Volume 35, Issue 9, September 1994
Index of content:

Quantization of a classical system on a coadjoint orbit of the Poincaré group in 1+1 dimensions
View Description Hide DescriptionThe connection is investigated between quantization by means of coherent states and geometric quantization, when the base manifold is a coadjoint orbit, and hence a homogeneous space, of the (1+1)‐dimensional Poincaré group. Coherent states of the Poincaré group stem from a representation that is square‐integrable modulo closed subgroup, and so they depend on a measurable section on the given homogeneous space. For each section that leads to a tight frame, a geometric prequantization is constructed, i.e., a Hermitian line bundle with metric connection. Conditions are given under which the two forms associated to the connection and to the coadjoint orbit structure of the base manifold coincide.

Spatial entropy of central potentials and strong asymptotics of orthogonal polynomials
View Description Hide DescriptionThe Boltzmann–Shannon informationentropy of quantum‐mechanical systems in central potentials can be expressed in terms of the entropyS _{ n } of the classical orthogonal polynomials. Here, an asymptotic formula for the entropy of general orthogonal polynomials on finite intervals is obtained. It is shown that this entropy is intimately related to the relative entropyI (ρ_{0},ρ) of the equilibrium measure ρ_{0}(x) and the weight function ρ(x) of the polynomials. To do so, the theory of strong asymptotics of orthogonal polynomials on compact sets is used.

Space of test functions for higher‐order field theories
View Description Hide DescriptionThe fundamental space ζ is defined as the set of entire analytic functions [test functions φ(z)], which are rapidly decreasing on the real axis. The variable z corresponds to the complex energy plane. The conjugate or dual space ζ’ is the set of continuous linear functionals (distributions) on ζ. Among those distributions are the propagators, determined by the poles implied by the equations of motion and the contour of integration implied by the boundary conditions. All propagators can be represented as linear combinations of elementary (one pole) functionals. The algebra of convolution products is also determined. The Fourier transformed space ζ̃ contains test functions φ̃(x). These functions are extra‐rapidly decreasing, so that the exponentially increasing solutions of higher‐order equations are distributions on ζ̃.

Symbols of operators and quantum evolution
View Description Hide DescriptionAn algebraic procedure to compute the higher order symbols of the Heisenberg quantum observables is explicitly implemented. As an application, it is proven in a very simple way that a time evolution under which the symbol of any Heisenberg quantum observable is exactly given by the evolution along the classical flow is necessarily generated by a quadratic Hamiltonian.

Phase‐space dynamics and Hermite polynomials of two variables and two indices
View Description Hide DescriptionThe theory of Hermite polynomials of two variables and two indices is discussed herein. Within the context of phase‐space formulation of classical and quantum mechanics, they play the same role as conventional Hermite polynomials in ordinary quantum mechanics. Finally their extension to m variables and m indices is analyzed.

Gauge potential decomposition, space‐time defects, and Planck’s constant
View Description Hide DescriptionA new geometrization of Planck’s constant is proposed, which also bases on the torsion in the space‐time. A quantity is introduced to describe the space‐time dislocations that appear due to torsion. It will be pointed out that there is U(1)‐like gauge invariance in this new geometrization. Using the gauge‐potential decomposition, the quantity is quantized in units of the Planck length. The quantum number is determined by Hopf indices and Brouwer degrees.

The untruncated Marinari–Parisi superstring
View Description Hide DescriptionIt is shown that the bosonic angular degrees of freedom in the one dimensional Marinari–Parisi superstring can be integrated out exactly in the Hamiltonian formulation. The resulting quantum mechanical Hamiltonian is that of a supersymmetric Calogero system plus a four fermions interaction. This extra interaction vanishes for all physical states with fermion number 0 or 1 where supersymmetry is manifest.

