Volume 36, Issue 6, June 1995
Index of content:

Gauge theory on a non‐simply connected domain and representations of canonical commutation relations
View Description Hide DescriptionA quantum system of a particle interacting with a (non‐Abelian) gauge field on the non‐simply connected domain M=R ^{2}\{a _{ n }}^{ N } _{ n=1} is considered, where a _{ n }, n=1,...,N, are fixed isolated points in R ^{2}. The gauge potential A of the gauge field is a p×p anti‐Hermitian matrix‐valued 1‐form on M, and may be strongly singular at the points a _{ n }, n=1,...,N. If A is flat, then the physical momentum and the position operators {P _{ j },q _{ j }}^{2} _{ j=1} of the particle satisfy the canonical commutation relations (CCR) with two degrees of freedom on a suitable dense domain of the Hilbert spaceL ^{2}(R ^{2};C ^{ p }). A necessary and sufficient condition for this representation to be the Schrödinger 2‐system is given in terms of the Wilson loops of the rectangles not intersecting a _{ n }, n=1,...,N. This also gives a characterization for the representation to be non‐Schrödinger. It is proven that, for a class of gauge potentials, which is not necessarily flat, P _{ j } is essentially self‐adjoint. Moreover, an example, which gives a class of non‐Schrödinger representations of the CCR with two degrees of freedom, is discussed in some detail.

An integral equation of Muskhelishvili type: Strong quantum electrodynamics
View Description Hide DescriptionAn equation that arises out of the bifurcation analysis of an improvement of the nonperturbative equations for the electron mass function in quenched quantum electrodynamics is analyzed. In the quasilinear approximation, the integral equation is solved by Mellin transformation, followed by the calculation of the Muskhelishvili index of the resultant singular integral operator.

A quantum mechanical arrow of time and the semigroup time evolution of Gamow vectors
View Description Hide DescriptionThe exponential decay (or growth) of resonances provides an arrow of time which is described as the semigroup time evolution of Gamow vector in a new formulation of quantum mechanics. Another direction of time follows from the fact that a state must first be prepared before observables can be measured in it. Applied to scattering experiments, this produces another quantum mechanical arrow of time. The mathematical statements of these two arrows of time are shown to be equivalent. If the semigroup arrow is interpreted as microphysical irreversibility and if the arrow of time from the prepared in‐state to its effect on the detector of a scattering experiment is interpreted as causality, then the equivalence of their mathematical statements implies that causality and irreversibility are interrelated.

Realizing 3‐cocycles as obstructions
View Description Hide DescriptionThe occurrence of a 3‐cocycle in quantum mechanics or quantum field theory has been interpreted somewhat paradoxically as a breakdown of the Jacobi identity. The main result of this paper is that the 3‐cocycle in chiralQCD arises as an obstruction which prevents the existence of a certain extension of one Lie algebra by another. This obstruction may be avoided by constructing a modified Lie algebra extension consisting of derivations on the algebra generated by the fields. However the 3‐cocycle then appears when an attempt is made to implement these derivations by commutation with unbounded operators in the canonical equal‐time formalism. Assuming the existence of these unbounded operators is what leads to the violation of the Jacobi identity.

Scattering by singular potentials. A. Wave function expansion
View Description Hide DescriptionThe series of the radial wave functions is developed by iterating a Volterra‐type integral equation with a modified Wentzel–Kramers–Brillouin problem supplying the reference system. The expansion proved to be convergent for exponentially singular scattering potentials in general while for the case of power singularity r ^{−p } it was convergent only at p≥4.

Quantum integrable systems and representations of Lie algebras
View Description Hide DescriptionIn this paper the quantum integrals of the Hamiltonian of the quantum many‐body problem with the interaction potential K/sinh^{2}(x) (Sutherland operator) are constructed as images of higher Casimirs of the Lie algebragl(N) under a certain homomorphism from the center of U (gl(N)) to the algebra of differential operators in N variables. A similar construction applied to the affine gl(N) at the critical level k=−N defines a correspondence between higher Sugawara operators and quantum integrals of the Hamiltonian of the quantum many‐body problem with the potential equal to constant times the Weierstrass function. This allows one to give a new proof of the Olshanetsky–Perelomov theorem stating that this Hamiltonian defines a completely integrable quantum system. A new expression is also given for eigenfunctions of the quantum integrals of the Sutherland operator as traces of intertwining operators between certain representations of gl(N).

