Volume 35, Issue 5, May 1994
Index of content:

On the regular Hilbert space representation of a Moyal quantization
View Description Hide DescriptionIt is shown that in the regular viz. phase space representation of a Moyal quantization real polynomials in the phase space variables become essentially self‐adjoint operators, and that functions in the positive cone of monomials of even power in the phase space variables yield positive operators. Conditions on the boundedness of Moyal operators are given. The classical limit of a Moyal operator is defined and used to relate classical with quantum entropy.

Asymptotic level state density for parabosonic strings
View Description Hide DescriptionMaking use of some results concerning the theory of partitions, relevant in number theory, the complete asymptotic behavior, for large N, of the level density of states for a parabosonic string is derived. It is also pointed out that there is a similarity between parabosonic strings and membranes.

Eigenvalues and eigenfunctions of the Dirac operator on spheres and pseudospheres
View Description Hide DescriptionThe Dirac equation for an electron on a curved space–time may be viewed as an eigenvalue problem for the Dirac operator on the spinor fields of the space–time. A general eigenvalue problem for the Dirac operator on a metric manifoldM in terms of spinor and tangent fields defined via the Clifford algebra is derived herein. Then it is shown how to solve the Diraceigenvalue problem on an imbedded submanifold N, of codimension one in M, by solving an eigenvalueequation on M. This is applied to the case of a sphere or pseudosphere imbedded in flat space. Eigenvalues and eigenfunctions for the Dirac operator on any sphere or pseudosphere are determined. In particular, when the pseudosphere is a space–time, the Dirac equation for a free lepton in this space–time can be solved. The resulting mass spectrum is discrete and depends on the curvature of the space–time.

Generalized supersymmetric quantum mechanics on Riemann surfaces with meromorphic superpotentials
View Description Hide DescriptionThe structure of generalized supersymmetry with mutually commuting involutions for the meromorphic supersymmetric quantum mechanics on the compact Riemann surface is introduced. For this purpose a Riemann surface is considered as a self‐conjugate complex manifold. On this basis the Witten index for the new involution is calculated.

Controlling quantum motion
View Description Hide DescriptionThe techniques of controlling the free evolution of a nonrelativistic charged particle by time‐dependent magnetic fields are proposed and the possibility of more general operations upon the Schrödinger wave packet is discussed. It is found that a properly programmed sequence of magnetic pulses can invert the free evolution process, forcing an arbitrary wave packet to ‘‘go back in time’’ to recover its past shape. Our manipulation prescriptions hold also for nonrelativistic particles of arbitrary spin.

Wigner function and quantum kinetic theory in curved space–time and external fields
View Description Hide DescriptionA new definition of the Wigner function for quantum fields coupled to curved space–time and an external Yang–Mills field is studied on the example of scalar and Dirac fields. The definition uses the formalism of the tangent bundles and is explicitly covariant and gauge invariant. Derivation of collisionless quantum kinetic equations is carried out for both quantum fields by using the first order formalism of Duffin and Kemmer. The evolution of the Wigner function is governed by the quantum corrected Liouville–Vlasov equation supplemented by the generalized mass–shell constraint. The structure of the quantum corrections is perturbatively found in all adiabatic orders. The lowest order quantum–curvature corrections coincide with the ones found by Winter.

Exchange relations and correlation functions for a quantum particle on the SU_{2} group manifold
View Description Hide DescriptionThe SO_{4} invariant quantum dynamics of a point particle moving on the 3 sphere (or equivalently, of the relative motion of the spherical top) is considered. Quantum exchange relations for different times are derived with an ‘‘R matrix’’ depending on the time difference and on the conservedangular momentum. Their implications for correlation functions are spelled out. The chiral exchange relations of Alekseev and Faddeev [Commun. Math. Phys. 141, 413–422 (1991)] are also extended to different times.

Supersymmetric quantum mechanics and the Korteweg–de Vries hierarchy
View Description Hide DescriptionThe connection between supersymmetric quantum mechanics and the Korteweg–de Vries (KdV) equation is discussed, with particular emphasis on the KdV conservation laws. It is shown that supersymmetric quantum mechanics aids in the derivation of the conservation laws, and gives some insight into the Miura transformation that converts the KdV equation into the modified KdV equation. The construction of the τ function by means of supersymmetric quantum mechanics is discussed.

