Volume 38, Issue 6, June 1997
Index of content:

Geometric approach to inverse scattering for the Schrödinger equation with magnetic and electric potentials
View Description Hide DescriptionWe consider the Hamiltonian of a quantum particle in a magnetic field and a potential in space dimensions If is of short range, then the highvelocity limit of the scattering operator uniquely determines the magnetic field and the potential If, in addition, longrange potentials are present, some knowledge of (the far out tail of) is needed to define a modified Dollard wave operator and a scattering operator Again its high velocity limit uniquely determines and Moreover, we give explicit error bounds which are inverse proportional to the velocity.

On the Casimir energy for a piece relativistic string
View Description Hide DescriptionThe Casimir energy for the transverse oscillations of a piecewise uniform closed string is calculated. The string consists of pieces of equal length, of alternating type I and type II material, and is taken to be relativistic in the sense that the velocity of sound always equals the velocity of light. By means of a new recursion formula we manage to calculate the Casimir energy for arbitrary integers Agreement with results obtained in earlier works on the string is found in all special cases. As basic regularization method we use the contour integration method. As a check, agreement is found with results obtained from the ζ function method (the Hurwitz function) in the case of low The Casimir energy is generally negative, and the more so the larger is the value of We illustrate the results graphically in some cases. The generalization to finite temperature theory is also given.

The generalized Koszul differential in the BRST quantization
View Description Hide DescriptionA new geometrical structure is proposed for a differential complex of the Koszul type. The generators of the complex are structured on more levels and new graduations are used. A generalized differential that can be split in many pieces acts on the generators. The first two items are identified with the Koszul operators from the sp(2) BRST quantization. This result suggests the possibility of the implementation of a symmetry bigger than sp(2). The third order Koszul differential is effectively constructed. As an application of this construction, the BRST charges and the extended Hamiltonian suitable for the sp(3) BRST quantization of a first rank gauge theory are presented.

Quasiexactly solvable spin Schrödinger operators
View Description Hide DescriptionThe algebraic structures underlying quasiexact solvability for spin Hamiltonians in one dimension are studied in detail. Necessary and sufficient conditions for a matrix secondorder differential operator preserving a space of wave functions with polynomial components to be equivalent to a Schrödinger operator are found. Systematic simplifications of these conditions are analyzed, and are then applied to the construction of new examples of multiparameter QES spin Hamiltonians in one dimension.

Multiperiodic coherent states and the Wentzel–Kramers–Brillouin exactness. II. Noncompact case and classical theories revisited
View Description Hide DescriptionWe show that the Wentzel–Kramers–Brillouin approximation gives the exact result in the trace formula of “” which is the noncompact counterpart of in terms of the “multiperiodic” coherent state. We revisit the symplectic 2forms on and and, especially, construct that on with the unitary form. We also revisit the exact calculation of the classical partition functions of them.

Scattering theory for finitely many sphere interactions supported by concentric spheres
View Description Hide DescriptionWe study stationary scattering theory for finitely many sphere interactions formally given by the Hamiltonian and its generalizations to the case of interactions of the second type and interactions with nonseparated boundary conditions. In a previous publication [J. Math. Phys. 29, 660–664 (1988)], it was shown that the selfadjoint Hamiltonian corresponding to may be defined as a limit in norm resolvent convergence of a family of local scaled shortrange Hamiltonians. In this paper we also study scattering theory corresponding to and show that the scattering quantities associated with converge to those of as

Constrained quantization on symplectic manifolds and quantum distribution functions
View Description Hide DescriptionA quantization scheme based on the extension of phase space with application of constrained quantization technique is considered. The obtained method is similar to the geometric quantization. For constrained systems the problem of scalar product on the reduced Hilbert space is investigated and a possible solution to this problem is given. Generalization of the Gupta–Bleulerlike conditions is done by minimization of quadratic fluctuations of quantum constraints. The scheme for the construction of generalized coherent states is considered and relation with Berezin quantization is found. The quantum distribution functions are introduced and their physical interpretation is discussed.