Transformation theory of the q‐oscillator
View Description Hide DescriptionA deformation of quantum mechanics obtained by replacing commutators by q‐commutators (where q is close enough to unity to be compatible with experiment) is studied. This idea is explored by studying the q‐harmonic oscillator in both configuration space and momentum space. The complete set of states in both representations as well as the transformation between x and p space is obtained. The states as well as the matrix elements lie in the SU_{ q }(2) algebra. To obtain expectation values and transition probabilities one must average over the SU_{ q }(2) algebra.

Dirac monopoles and the Seiberg–Witten monopole equations
View Description Hide DescriptionA simple solution of the Seiberg–Witten monopole equations is given.

Path integrals in polar variables with spontaneously broken symmetry
View Description Hide DescriptionThe path‐integral formalism is applied to a two‐dimensional complex quantum field theory with a normal‐ordered Hamiltonian and spontaneously broken symmetry in the classical ground state. In most systems the spurious high‐frequency divergences arising naively from the path‐integral treatment can be avoided by deriving this treatment carefully from the operator formalism. But here, in order to justify the Gaussian approximation at the one‐loop order, one needs to express the complex field in polar variables. To avoid having to deal with the polar form of an operator conventional holomorphic variables are used to derive a path‐integral formulation entirely in terms of a c‐number complex field in position space. Then the holomorphic variables are dropped, the c‐number field is re‐expressed in polar form, and Fourier analysis is applied to the polar variables. In the one‐loop approximation the usual grand partition function is recovered with the first two ultraviolet divergences already subtracted out from the sum over zero‐point energies.

From geometric quantization to Moyal quantization
View Description Hide DescriptionIt is shown how the Moyal product of phase‐space functions, and the Weyl correspondence between symbols and operator kernels, may be obtained directly using the procedures of geometric quantization, applied to the symplectic groupoid constructed by ‘‘doubling’’ the phase space.

Scattering on small three‐dimensional, nonspherically symmetric potentials
View Description Hide DescriptionThree‐dimensional anisotropic inhomogeneous scatterer is considered, which is modeled by the Schrödinger operator with a pseudodifferential potential. Definition of the scattering amplitude is given based on the scattering of the plane wave. The amplitude, as well as the potential, depends on five real parameters. The correspondence between the scatterer (potential) and the scattering amplitude is discussed. It is stated, in particular, that any small nontrivial potential produces a nonzero scattering amplitude.

Exact solutions of Heisenberg equations for multiphoton Jaynes–Cummings model
View Description Hide DescriptionThe Heisenberg operator solutions for the multiphoton Jaynes–Cummings model will be presented. By solving the operator differential equations, the exact solutions for the photon annihilation operator a(t), the atomic dipole‐moment operator σ^{−}(t), and the population difference operator σ_{ z }(t) are obtained. The time dynamical behaviors for the operators are compared with the results obtained by the density operator methods.

Induced QCD from the noncommutative geometry of a supermanifold
View Description Hide DescriptionThe noncommutative geometry of a two‐leaf Parisi–Sourlas supermanifold in Connes’ formalism using different K cycles over the Grassmann algebra‐valued functions on the supermanifold is studied. We find that the curvature of the trivial noncommutative vector bundle defines in the simplest case the super Yang–Mills action coupled to a scalar field. By considering a modified Dirac operator and a suitable limit of its parameters we then obtain an action that turns out to be the continuum limit of the induced QCD in the Kazakov–Migdal model.

Scattering theory for the Belavkin equation describing a quantum particle with continuously observed coordinate
View Description Hide DescriptionIn the article, the complete investigation is given of the large times behavior of the solutions of the Belavkin quantum filtering equation describing a quantum particle with continuously observed coordinate. It turns out that these solutions have an extraordinary property, namely: as t → ∞, any solution will tend to a Gaussian function (for almost all realizations of innovating Wiener process). As a consequence, it follows that the dispersion of coordinate will always tend to the same limit, as t → ∞, not depending on initial data.

