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Volume 35, Issue 2, February 1994

Proof of Jacobi identity in generalized quantum dynamics
View Description Hide DescriptionIt is proven that the Jacobi identity for the generalized Poisson bracket is satisfied in the generalization of Heisenberg picture quantum mechanics recently proposed by one of the authors. The identity holds for any combination of fermionic and bosonic fields, and requires no assumptions about their mutual commutativity.

Adiabatic and post‐adiabatic representations for multichannel Schrödinger equations
View Description Hide DescriptionThe properties of the adiabatic representation of a multichannel Schrödinger equation are analyzed by exploiting the Hamiltonian and symplectic nature of the coefficient and transformation matrices, respectively. Use of this algebraic structure of the problem is shown to be in line with an approach developed by Fano and Klar [Klar and Fano, Phys. Rev. Lett. 37, 1132 (1976); Klar, Phys. Rev. A 15, 1452 (1977)] in their introduction of the postadiabatic potentials. The formal calculations due to Klar and Fano which halve the order of the matrices involved are given a rigorous mathematical background and described in a more general setup from the viewpoint of the theory of Hamiltonian and symplectic linear operators. An infinite sequence of postadiabatic representations is constructed and an algorithm for the choice of a symplectic transformation matrix for each representation is proposed. The interaction of fluorine atoms with hydrogen halides is considered as an example: In these cases, it is found that the first‐postadiabatic representation shows lower coupling than the adiabatic one, and this provides a proper choice for a decoupling approximation. The present results, and in particular the recipes for obtaining the eigenvalues and eigenvectors of relevant matrices manipulating matrices of half the size, offer interesting perspectives for the numerical integration of multichannel Schrödinger equations.

Dynamical evolution of the projector induced from the generalized squeezed state
View Description Hide DescriptionThe quantum and classical time evolution of the approximate simultaneous observable on phase space (it is merely called a projector) induced from the generalized squeezed state is considered herein. The purpose of this article is to give a trace norm estimate of the difference between these two types of time evolutions of the projectors. As a consequence of this result, it is shown that this trace norm estimate converges to 0 as ℏ→0.

A generalized time‐dependent relativistic wave equation
View Description Hide DescriptionA time‐dependent, spin‐1/2, relativistic wave equation is introduced, so that some difficulties associated with the free particle Dirac equation may be resolved in a natural and systematic way. It has been found possible to establish a connection between the fluctuation of the velocity and a mass function which describes the fluctuation of the rest mass, derived from the Dirac theory. The time‐dependent equation obtained by substituting the mass function for the constant mass term allows the description of spin fluctuations, for example, as time‐dependent transitions between nonstationary states.

Relativistic density‐functional theory of a single‐electron problem
View Description Hide DescriptionThe kinetic energy as a functional of the particle‐number and current densities and the corresponding functional derivatives of this energy are discussed and constructed for a relativistic single‐electron problem. By an additional assumption of slowly varying spatial potentials, a Klein–Gordon equation is established from which the ground‐state electron density can be calculated, and an approximate form of the kinetic energy functional obtained. While there are local errors of O(1/c ^{2}) introduced by the mentioned assumption, it is shown that in calculating the energy of the system these errors average to zero, leaving errors only of O(1/c ^{4}).

Comments on the Gribov ambiguity
View Description Hide DescriptionThe existence of Gribov ambiguities in SU(m)×U(1) gauge theories over the n‐spheres is discussed herein. The goal of this article is achieved by showing that there is exactly one conjugacy class of groups of gauge transformations for the theories given above. This implies that these transformation groups are conjugate to the ones of the trivial SU(m)×U(1) fiber bundles over the n‐spheres. By using properties of the space of maps Map_{*}(S ^{ n },G) where G is one of U(1), SU(m), we are able to determine the homotopy type of the groups of gauge transformations in terms of the homotopy groups of G. The nontriviality of these homotopy groups gives the desired result.

The polygon quantum‐billiard problem
View Description Hide DescriptionThe present work addresses the quantum polygon billiard problem with attention given to analytic and degeneracy properties of energy eigenstates. The ‘‘polygon ground‐state theorem’’ is proven which states that the only polygons that contain respective ground states that are analytic in the closed domain of the entire polygon are the ‘‘elemental polygons’’ (defined in the text). The ‘‘polygon first excited‐state theorem’’ is established which states that for every N‐sided regular polygon, N equivalent first excited states exist, each of which contains a nodal curve that is a line of mirror symmetry of the related polygon. A vector description of nodal diagonal eigenstates is introduced to establish the second component of this theorem which indicates that the space of first excited states for the N‐sided regular polygon is spanned by any two of these N nodal‐diagonal eigenstates (i.e., the first excited state is twofold degenerate). At various levels of the discussion attention is drawn to the correspondence between the quantum and classical solutions for these configurations. Thus, for example, correspondence is demonstrated in the common source of integrability in the classical and quantum domains for three inclusive, nonoverlapping classes of polygons. Stemming from the preceding conclusions, a discussion is included on the possibility of employing transverse magnetic (TM) modes in a metal waveguide to determine if an equivalent quantum billiard configuration is chaotic.

Long‐range interaction of electromagnetic field induced by Chern–Simons field
View Description Hide DescriptionThe canonical quantization to a Ginzburg–Landau‐like theory is shown by considering the potentials of the electromagnetic field and the Chern–Simons field independently. The operator of matter fields which carries fractional spin and obeys the fractional statistics is constructed. The transportation of flux between those of the corresponding gauge field and Chern–Simons field is found. This leads to a striking conclusion that the electromagnetic field induced a topological term, which gives photon mass and leads to a Meissner effect different than happens in ordinary Ginzburg–Landau theory. The induction of a Chern–Simons term for the electromagnetic field is considered the origin of fractional statistics in general, through the transportation of flux is equivalent to the transportation of the long‐distance interaction in the text. That the Chern–Simons field is hidden in the constructed matter field gives rise to a theoretical interpretation of the off‐diagonal long‐range order in the theory.

Stability of oscillators driven by ergodic processes
View Description Hide DescriptionThe evolution of a quantum harmonic oscillator, forced by a stationary ergodic process, is considered. Under suitable conditions on the ‘‘dynamics’’ of this process, it is shown that generically the spectrum of the associated quasienergy operator is continuous. These results also apply to kicked harmonic oscillators where the amplitudes of the kicks are governed by a stationary ergodic process.

Resolution of simple singularities yielding particle symmetries in a space–time
View Description Hide DescriptionA finite subgroup of the conformal group SL(2,C) can be related to invariant polynomials on a hypersurface in C ^{3}. The latter then carries a simple singularity, which resolves by a finite iteration of basic cycles of deprojections. The homological intersection graph of these cycles is the Dynkin graph of an ADE Lie group, i.e., a Lie group from the cartan series A, D, or E. The deformation of the simple singularity corresponds to ADE symmetry breaking. A (3+1)‐dimensional topological model of observation is constructed, transforming consistently under SL(2,C), as an evolving three‐dimensional system of world tubes, which connect ‘‘possible points of observation.’’ The existence of an initial singularity for the four‐dimensional space–time is related to its global topological structure. Associating the geometry of ADE singularities to the vertex structure of the topological model puts forward the conjecture on a likewise relation of inner symmetries of elementary particles to local space–time structure.

