples.