Volume 36, Issue 10, October 1995
Index of content:

Representation of the propagator and Schwinger functions of Dirac fields in terms of Brownian motions
View Description Hide DescriptionA mathematical path integral representation of the propagator and Schwinger functions of the Dirac fields on R ^{4} interacting with a smooth non‐Abelian gauge field is presented. The representation is essentially in terms of expectations with respect to the Brownian motion on R ^{4}.

Euclidean random fields obtained by convolution from generalized white noise
View Description Hide DescriptionWe study Euclidean random fields X over R ^{ d } of the form X=G*F, where F is a generalized white noise over R ^{ d } and G is an integral kernel. We give conditions for the existence of the characteristic functional and moment functions and we construct a convergent lattice approximation of X. Finally, we perform the analytic continuation of the moment functions and the characteristic functional of X, obtaining the corresponding relativistic functions on Minkowski space.

The L‐S coupling revisited: A general solution for the 4 electron system
View Description Hide DescriptionThe relation between the unitary group SU(n) and the symmetric group S _{ n } is treated using the induced characters of S _{ n }. By means of this approach the L‐S coupling for four electrons is solved.

Classical and quantum sl(1‖2) superalgebras, Casimir operators and quantum chain Hamiltonians
View Description Hide DescriptionWe examine in this paper the two parameter deformed superalgebra U_{ qs }(sl(1‖2)) and use the results in the construction of quantum chain Hamiltonians. This study is done both in the framework of the Serre presentation and in the R‐matrix scheme of Faddeev, Reshetikhin, and Takhtajan (FRT). We show that there exists an infinite number of Casimir operators, indexed by integers p≥2 in the undeformed case and by p∈Z in the deformed case, which obey quadratic relations. The construction of the dual superalgebra of functions of SL _{ qs }(1‖2) is also given and higher tensor product representations are discussed. Finally, we construct quantum chain Hamiltonians based on the Casimir operators. In the deformed case we find two Hamiltonians which describe deformed t−J models.

A Z_{2} structure in the configuration space of Yang–Mills theories
View Description Hide DescriptionWe argue for the presence of a Z _{2} topological structure in the space of static gauge–Higgs field configurations of SU(2n) and SO(2n) Yang–Mills theories. We rigorously prove the existence of a Z _{2} homotopy group of mappings from the two‐dimensional projective sphere R P ^{2} into SU(2n)/Z _{2} and SO(2n)/Z _{2}Lie groups, respectively. Consequently the symmetric phase of these theories admits infinite surfaces of odd‐parity static and unstable gauge field configurations which divide into two disconnected sectors with integer Chern–Simons numbers n and n+1/2, respectively. Such a Z _{2} structure persists in the Higgs phase of the above theories and accounts for the existence of CS=1/2 odd‐parity saddle point solutions to the field equations which correspond to spontaneous symmetry breaking mass scales.

Charged particles in a magnetic field on a cone
View Description Hide DescriptionWe analyze the problem of two charged particles moving on a cone in the presence of a static, uniform magnetic field plus a Coulomb interaction. In the specific case where the deficit angle δφ=π, closed expressions for the wave functions and energies are obtained.

Construction of a transformation operator
View Description Hide DescriptionWe shall be concerned with the explicit construction of the kernel of transformation operator used in the scattering theory associated with the Gelfand–Levitan–Marchenko equations. The kernel of the transmutation is well known to play an important role in the inverse scatteringtheory, and nonlinear integrable systems such as KdV equations. Our method shall use operational calculus.

On the solution of the inverse scattering problem, at fixed energy, for the class of Yukawian potentials
View Description Hide DescriptionIn this paper we prove that if in the Fourier–Legendre expansion of the scattering amplitude the partial‐waves f _{ l } satisfy a Hausdorff‐type condition and are exponentially bounded in l, then this expansion converges uniformly to a function f(θ) (θ=u+iv is the complexified scattering angle) analytic in a strip contained in the θ‐plane; furthermore we prove that this function has a holomorphic extension to a ‘‘cut‐domain.’’ From the discontinuity of f(θ) across the cuts we can reconstruct the spectral density σ(μ), μ=k{2(cosh v−1)}^{1/2}, associated with the class of Yukawian potentials. We give also an inversion algorithm which is able to reconstruct the density σ(μ), via an iterative procedure, starting from the partial‐waves {f _{ l }}^{∞} _{0}. Since the data f _{ l } are necessarily affected by noise, and we know only a finite number of them, then the data set is given by {f ^{(ε)} _{ l }}^{ N } _{0}, where ε characterizes the noise. It follows that there is a certain degree of ‘‘irreversibility’’ in the inverse problem, due to its ill‐posedness. All these questions, as well as the various problems connected with the convergence of the algorithm, are analyzed in details. Finally, the numerical efficiency of the algorithm is tested on some numerical examples.