High‐energy asymptotics of resonances for three‐dimensional Schrödinger operator with screened Coulomb potential
View Description Hide DescriptionResonances for one‐particle three‐dimensional Schrödinger operators with Coulomb potential perturbed by a spherically symmetric compactly supported function q(r) are studied. The mth derivative of q(r) has a jump on the boundary of the support (m≥0). Resonances are defined as poles of an analytical continuation of the quadratic form of the resolvent to the second Riemann sheet through the branch cut along the continuous spectrum. It is shown that there exists an infinite set of resonance poles which splits into an infinite sequence of infinite series corresponding to different values of an angular momentum. Resonances in each series have the only point of accumulation at infinity. The main result of the work is an asymptotic formula for resonances in each series. It follows from this formula that high‐energy resonances of the above operator are asymptotically close to those of the Schrödinger operator with the same potential q(r) but without Coulomb term. It was shown in one of the previous works of the author that, in contrast with the non‐Coulomb case, some perturbations q(r) of the Coulomb potential can produce a sequence of resonances converging to zero. The result of this work together with the above result shows that the Coulomb part of the potential, while dramatically changing the geometry of resonances at low energies, does not destroy their asymptotic behavior at high energies.

Wave scattering in a strip
View Description Hide DescriptionThe scattering problem of the discrete random Schrödinger equation in a strip is studied. The wave is considered to be incoming from one side of the disordered section. The main result is that transmission of a wave packet is improbable.

Nonuniqueness in inverse acoustic scattering on the line
View Description Hide DescriptionThe generalized one‐dimensional Schrödinger equation d ^{2}φ/dx ^{2}+k ^{2} H(x)^{2}φ =P(x)φ is considered. The nonuniqueness is studied in the recovery of the function P(x) when the scattering matrix, H(x), and the bound state energies and norming constants are known. It is shown that when the reflection coefficient is unity at zero energy, there is a one‐parameter family of functions P(x) corresponding to the same scattering data. An explicitly solved example is provided. The construction of H(x) from the scattering data is also discussed when H(x) is piecewise continuous, and two explicitly solved examples are given with H(x) containing a jump discontinuity.

Solvable (nonrelativistic, classical) n‐body problems in multidimensions. I
View Description Hide DescriptionSeveral solvable n‐body problems are exhibited. They are characterized by equations of motion of Newtonian type, m _{ jr↘̈ } _{ j } = F↘_{ j }, j=1,...,n, with the ‘‘forces’’ F↘_{ j } given as explicit functions of the ‘‘particle coordinates’’ r↘_{ k } and their velocities r↘̇_{ k }, k=1,...,n; the forces F↘_{ j } generally also depend on some free parameters, so that in each case there is actually a class of solvable models. The particle coordinates r↘_{ j }, and the forces F↘_{ j }, are vectors in N‐dimensional space, with N=1,2,3. In this article we focus mainly on few‐body problems (n=1,2,3,4) in two‐ and three‐dimensional space (N=2,3); all these models are rotation‐invariant, and some are also translation‐invariant, but they are generally not Galilei‐invariant. The solution of the initial‐value problem is given in each case. We also exhibit one‐dimensional problems (N=1), including cases with n≥4; these models are generally translation‐invariant, and some are also Galilei‐invariant.

Nonlinear theories of spin‐2 field in terms of Fierz variables
View Description Hide DescriptionExceptional nonlinear Lagrangians for a spin‐2 field in terms of Fierz variables are obtained using the method proposed by Lax. They are equivalent to the Born–Infeld Lagrangian for a spin‐1 field theory and are free from unbounded growth of wave velocities.

Time‐dependent Lagrangian systems: A geometric approach to the theory of systems with constraints
View Description Hide DescriptionA geometric approach to the theory of time‐dependent regular Lagrangian systems with constraints is presented using the framework of the exact contact manifold (T Q×R,Θ_{ L }). The main subject of the article concerns the properties of the time‐dependent nonholonomic constraints and the geometric approach to the method of the Lagrange multipliers. It is shown that every constraint determines a contact one‐form, a vertical vector field, and a nonvertical vector field. The explicit form of the vector field representing the constrained dynamics is obtained and, finally, the properties of all these one‐forms and vector fields are discussed.

