Volume 40, Issue 11, November 1999
 QUANTUM PHYSICS; PARTICLES AND FIELDS


On the Schrödinger equation with steplike potentials
View Description Hide DescriptionThe onedimensional Schrödinger equation is considered when the potential is asymptotic to a positive constant on the right half line in a certain sense. The zeroenergy limits of the scattering coefficients are obtained under weaker assumptions than used elsewhere, and the continuity of the scattering coefficients from the left are established. The scattering coefficients for the potential are expressed in terms of the corresponding coefficients for the pieces of the potential on the positive and negative half lines. The number of bound states for the whole potential is related to the number of bound states for the two pieces. Finally, an improved result is given on the smallenergy asymptotics of reflection coefficients for potentials supported on a half line.

On the Batalin, Fradkin, Fradkina, and Tyutin quantization of first order systems
View Description Hide DescriptionBy using the field–antifield formalism, we show that the method of Batalin, Fradkin, Fradkina, and Tyutin (BFFT) to convert Hamiltonian systems submitted to second class constraints introduces compensating fields which do not belong to the BRST cohomology at ghost number one. This assures that the gauge symmetries which arise from the BFFT procedure are not obstructed at quantum level. An example where massive electrodynamics is coupled to chiral fermions is considered. We solve the quantum master equation for the model and show that the respective counterterm has a decisive role in extracting anomalous expectation values associated with the divergence of the Noether chiral current.

A path space formula for Gauss vectors in Chern–Simons quantum electrodynamics
View Description Hide DescriptionCanonical quantization of a Chern–Simons gauge field minimally coupled to a spinor field is studied as an indefinite metric quantum field theory in the usual covariant gauges, by using a lattice cutoff. For this model, we show that positivity for the indefinite metric, Gauss’ Law, gauge invariance and Osterwalder–Schrader positivity for a selfadjoint Hamiltonian are equivalent. In addition, the pathspace formula for the Osterwalder–Schrader semigroup is constructed in terms of a Euclidean scalar, massive, Gaussian random field.

Multidimensional integrable Schrödinger operators with matrix potential
View Description Hide DescriptionThe Schrödinger operators with matrix rational potential, which are Dintegrable, i.e., can be intertwined with the pure Laplacian, are investigated. Corresponding potentials are uniquely determined by their singular data which are a configuration of the hyperplanes in with prescribed matrices. We describe some algebraic conditions (matrix locus equations) on these data, which are sufficient for Dintegrability. As the examples some matrix generalizations of the Calogero–Moser operators are considered.

Evaluation of multiloop diagrams via lightcone integration
View Description Hide DescriptionWe present a systematic method to determine the dominant regions of internal momenta contributing to any twobody highenergy nearforward scattering diagram. Such a knowledge is used to evaluate leading highenergy dependences of loop diagrams. It also gives a good idea where dominant multiparticle cross sections occur.

Quasispin gradedfermion formalism and branching rules
View Description Hide DescriptionThe gradedfermion algebra and quasispin formalism are introduced and applied to obtain the branching rules for the “twocolumn” tensor irreducible representations of for the case In the case all such irreducible representations of are shown to be completely reducible as representations of This is also shown to be true for the case except for the “spinsinglet” representations, which contain an indecomposable representation of with composition length 3. These branching rules are given in fully explicit form.

Construction of relativistic quantum fields in the framework of white noise analysis
View Description Hide DescriptionWe construct a class of Euclidean invariant distributions indexed by a function H holomorphic at zero. These generalized functions can be considered as generalized densities w.r.t. the white noise measure, and their moments fulfill all Osterwalder–Schrader axioms, except for reflection positivity. The case where is a Lévy characteristic is considered in Rev. Math. Phys. 8, 763 (1996). Under this assumption the moments of the Euclidean invariant distributions can be represented as moments of a generalized white noise measure Here we enlarge this class by convolution with kernels G coming from Euclidean invariant operators G. The moments of the resulting Euclidean invariant distributions also fulfill all Osterwalder–Schrader axioms except for reflection positivity. For no nontrivial case we succeeded in proving reflection positivity. Nevertheless, an analytic extension to Wightman functions can be performed. These functions fulfill all Wightman axioms except for the positivity condition. Moreover, we can show that they fulfill the Hilbert space structure condition and therefore the modified Wightman axioms of indefinite metric quantum field theory [Dynamics of Complex and Irregular Systems (World Scientific, Singapore, 1993)].