Quantum chains with a Catalan tree pattern of conserved charges: The Δ=−1 XXZ model and the isotropic octonionic chain
View Description Hide DescriptionA class of quantum chains possessing a family of local conserved charges with a Catalan tree pattern is studied. Recently, we have identified such a structure in the integrable SU(N)‐invariant chains. In the present work we find sufficient conditions for the existence of a family of charges with this structure in terms of the underlying algebra. Two additional systems with a Catalan tree structure of conserved charges are found. One is the spin 1/2 XXZ model with Δ=−1. The other is a new octonionic isotropic chain, generalizing the Heisenberg model. This system provides an interesting example of an infinite family of noncommuting local conserved quantities.

Quantized Neumann problem, separable potentials on S ^{ n } and the Lamè equation
View Description Hide DescriptionThe paper studies spectraltheory of Schrödinger operators H=ℏ^{2}Δ+V on the sphere from the standpoint of integrability and separation. Our goal is to uncover the fine structure of spec H, i.e., asymptotics of eigenvalues and spectral clusters, determine their relation to the underlying geometry and classical dynamics and apply this data to the inverse spectral problem on the sphere. The prototype model is the celebrated Neumann Hamiltonian p ^{2}+V with quadratic potential V on S ^{ n }. We show that the quantum Neumann Hamiltonian (Schrödinger operator H) remains an integrable and find an explicit set of commuting integrals. We also exhibit large classes of separable potentials {V} based on ellipsoidal coordinates on S ^{ n }. Several approaches to spectraltheory of such Hamiltonians are outlined. The semiclassical problem (small ℏ) involves the EKB(M)‐quantization of the classical Neumann flow along with its invariant tori, Maslov indices, etc., all made explicit via separation of variables. Another approach exploits Stäckel–Robertson separation of the quantum Hamiltonian and reduction to certain ODE problems: the Hill’s and the generalized Lamèequations. The detailed analysis is carried out for S ^{2}, where the ODE becomes the perturbed classical Lamè equation and the Schrödinger eigenvalues are expressed through the Lamè eigendata.

Continuous histories and the history group in generalized quantum theory
View Description Hide DescriptionWe treat continuous histories within the histories approach to generalized quantum mechanics. The essential tool is the ‘‘history group:’’ the analog, within the generalized history scheme, of the canonical group of single‐time quantum mechanics.

The structure of decoherence functionals for von Neumann quantum histories
View Description Hide DescriptionGell‐Mann and Hartle have proposed a significant generalization of quantum theory in which decoherence functionals perform a key role. Isham–Linden–Schreckenberg have given a penetrating analysis of decoherence functionals for L(H), where H is finite dimensional (dimension greater than 2). In this note their conjecture on the significance of boundedness is verified. In particular, it is shown that when d is a bounded decoherence functional associated with a von Neumann algebraA, then, provided A has no direct summand of type I _{2}, d can be represented as the difference between semi‐innerproducts on A.

Gauging discrete symmetries
View Description Hide DescriptionRecent developments in quantum gravitytheory have led to the suggestion that various discrete symmetries, in particular charge–parity (CP), should be ‘‘gauged,’’ that is, interpreted as elements of some connected Lie group. As the parity operator is related to a space–time isometry, however, it is far from clear that this suggestion has any real meaning. We give a simple geometric construction in terms of which it is meaningful to ‘‘gauge’’ discrete symmetries,provided that certain nontrivial conditions are satisfied by the space–time manifold, by the gauge group, and by the discrete symmetry itself.

Asymptotics of radial wave equations
View Description Hide DescriptionThe Langer modification is an improvement in the WKB analysis of the radial Schrödinger equation. We derive a generalization of the Langer modification to any radial operator. For differential operators we write the modified classical symbols explicitly and show that the WKB wavefunctions with the modification have the exact limiting behavior for small radius. Unlike in the Schrödinger case, generally the modified radial analysis is not equivalent to the WKB analysis of the full problem before reduction by the spherical symmetry.