Quantum logic and the histories approach to quantum theory
View Description Hide DescriptionAn extensive analysis is made of the Gell‐Mann and Hartle axioms for a generalized ‘histories’ approach to quantum theory. Emphasis is placed on finding analogs of the lattice structure employed in standard quantum logic. Particular attention is given to ‘quasitemporal’ theories in which the notion of time‐evolution in conventional Hamiltonian physics is replaced by something that is much broader; theories of this type are expected to arise naturally in the context of quantum gravity and quantum field theory in a curved space–time. The quasitemporal structure is coded in a partial semigroup of ‘temporal supports’ that underpins the lattice of history propositions. Nontrivial examples include quantum field theory on a non‐globally‐hyperbolic space–time, and a possible cobordism approach to a theory of quantum topology. A key result is the demonstration that the set of history propositions in standard quantum theory can be realized in such a way that each such proposition is represented by a genuine projection operator. This gives valuable insight into the possible lattice structure in general history theories.

U(2) as the electroweak gauge group
View Description Hide DescriptionUsing the fiber bundle formalism, the consequences of taking U(2) instead of SU(2)×U(1) as the electroweak gauge group are studied. This choice motivates using electric charge as the defining characteristic of particles: isospin and hypercharge quantum numbers are not assigned to particles. This in turn leads to a definition of matter fields in which quarks and leptons transform by the same U(2) representation; the representation being determined by electric charge quantum numbers only. As a result, all matter fields have equivalent coupling constants. Indeed, in this model,quarks have integer charge. Nevertheless, the quark and lepton currents as well as the gauge boson mass ratio are identical with the Standard Model predictions.

Generating mass without the Higgs particle
View Description Hide DescriptionThe conformal group is added to the gauge group of a model of elementary particles based on the fiber bundle formalism. The inertial mass of a particle is interpreted as a manifestation of its interaction with the gauge bosons associated with the generators of the translation and special conformal subgroups. The generation of mass by this mechanism (together with the fact that the Higgs field is not necessarily required to effect bundle reduction) makes the Higgs and the Yukawa terms in the Lagrangian unnecessary: the Lagrangian then consists only of a Yang–Mills term and a covariantly free matter field term. A particular choice of gauge reproduces the usual mass terms for both fermions and gauge bosons. The ‘‘no‐go theorem’’ is not violated by the construction. The Poincaré generators commute with the internal symmetry generators after bundle reduction, and there is no mass splitting within symmetry multiplets. However, the left‐handed/right‐handed asymmetry allows the neutrino to remain massless; the definition of matter fields allows the up and down quarks to acquire different mass couplings; and the gauge bosons have different mass couplings determined by the inner product on the Lie algebra of the broken symmetry subgroup. The mass ratios of the gauge bosons—at tree level—are precisely those predicted by the Standard Model.

A nonprincipal value prescription for the temporal gauge
View Description Hide DescriptionA nonprincipal value prescription is used to define the spurious singularities of Yang–Mills theory in the temporal gauge. Typical one‐loop dimensionally regularized temporal‐gauge integrals in the prescription are explicitly calculated, and a regularization for the spurious gauge divergences is introduced. The divergent part of the one‐loop self‐energy is shown to be local and has the same form as that in the spatial axial gauge with the principal‐value prescription. The renormalization of the theory is also briefly mentioned.

Circular‐sector quantum‐billiard and allied configurations
View Description Hide DescriptionThe circular‐sector quantum‐billiard problem is studied. Numerical evaluation of the zeros of first‐order Bessel functions finds that there is an abrupt change in the nodal‐line structure of the first excited state of the system (equivalently, second eigenstate of the Laplacian) at the critical sector‐angle θ_{ c }=0.354π. For sector‐angle θ_{0}, in the domain 0<θ_{0}<θ_{ c }, the nodal curve of the first excited state is a circular‐arc segment. For θ_{ c }<θ_{0}≤π, the nodal curve of the first excited state is the bisector of the sector. Otherwise nondegenerate first excited states become twofold degenerate at the critical‐angle θ_{ c }. The ground‐ and first‐excited‐state energies (E _{ G },E _{1}) increase monotonically as θ_{0} decreases from its maximum value, π. A graph of E _{1} vs θ_{0} reveals an inflection point at θ_{0}=θ_{ c }, which is attributed to the change in Bessel‐function contribution to the development of E _{1}. A proof is given for the existence of a common zero for two Bessel functions whose respective orders differ by a noninteger. Application of these results is made to a number of closely allied quantum‐billiard configurations.