An inverse boundary problem for the Schrödinger operator with magnetic field
View Description Hide DescriptionThe problem under consideration in this article is the inverse boundary spectral problem for the Schrödinger operator with magnetic field. The operator is considered on a compact Riemannian manifold (with nonzero boundary) so that the corresponding unperturbed operator is the Laplace–Beltrami operator on the manifold. It is shown that boundary spectral data determine the potential and the boundary impedance uniquely, while the magnetic field may be found to within the group of gauge transformations.

Invariant spinors and reduced Dirac equations under subgroups of the Euclidean group in four‐dimensional Euclidean space
View Description Hide DescriptionThe most general Dirac spinors globally invariant under the representatives of the conjugacy classes of the connected part to the identity of the Euclidean group in four dimensions are determined. The Dirac equation is reduced under each subgroup representative and a relation to the reduction by symmetry of a linear system of the self‐dual Yang–Mills equations is given.

Berry phases for Landau Hamiltonians on deformed tori
View Description Hide DescriptionParametrized families of Landau Hamiltonians are introduced, where the parameter space is the Teichmüller space (topologically the complex upper half plane) corresponding to deformations of tori. The underlying SO(2,1) symmetry of the families enables an explicit calculation of the Berry phases picked up by the eigenstates when the torus is slowly deformed. It is also shown that apart from these phases that are local in origin, there are global non‐Abelian ones too, related to the hidden discrete symmetry group Γ_{ϑ} (the theta group, which is a subgroup of the modular group) of the families. The induced Riemannian structure on the parameter space is the usual Poincare metric on the upper half plane of constant negative curvature. Due to the discrete symmetry Γ_{ϑ} the geodesic motion restricted to the fundamental domain of this group is chaotic.

On Wick algebras with braid relations
View Description Hide DescriptionThe general theory of deformed commutation relations for creation and annihilation operators of particles equipped with arbitrary braid statistics (Ψ‐statistics) is studied in terms of Wick algebras. The construction and examples of such algebras are given.

Two‐body scattering on a graph and application to simple nanoelectronic devices
View Description Hide DescriptionThe two‐body system on a graph with one junction is considered. The effective three‐body scattering problem turns out to be exactly solvable for pointwise interactions. Additional degrees of freedom corresponding to a dynamics of some structure (e.g., an atomic cluster) located in the junction (point of common contact) of three thin electrodes are considered. These degrees of freedom bring effective energy‐dependent interaction into the effective Schrödinger equation in the scattering channel. The wave function of the system is constructed in the explicit form using the extension theory methods. The obtained results are applied to the qualitative description of a simple three‐electrode nanoelectronic device. The perturbation theory approach based on the analysis of the Liouville equation is suggested for calculation of the conductivity for such a device in terms of the obtained wave function.

Harmonic analysis on spannors
View Description Hide DescriptionThe spannor representation of the universal covering group of the conformal group of Minkowski space–time is studied herein. It is an example of an indecomposable representation. Various parallelizations for spannors are described. Normalized K̃‐finite basis fields for the spannors are introduced, and a computation of the actions of the scale generator and other generators of the conformal group on these basis fields is given. The irreducibility of all irreducible composition factors is established, and it is decided which ones are infinitesimally unitary. Two sets of generators for maximal Abelian subalgebras of the enveloping algebra of the conformal group are introduced. The action of one of the sets on K̃‐finite basis fields in the spannor representation is studied; this is important for understanding of the problem of degeneracy in the spannor representation, since K̃ types occur with multiplicity 2. Many calculations make use of the lowest (highest) weight module structures of the unitarizable and positive (negative) energy, irreducible composition factors; and, thus, many of our results and techniques can be used to initiate a study of the quantum deformations of the associated Lie algebra representations. Results on the Poincaré content of the unitarizable, irreducible composition factors are stated, and unitary equivalences between certain of these factors are established.