A factorization of a special type matrix into Jost matrices
View Description Hide DescriptionAn effective algebraic approach to matrix factorization into Jost matrices is developed for the case of a matrix that can be diagonalized by means of a rotation by an angle whose tangent is a rational function of The Jost matrix is given as a solution of the boundary value Riemann–Hilbert problem.

Fock representations of exchange algebras with involution
View Description Hide DescriptionAn associative algebra with exchange properties generalizing the canonical (anti)commutation relations is considered. We introduce a family of involutions in and construct the relative Fock representations, examining the positivity of the metric. As an application of the general results, we rigorously prove unitarity of the scattering operator of integrable models in spacetime dimensions. In this context the possibility of adopting various involutions in the Zamolodchikov–Faddeev algebra is also explored.

Perturbation theory and the classical limit of quantum mechanics
View Description Hide DescriptionWe consider the classical limit of quantum mechanics from the viewpoint of perturbation theory. The main focus is time dependent perturbation theory, in particular, the time evolution of a harmonic oscillatorcoherent state in an anharmonic potential. We explore in detail a perturbation method introduced by Bhaumik and DuttaRoy [J. Math. Phys. 16, 1131 (1975)] and resolve several complications that arise when this method is extended to second order. A classical limit for coherent states used by the above authors is then applied to the quantum perturbation expansions and, to second order, the classical Poincaré–Lindstedt series is retrieved. We conclude with an investigation of the connection between the classical limits of time dependent and time independent perturbation theories, respectively.

Canonical quantization of photons in a Rindler wedge
View Description Hide DescriptionPhotons and thermal photons are studied in the Rindler wedge employing Feynman’s gauge and canonical quantization. A Gupta–Bleulerlike formalism is explicitly implemented. Nonthermal Wightman functions and related (Euclidean and Lorentzian) Green functions are explicitly calculated and their complex time analytic structure is carefully analyzed using the Fulling–Ruijsenaars master function. The invariance of the advanced minus retarded fundamental solution is checked and a Ward identity discussed. It is suggested that the KMS condition can be implemented to define thermal states also dealing with unphysical photons. Following this way, thermal Wightman functions and related (Euclidean and Lorentzian) Green functions are built up. Their analytic structure is carefully examined employing a thermal master function as in the nonthermal case and other corresponding properties are discussed. Some subtleties arising dealing with unphysical photons in the presence of the Rindler conical singularity are pointed out. In particular, a oneparameter family of thermal Wightman and Schwinger functions with the same physical content is proved to exist due to a remaining (nontrivial) static gauge ambiguity. A photon version of the Bisognano–Wichmann theorem is investigated in the case of photons propagating in the Rindler Wedge employing Wightman functions. In spite of the found ambiguity in defining Rindler Green functions, the coincidence of Rindler Wightman functions and Minkowski Wightman functions is proved dealing with test functions related to physical photons and Lorentzphotons.

Langevin approach for Abelian topological gauge theory
View Description Hide DescriptionAn Abelian topological action is constructed from the quantization of Seiberg–Witten monopole equations as “Langevin equations.” The starting point is an analogous action to the Labastida–Pernici’s nonsupersymmetric action for Donaldson theory. As the local symmetry of the action is first stage reducible, the quantum action is obtained by using Batalin–Vilkovisky quantization procedure. We can also obtain offshell quantum action and BRST transformation.