Derivative expansion of the effective action for quantum electrodynamics in 2+1 and 3+1 dimensions
View Description Hide DescriptionThe derivative expansion of the oneloop effective action in and (quantum electrodynamics) is considered. The first term in such an expansion is the effective action for a constant electromagnetic field. An explicit expression for the next term containing two derivatives of the field strength but exact in the magnitude of the field strength, is obtained. The general results for both fermion and scalar electrodynamics are presented. The cases of pure electric and pure magnetic external fields are considered in detail. The Feynman technique for the perturbative expansion of the oneloop effective action in the number of derivatives is developed.

Monopoles and harmonic maps
View Description Hide DescriptionRecently Jarvis has proved a correspondence between monopoles and rational maps of the Riemann sphere into flag manifolds. Furthermore, he has outlined a construction to obtain the monopole fields from the rational map. In this paper we examine this construction in some detail and provide explicit examples for spherically symmetric monopoles with various symmetry breakings. In particular we show how to obtain these monopoles from harmonic maps into complex projective spaces. The approach extends in a natural way to monopoles in hyperbolic space and we use it to construct new spherically symmetric hyperbolic monopoles.

Adiabatic evolution for systems with infinitely many eigenvalue crossings
View Description Hide DescriptionWe formulate an adiabatic theorem adapted to models that present an instantaneous eigenvalue experiencing an infinite number of crossings with the rest of the spectrum. We give an upper bound on the leading correction terms with respect to the adiabatic limit. The result requires only differentiability of the considered projector, and some geometric hypothesis on the local behavior of the eigenvalues at the crossings.

New solvable and quasiexactly solvable periodic potentials
View Description Hide DescriptionUsing the formalism of supersymmetric quantum mechanics, we obtain a large number of new analytically solvable onedimensional periodic potentials and study their properties. More specifically, the supersymmetric partners of the Lamé potentials are computed for integer values For all cases (except we show that the partner potential is distinctly different from the original Lamé potential, even though they both have the same energy band structure. We also derive and discuss the energy band edges of the associated Lamé potentials which constitute a much richer class of periodic problems. Computation of their supersymmetric partners yields many additional new solvable and quasiexactly solvable periodic potentials.

Random Schrödinger operators arising from lattice gauge fields. I. Existence and examples
View Description Hide DescriptionWe consider new models of ergodic Schrödinger operators whose existence relies on a cohomological theorem of Feldman and Moore in ergodic theory. These operators generalize the Harper operator which describes the case of a constant magnetic field. An example is the case when the magnetic field is given by independent random variables attached to the lattice plaquettes. A generalization of the Feldman–Moore theorem by Lind to nonAbelian groups also allows us to consider Schrödinger operators obtained from nonAbelian lattice gauge fields. The existence result extends to more general graphs like to operators on tilings and to higher dimensions. We compute some moment expansions for the density of states. For example, for independent, identically and uniformly distributed magnetic fields, a model which has been studied at least since 1970, and whose existence can also be seen without involving the abovementioned existence theorem, we show that the nth moment is the number of closed paths in the twodimensional lattice starting at the origin for which the winding number vanishes at each plaquette point. This goes beyond the Brinkman–Rice selfretracing path approximation. Other examples are a higher dimensional example, a onedimensional Anderson model which can be treated in this framework, as well as the Hofstadter model with constant magnetic field, where one averages over all possible magnetic fields. We also reprove a result of Jitomirskaya–Mandelshtam stating that the deterministic Aharonov–Bohm model is a compact perturbation of the free Laplacian.