Redundancy of the quantum level gauge condition
View Description Hide DescriptionThe manifold Γ̂ defined by the equations of motion (EM) of the gauge and ghost fields w.r.t. the gauge‐fixed action is regarded as a supercanonical line bundle over the manifold Γ defined by the EM of the gauge fields only w.r.t. the classical action. In this language the BRST operator Q is a local section of the bundle Γ̂. Using this we give a local expression for Q as being, on the other hand, the nilpotent exterior derivative on Γ with the ghost field Ψ as its generator. This fiber bundle setup allows us to prove that any ‘‘second level’’ gauge condition, i.e., a gauge condition on Γ̂ is equivalent to a gauge on the base manifold Γ and thus does not break the BRST symmetry of the quantized theory.

Uncertainty relation in quantum mechanics with quantum group symmetry
View Description Hide DescriptionThe commutation relations, uncertainty relations, and spectra of position and momentum operators were studied within the framework of quantum group symmetric Heisenberg algebras and their (Bargmann) Fock representations. As an effect of the underlying noncommutative geometry, a length and a momentum scale appear, leading to the existence of nonzero minimal uncertainties in the positions and momenta. The usual quantum mechanical behavior is recovered as a limiting case for not too small and not too large distances and momenta.

The exact solutions of the equation for a composite particle in the external electromagnetic fields
View Description Hide DescriptionExact analytic solutions are provided for a class of equations describing a composite scalar particle in constant electric and magnetic fields, and also in plane electromagnetic fields. The external fields are taken in both classical and quantum cases. A detailed physical analysis of the results is given.

A connection between quantum dynamics and approximation of Markov diffusions
View Description Hide DescriptionIt is observed that the existence of an attracting set in the class of solutions to the stochastic Lagrangianvariational principle leads to a natural problem of convergence of diffusions in the Carlen class. It is then shown how dynamical properties enable one to prove some convergence results in the two‐dimensional Gaussian case.

Solution of the central field problem for a Duffin–Kemmer–Petiau vector boson
View Description Hide DescriptionIn view of current interest in the use of the Duffin–Kemmer–Petiau (DKP) relativistic equation, the problem of the vector DKP boson in a central field is resolved and the system of first‐order coupled differential radial equations needed for an exact calculation of the eigenvalues as well as the full ten‐component spinor is derived. This is of practical importance for problems involving massive vector bosons in central fields. This formalism is applied to the free‐particle, spherically symmetric square well and Coulomb problems.

Knowledge about noncommuting quantum observables by means of Einstein–Podolsky–Rosen correlations
View Description Hide DescriptionIt is well known that quantum physics forbids the simultaneous measurement of two noncommuting projection operators. However, sometimes there exists a third projection (mirror projection) which may be measured together with the second one and whose outcomes are completely correlated, in the Einstein–Podolsky–Rosen sense, with the outcomes of the first projection. In this article the states for which such a mirror projection exists are characterized and a procedure for constructing the mirror is given. In particular, it is shown that for two noncommuting projections there are always states for which the mirror projection exists, but also states for which it does not exist.

Unstable geodesics and topological field theory
View Description Hide DescriptionA topological field theory is used to study the cohomology of mapping space. The cohomology is identified with the Becchi–Rouet–Stora–Tyutin cohomology realizing the physical Hilbert space and the coboundary operator given by the calculations of tunneling between the perturbative vacua. The method is illustrated by a simple example.

Invariants of Lagrangians and their classifications
View Description Hide DescriptionA complete classification of natural transformations of finite order of Lagrangians into energies over n‐dimensional manifolds, where n≥2, is given herein. The classification of natural transformations of finite order of vector fields and Lagrangians into energies and constants of the motion is also obtained for n≥3. By using these results, a classification of Lagrangians into Poincaré–Cartan one‐forms and Legendre transformations is obtained.