The reduction of five dimensional Chern–Simons theories
View Description Hide DescriptionThe effects of gauge symmetries on Lagrangiangauge theories which include the Chern–Simons term are studied herein. It is found that the Chern–Simons term appended to a five dimensional Yang–Mills theory makes nontrivial contributions to the classical fieldequations in four and two dimensions depending on the gauge group, but makes no contribution to the field equations in three dimensions, regardless of the gauge group. In addition to the reduction of Yang–Mills–Chern–Simons, the pure Chern‐Simons theory is reduced in five dimensions—a topological field theory—to two and three dimensions. The potential physical observability of solutions to the resulting field equations is examined by determining if a nontrivial spontaneous symmetry breaking is permitted.

Duality in long‐range Ising ferromagnets
View Description Hide DescriptionIt is proved that for a system of spins σ_{ i }=±1 having an interaction energy −∑K _{ ij }σ_{ i }σ_{ j } with all the K _{ ij } strictly positive, one can construct a dual formulation by associating a dual spin S _{ ijk }=±1 to each triplet of distinct sites i, j, and k. The dual interaction energy reads −∑_{(ij)} D _{ ij }Π _{ k≠i,j } S _{ ijk } with tanh(K _{ ij })=exp(−2D _{ ij }), and it is invariant under local symmetries. The gauge‐fixing procedure, identities relating averages of order and disorder variables, and representations of various quantities as integrals over Grassmann variables are discussed. The relevance of these results for Polyakov’s approach of the 3‐D Ising model is briefly discussed.

Geometry of canonical correlation on the state space of a quantum system
View Description Hide DescriptionA Riemannian metric is defined on the state space of a finite quantum system by the canonical correlation (or Kubo–Mori/Bogoliubov scalar product). This metric is infinitesimally induced by the (nonsymmetric) relative entropy functional or the von Neumann entropy of density matrices. Hence its geometry expresses maximal uncertainty. It is proven that the metric is monotone under stochastic mappings, however, an example shows that it is not the only such Riemannian metric. This fact is remarkable because in the probabilistic case, the Markovian monotonicity property characterizes the Fisher information metric. The essential difference appears in the curvatures of a classical state space and a quantum one. A conjecture is made that the scalar curvature is monotone with respect to the ‘‘more mixed’’ (statistical) partial order of density matrices. Furthermore, an informationinequality resembling the Cramér–Rao inequality of classical statistics is established. The inequality provides a lower bound for the canonical correlation matrix (of an unbiased observable) and it is saturated when a (partial) observation level and the corresponding family of coarse‐grained states are considered.

Stochastic renormalization group approach to random point processes. A fractal generalization of Poisson statistics
View Description Hide DescriptionA generalization of the Shlesinger–Hughes stochastic renormalization method is suggested for random point processes. A decimation procedure is introduced in terms of two parameters: the probability α that a decimation step takes place and the probability β that during a decimation step a random dot is removed from the process. At each step a random number of dots are removed; by this procedure a chain of point processes is generated. The renormalized point process is a superposition of the intermediate processes attached to the different steps. For a statistical ensemble the fraction L _{ q } of systems for which q decimation steps occur is a power function of the fraction ρ_{ q } of points which survive q decimation steps L _{ q } = (ρ_{ q })^{1−d f } where d _{ f }=1−ln α/ln(1−β) is a fractal exponent smaller than unity, d _{ f }<1. If the random points are initially independent then a complete analysis is possible. In this case explicit expressions for the renormalized Janossy densities, the joint densities, and the generating functional of the process are derived. Even though the initial process is made up of independent random points the points in the renormalized process are correlated. The probability of the number of points is a superposition of Poissonians corresponding to the different steps; however, it is generally non‐Poissonian. It may be considered as a fractal generalization of Poisson statistics. All positive moments of the number of points exist and are finite and thus the corresponding probability does not have a long tail; the fractal features are displayed by the dependence of the probability on the initial average number of points λ: for λ→∞ the probability has an inverse power tail in λ modulated by a periodic function in ln λ with a period −ln(1−β). The new formalism is of interest for describing the lacunary structures corresponding to the final stages of chemical processes in low dimensional systems and for the statistics of rare events.