Nonperturbative treatment of singular potentials
View Description Hide DescriptionA nonperturbative but absolutely convergent algorithm is applied to the determination of the eigenfunctions of Hamiltonian with a singular potential of the form H=−d ^{2}/dx ^{2}+x ^{2}+λ/x ^{α} in the domain [0, ∞] which obey Dirichlet boundary conditons. The formal structure of the algorithm is identical to that of the Lanczos algorithm when it is formally extended to self‐adjoint operators.

On the possibility of localized wave propagation using a classical two‐fluid model of superconductors
View Description Hide DescriptionIn this paper, the classical two‐fluid model for superconductors is used to determine the time‐dependent partial differential equations(PDEs) which govern wave propagation in superconducting media. These equations are then solved exactly in both two and three dimensions using localized wave solutions rather than the traditional eigenfunction solutions. We have applied these localized wave solutions to the problem of a symmetric superconducting slab, neglecting the normally conducting current density, and found that the resultant focus wave mode magnetic field solution expels flux from the interior of the slab and can regain its initial amplitude as it travels along the surface of the slab. A comparison of the transverse part of the localized wave solution with the transverse part of the more usual plane wave solution shows remarkable agreement.

Symplectic manifolds, coherent states, and semiclassical approximation
View Description Hide DescriptionThe symplectic structure and Hamiltonian dynamics for a class of Grassmannian manifolds are described here. Using the two‐dimensional sphere (S ^{2}) and disc (D ^{2}) as illustrative cases, we write their path integral representations using coherent state techniques. These path integrals can be evaluated exactly by semiclassical methods, thus providing examples of the localization formula. Along the way, we also give a local coordinate description for a class of Grassmannians.

A geometrical formulation of Abelian gauge structure in non‐Abelian gauge theories and disconnected gauge group
View Description Hide DescriptionA geometrical formalism of non‐Abelian gauge theory with a topological term is constructed here with special reference to the role of Abelian gauge structures in non‐Abelian theories. It is shown that when fermionic currents are written in chiral forms, we can take into account the disconnected gauge group which helps us to formulate a non‐Abelian gauge structure so that the theory becomes asymptotically free.

Eigenstates of Calogero–Sutherland–Moser model and generalized Schur functions
View Description Hide DescriptionThe eigenvalue problem for the Calogero–Sutherland–Moser model is solved exactly. The complete set of eigenfunctions is given by the homogeneous Laurent polynomials. Its subset consisting of polynomial solutions is expressed by the generalized Schur functions which correspond to the λ‐deformed general linear group.

Generalized intelligent states and squeezing
View Description Hide DescriptionThe Robertson–Schrödinger uncertainty relation for two observables A and B is shown to be minimized in the eigenstates of the operator λA+iB, λ being a complex number. Such states, called generalized intelligent states (GIS), can exhibit arbitrarily strong squeezing of A or B. The time evolution of GIS is stable for Hamiltonians which admit linear in A and B invariants. Systems of GIS for the SU(1,1) and SU(2) groups are constructed and discussed. It is shown that SU(1,1) GIS contain all the Perelomov coherent states (CS) and the Barut and Girardello CS while the spin CS are a subset of SU(2) GIS. CS for an arbitrary semisimple Lie group can be considered as a GIS for the quadratures of the Weyl generators.

Dirac equation and Clifford algebra
View Description Hide DescriptionThe Dirac equation is rewritten in the framework of a Clifford algebra, revealing some new features, of which the most interesting is the U(1)×SU(2) symmetry of the corresponding Lagrangian.