A chiral spin theory in the framework of an invariant evolution parameter formalism
View Description Hide DescriptionWe present a formulation for the construction of firstorder equations which describe particles with spin, in the context of a manifestly covariant relativistic theory governed by an invariant evolution parameter; one obtains a consistent quantized formalism dealing with offshell particles with spin. Our basic requirement is that the secondorder equation in the theory is of the Schrödinger–Stueckelberg type, which exhibits features of both the Klein–Gordon and Schrödinger equations. This requirement restricts the structure of the firstorder equation, in particular, to a chiral form. One thus obtains, in a natural way, a theory of chiral form for massive particles, which may contain both left and right chiralities, or just one of them. We observe that by iterating the firstorder system, we are able to obtain secondorder forms containing the transverse and longitudinal momentum relative to a timelike vector used to maintain covariance of the theory. This timelike vector coincides with the one used by Horwitz, Piron, and Reuse to obtain an invariant positive definite space–time scalar product, which permits the construction of an induced representation for states of a particle with spin. We discuss the currents and continuity equations. The transverse and longitudinal aspects of the particle are complementary, and can be treated in a unified manner using a tensor product Hilbert space. Introducing the electromagnetic field we find an equation which gives rise to the correct gyromagnetic ratio, and is fully Hermitian under the proposed scalar product. Finally, we show that the original structure of Dirac’s equation and its solutions is obtained in the highly constrained limit in which is proportional to on mass shell. The chiral nature of the theory is apparent. We define the discrete symmetries of the theory, and find that they are represented by states which are pure left or right handed.

Solution of the Dirac equation in the field of a magnetic monopole
View Description Hide DescriptionThe Dirac equation for a particle subject to a Coulomb potential, a scalar potential, and the potential of a magnetic monopole is solved by separation of variables using the spinweighted spherical harmonics and the bound states are obtained. It is shown that the separation constants are the eigenvalues of the component and the square of the total angular momentum, which includes that of the electromagnetic field and the spin of the particle. We find that, under certain conditions, there exist solutions where the spin is in the outward or inward radial direction.

On the integrability of pure Skyrme models in two dimensions
View Description Hide DescriptionWe point out that some recently studied pure skyrme models in (2+0) dimensions are completely integrable. We discuss some implications of this fact.

Summational invariants
View Description Hide DescriptionGeneral summational invariants, i.e., conservation laws acting additively on asymptotic particle states, are investigated within a classical framework for point particles with nontrivial scattering.

Effect algebras and statistical physical theories
View Description Hide DescriptionThe dichotomic physical quantities of a physical system can be naturally hosted in a mathematical structure, called effect algebra, of which orthomodular posets and Boolean algebras are particular examples. We examine how effect algebras arise inside statistical physical theories and, conversely, we study to what extent an effect algebra can be taken as a primitive structure on which a satisfactory statistical physical model equipped with a convex set of states can be constructed.

Statistical solutions of the Navier–Stokes equations on the phase space of vorticity and the inviscid limits
View Description Hide DescriptionUsing the methods of Foias [Sem. Math. Univ. Padova 48, 219–343 (1972); 49, 9–123 (1973)] and Vishik–Fursikov [Mathematical Problems of Statistical Hydromechanics (Kluwer, Dordrecht, 1988)], we prove the existence and uniqueness of both spatial and space–time statistical solutions of the Navier–Stokes equations on the phase space of vorticity. Here the initial vorticity is in Yudovich space and the initial measure has finite mean enstrophy. We show under further assumptions on the initial vorticity that the statistical solutions of the Navier–Stokes equations converge weakly and the inviscid limits are the corresponding statistical solutions of the Euler equations.

The Meissner effect and the Ginzburg–Landau equations in the presence of an applied magnetic field
View Description Hide DescriptionIt is shown that a way of phenomenological description of the Meissner effect in a superconductor under an applied magnetic field is to consider the minimizing problem of the Gibbs free energy under the constraints of complete expulsion of magnetic field from the superconducting states.

A constraint algorithm for singular Lagrangians subjected to nonholonomic constraints
View Description Hide DescriptionWe construct a constraint algorithm for singular Lagrangian systems subjected to nonholonomic constraints which generalizes that of Dirac for constrained Hamiltonian systems.