Green’s‐function approach to two‐ and three‐dimensional delta‐function potentials and application to the spin‐1/2 Aharonov–Bohm problem
View Description Hide DescriptionDelta‐function potentials in two‐ and three‐dimensional quantum mechanics are analyzed by incorporating the self‐adjoint extension method within the Green’s‐function method. The energy‐dependent Green’s functions for free particle plus delta‐function potential systems are explicitly determined and similar calculations are carried out for the spin‐1/2 Aharonov–Bohm system. It is found that the time‐dependent propagator for the latter cannot be evaluated analytically except for a particular value of the self‐adjoint extension parameter. It corresponds to the case in which the singular solution alone contributes for the partial wave defined by ‖m+α‖<1.

Gauge fields on the quantum net
View Description Hide DescriptionWe continue our study of the quantum net, at a coarser resolution than previously. We adapt and extend various ideas of Finkelstein to delineate the structure of the space of defects, and develop a formal dynamics along the lines of the Schwinger action principle. Formal variation of the defect and the application of the correspondence principle developed earlier lead immediately to macroscopic continuum Lagrangian densities taking the form of power series in the net constant, whose lowest‐order terms reproduce exactly (up to constant factors): the Yang–Mills form, plus the Feynman gauge fixing term, for SU(2) and SU(3) gauge theories, and the Einstein–Hilbert form for gravity. The algebraic manipulations, though formal, are entirely elementary. The theory is, as yet, local, unghosted, and not second‐quantized. No U(1) gauge symmetry, or Higgs‐like field, is apparent.

Semiclassical approximation for Schrödinger operators on a two‐sphere at high energy
View Description Hide DescriptionLet H=−Δ _{S} +V be a Schrödinger operator acting in L ^{2}(S), with S the two‐dimensional unit sphere, Δ _{S} the spherical Laplacian, and V a smooth potential. Approximate eigenfunctions and eigenvalues for H are obtained involving expansions in inverse powers of the classical angular momentum variables, provided that these variables are in a region of phase space where the corresponding classical Hamiltonian is nearly integrable. The analysis is carried out in a Bargmann representation, where Δ _{S} becomes a quadratic expression in the sum of two quantum harmonic oscillator Hamiltonians, and V becomes a pseudodifferential operator.

On the new approach to variable separation in the time‐dependent Schrödinger equation with two space dimensions
View Description Hide DescriptionWe suggest an effective approach to separation of variables in the Schrödinger equation with two space variables. Using it we classify inequivalent potentials V(x _{1},x _{2}) such that the corresponding Schrödinger equations admit separation of variables. Besides that, we carry out separation of variables in the Schrödinger equation with the anisotropic harmonic oscillator potential V=k _{1} x ^{2} _{1}+k _{2} x ^{2} _{2} and obtain a complete list of coordinate systems providing its separability. Most of these coordinate systems depend essentially on the form of the potential and do not provide separation of variables in the free Schrödinger equation (V=0).

Coisotropic regularization of singular Lagrangians
View Description Hide DescriptionWe present an alternative approach to the usual treatments of singular Lagrangians. It is based on a Hamiltonian regularization scheme inspired on the coisotropic embedding of presymplectic systems. A Lagrangian regularization of a singular Lagrangian is a regular Lagrangian defined on an extended velocity phase space that reproduces the original theory when restricted to the initial configuration space. A Lagrangian regularization does not always exists, but a family of singular Lagrangians is studied for which such a regularization can be described explicitly. These regularizations turn out to be essentially unique and provide an alternative setting to quantize the corresponding physical systems. These ideas can be applied both in classical mechanics and field theories. Several examples are discussed in detail.

Angular momentum and space–time hyperboloids
View Description Hide DescriptionFor a classical field theory, in Minkowski space–time, the field momentum and angular momentum are defined as an integral of the appropriate density over an asymptotically null hyperboloid. We prove that for the electromagnetic field these quantities are well defined even for charged sources where infrared divergences are a problem for defining the angular momentum. Moreover, it is also shown that if the angular momentum density is integrated over a null plane, the angular momentum is not well defined for charged sources.