Nonstandard Feynman path integral for the harmonic oscillator
View Description Hide DescriptionUsing Nonstandard Analysis, we will provide a rigorous computation for the harmonic oscillator Feynman path integral. The computation will be done without having prior knowledge of the classical path. We will see that properties of classical physics falls out naturally from a purely quantum mechanical point of view. We will assume that the reader is familiar with Nonstandard Analysis.

Bose–Einstein condensation in an external potential at zero temperature: Solitarywave theory
View Description Hide DescriptionFor a trapped, dilute atomic gas of shortrange, repulsive interactions at extremely low temperatures, when Bose–Einstein condensation is nearly complete, some special forms of the timedependent condensate wave function and the pairexcitation function, the latter being responsible for phonon creation, are investigated. Specifically, (i) a class of external potentials that allow for localized, shapepreserving solutions to the nonlinear Schrödinger equation for the condensate wave function, each recognized as a solitary wave moving along an arbitrary trajectory, is derived and analyzed in any number of space dimensions; and (ii) for any such external potential and condensate wave function, the nonlinear integrodifferential equation for the pairexcitation function is shown to admit solutions of the same nature. Approximate analytical results are presented for a sufficiently slowly varying trapping potential. Numerical results are obtained for the condensate wave function when is a timeindependent, spherically symmetric harmonic potential.

A generalization of Wigner’s unitary–antiunitary theorem to Hilbert modules
View Description Hide DescriptionLet H be a Hilbert module over a matrix algebraA. It is proved that any function which preserves the absolute value of the (generalized) inner product is of the form where φ is a phasefunction and U is an Alinear isometry. The result gives a natural extension of Wigner’s classical unitary–antiunitary theorem for Hilbert modules.

Two new potentials for the free particle model
View Description Hide DescriptionIn this work, is proposed a very simple method for obtaining the generalized potential associated with a known standard potential. The procedure is straightforward because it only uses two Ricattitype relationships as enough condition to find a generalized potential; one particular equation is needed to identify the specific potential under study and one general Ricatti relationship is used to find the corresponding generalized potential. Moreover, the method is completely general due to the fact that an arbitrary potential has been considered in its development for which the procedure can also be used to find new potentials which could be needed in the modeling of important quantum interactions. The usefulness of the proposed approach, is shown with the treatment of the three and onedimensional potential for the free particle model. This work example leads to two new potentials whose Hamiltonians are isospectral when they are compared with the former Hamiltonian.

Derivation of the wave function collapse in the context of Nelson’s stochastic mechanics
View Description Hide DescriptionThe von Neumann collapse of the quantum mechanical wave function after a position measurement is derived by a purely probabilistic mechanism in the context of Nelson’s stochastic mechanics.

Estimates for the spectral shift function of the polyharmonic operator
View Description Hide DescriptionThe Lifshits–Krein spectral shift function is considered for the pair of operators and in here is a multiplication operator. The estimates for this spectral shift function are obtained in terms of the spectral parameter and the integral norms of These estimates are in a good agreement with the ones predicted by the classical phase space volume considerations.

Symplectic Dirac–Kähler fields
View Description Hide DescriptionFor the description of space–time fermions, Dirac–Kähler fields (inhomogeneous differential forms) provide an interesting alternative to the Dirac spinor fields. In this paper we develop a similar concept within the symplectic geometry of phase spaces. Rather than on space–time, symplectic Dirac–Kähler fields can be defined on the classical phase space of any Hamiltonian system. They are equivalent to an infinite family of metaplectic spinor fields, i.e., spinors of in the same way an ordinary Dirac–Kähler field is equivalent to a (finite) multiplet of Dirac spinors. The results are interpreted in the framework of the gauge theory formulation of quantum mechanics which was proposed recently. An intriguing analogy is found between the lattice fermion problem (species doubling) and the problem of quantization in general.

Worldline Green functions with momentum and source conservations
View Description Hide DescriptionBased on the generating functional method with an external source function, a useful constraint on the source function is proposed for analyzing the one and twoloop worldline Green functions. The constraint plays the same role as the momentum conservation law of a certain nontrivial form, and transforms ambiguous Green functions into the uniquely defined Green functions. We also argue reparametrizations of the Green functions defined on differently parameterized worldline diagrams.