Elastic waves in discontinuous media: Three‐dimensional scattering
View Description Hide DescriptionThis report contains an exact study of elastic wave propagation and its scattering in discontinuous media where hard reflectors are onionlike sets of surfaces. In order to reformulate the problem as a finite set of boundary integralequations, the wave motion between reflectors is represented by means of elastic potentials which involve vectorial densities on the surfaces. In the external medium, an outgoing asymptotic condition generalizes the Silver–Müller (and the Sommerfeld) condition to the case of coupled waves (S and Pwaves) moving with different velocities. The uniqueness of the Green’s function, which guarantees the uniqueness of the direct problem solution, is proven. For any incident wave and arbitrary number of surfaces, the transmission and scattering problems are studied, with and without the simplification obtained by assuming constant Poisson ratios. According to the parameter ranges, the equations which are obtained are well posed, either as second kind Fredholm equations, or because they reduce to the inverse of the sum of the identity operator and a ‘‘small norm’’ bounded operator. The results can be used to describe rigorously the three‐dimensional scattering of elasticwaves in the frequency domain for any kind of incident wave function (P,S,...) as well as the response to a localized source.

Hierarchical structures and asymmetric stochastic processes on p‐adics and adeles
View Description Hide DescriptionThe space of states of some phenomena, in physics and other sciences, displays a hierarchical structure. When that is the case, it is natural to label the states by a p‐adic number field. Both the classification of the states and their relationships are then based on a notion of distance with ultrametric properties. The dynamics of the phenomena, that is, the transition between different states, is also a function of the p‐adic distance d _{ p }. However, because the distance is a symmetric function, probabilistic processes which depend only on d _{ p } have a uniform invariant probability measure, that is, all states are equally probable at large times. This being a severe limitation for cases of physical interest, processes with asymmetric transition functions have been studied. In addition to the dependence on the ultrametric distance, the asymmetric transition functions are allowed to depend also on the probability of the target state, leading to any desired invariant probability measure. When each state of a physical system is associated to several distinct hierarchical structures or parametrizations, an appropriate labeling set is the ring of adeles. Stochastic processes on the adeles are also constructed.

Existence theorems for 90° vortex–vortex scattering
View Description Hide DescriptionA Cauchy problem is formulated with data that describe the scattering of two vortices in a superconductor, and the following is proven: First, a unique global finite‐energy solution exists. Second, the symmetry of the solution rules out all cases other than 0°, 90°, or 180° scattering. Third, a local analytic solution exists near the origin. The leading terms of this solution show that, of the three cases, 90° scattering is realized.

R‐matrix approach to lattice integrable systems
View Description Hide DescriptionAn r‐matrix formalism is applied to the construction of the integrable lattice systems and their bi‐Hamiltonian structure. Miura‐like gauge transformations between the hierarchies are also investigated. In the end the ladder of linear maps between generated hierarchies is established and described.

Properties of solutions of the Kadomtsev–Petviashvili I equation
View Description Hide DescriptionThe Kadomtsev–Petviashvili I (KPI) equation is considered as a useful laboratory for experimenting with new theoretical tools able to handle the specific features of integrable models in 2+1 dimensions. The linearized version of the KPI equation is first considered by solving the initial value problem for different classes of initial data. Properties of the solutions in different cases are analyzed in details. The obtained results are used as a guideline for studying the properties of the solution u(t,x,y) of the Kadomtsev–Petviashvili I (KPI) equation with given initial data u(0,x,y) belonging to the Schwartz space. The spectral theory associated to KPI is studied in the space of the Fourier transform of the solutions. The variables p={p _{1},p _{2}} of the Fourier space are shown to be the most convenient spectral variables to use for spectral data. Spectral data are shown to decay rapidly at large p but to be discontinuous at p=0. Direct and inverse problems are solved with special attention to the behavior of all the quantities involved in the neighborhood of t=0 and p=0. It is shown in particular that the solution u(t,x,y) has a time derivative discontinuous at t=0 and that at any t≠0 it does not belong to the Schwartz space no matter how small in norm and rapidly decaying at large distances the initial data are chosen.