Volume 39, Issue 9, September 1998
 QUANTUM PHYSICS; PARTICLES AND FIELDS


On the number of bound states for the onedimensional Schrödinger equation
View Description Hide DescriptionThe number of bound states of the onedimensional Schrödinger equation is analyzed in terms of the number of bound states corresponding to “fragments” of the potential. When the potential is integrable and has a finite first moment, the sharp inequalities are proved, where p is the number of fragments, N is the total number of bound states, and is the number of bound states for the fragment. When the question of whether or is investigated in detail. An illustrative example is also provided.

Symplectic structure for Gaussian diffusions
View Description Hide DescriptionUsing information exclusively about classical Hamiltonian boundary value problems (with quadratic timedependent potentials) we construct probability measures providing a symplectic structure associated with some inhomogeneous Gaussian diffusion processes introduced earlier [T. Kolsrod and J. C. Zambrini, J. Math. Phys. 33, 1301 (1992)].

The effect of configuration mixing on the firstorder moments of polarized hydrogen lines
View Description Hide DescriptionWe extend previous work on the algebraic firstorder moments of polarization profiles of hydrogen lines in the presence of external fields, to include the effect of configuration mixing induced by the applied fields. An algebraic formula for hydrogen line shifts due to the quadratic Stark effect is derived, under the assumption that the applied electric field is weak enough that line quenching induced by field ionization can be neglected. The derivation of such a formula is made possible by an aimed use of standard techniques of projectionoperator algebra and of the fundamental implications of the spectral theorem. The results of this paper should help in decreasing the amount of numerical work needed in the calculation of line shifts of hydrogen lines induced by the presence of electric fields, as the preliminary numerical calculation of the Hamiltonian eigenvalues and eigenvectors is avoided, thus making calculations tractable for lines of higher hydrogen series than considered so far, e.g., in applications of pressurebroadening theory.

A geometrical angle on Feynman integrals
View Description Hide DescriptionA direct link between a oneloop Npoint Feynman diagram and a geometrical representation based on the Ndimensional simplex is established by relating the Feynman parametric representations to the integrals over contents of dimensional simplices in nonEuclidean geometry of constant curvature. In particular, the fourpoint function in four dimensions is proportional to the volume of a threedimensional spherical (or hyperbolic) tetrahedron which can be calculated by splitting into birectangular ones. It is also shown that the known formula of reduction of the Npoint function in dimensions corresponds to splitting the related Ndimensional simplex into N rectangular ones.

A proof of the dynamical version of the Bombieri–Taylor conjecture
View Description Hide DescriptionAn adaptation of the Bombieri–Taylor conjecture on diffraction has been applied to some timedependent quantum systems. Here the problem is mathematically stated and a proof is presented. Simple examples give an idea as to what extent the conjecture is valid.

The general decomposition theory of SU(2) gauge potential, topological structure and bifurcation of SU(2) Chern density
View Description Hide DescriptionBy means of the geometricalgebra the general decomposition of SU(2) gauge potential on the sphere bundle of a compact and oriented fourdimensional manifold is given. Using this decompositiontheory the SU(2) Chern density has been studied in detail. It shows that the SU(2) Chern density can be expressed in terms of the δfunction δ(φ). One can also find that the zero points of the vector fields φ are essential to the topological properties of a manifold. It is shown that there exists the crucial case of branch process at the zero points. Based on the implicit function theorem and the Taylor expansion, the bifurcation of the Chern density is detailed in the neighborhoods of the bifurcation points of φ. It is pointed out that, since the Chern density is a topological invariant, the sum topological chargers of the branches will remain constant during the bifurcation process.

The elliptic quantum algebra and its bosonization at level one
View Description Hide DescriptionWe extend the work of Foda et al. and propose an elliptic quantum algebra Similar to the case of our presentation of the algebra is based on the relation where R and are symmetric R matrices with the elliptic moduli chosen differently, and a scalar factor is also involved. With the help of the results obtained by Asai et al., we realize type I and type II vertex operators in terms of bosonic free fields for the symmetric Belavin model. We also give a bosonization for the elliptic quantum algebra at level one.

Coupling constant thresholds of perturbed periodic Hamiltonians
View Description Hide DescriptionWe consider Schrödinger operators of the form on (ν=1, 2, or 3) with periodic, short range, and λ a real nonnegative parameter. Then the continuous spectrum of has the typical band structure consisting of intervals, separated by gaps. In the gaps there may be discrete eigenvalues of that are functions of the parameter λ. Let be a gap and an eigenvalue of We study the asymptotic behavior of as λ approaches a critical value called a coupling constant threshold, at which the eigenvalue either emerges from or is absorbed into the continuous spectrum. A typical question is the following: Assuming as is for some and or is there an expansion in some other quantity? As one expects from previous work in the case the answer strongly depends on ν.

JWKB connectionformula problem revisited again via Borel summation
View Description Hide DescriptionSilverstone [Phys. Rev. Lett. 55, 2523–2526 (1985)] has expressed the opinion that the traditional version of the JWKB connection formulas associated with a classical turning point is incorrect and that the correct version follows from the Borelbased summability of the asymptotic expansions for the Airy functions. In the present paper we show that this assertion is incorrect. After the Borel summation of the asymptotic expressions for the Airy functions has actually been performed explicitly, there are no connection problems, and thus no connection formulas, but exact relations expressing the functions and in terms of the Whittaker function, although with two defects: Due to the derivation there appear false restrictions on and there is an artificial cut that coincides with the Stokes line along the real zaxis. We also show that these exact relations can be obtained more appropriately in a direct way with the aid of standard formulas available in handbooks, whereby the defects mentioned do not appear. On the other hand, when one uses truncated asymptotic series, the Stokes phenomenon gives rise to connection problems, and the connection formulas, which one uses for handling these connection problems, are inherently onedirectional. This property of the connection formulas is demonstrated and discussed anew, in general terms as well as with special regard to the Airy functions.

Derivative expansion of oneloop effective actions for YangMills fields
View Description Hide DescriptionLow energy effective actions are computed by integrating out degrees of freedom heavier than some energy scale of physical interest. At the oneloop level, the low energy effective action can be computed in terms of the functional trace of the heat kernel associated with the operator appearing in the part of the action quadratic in the heavy fields. Here, a technique for computing this functional trace is illustrated in the case of heavy scalar fields coupled to a YangMills background. It makes use of the similarity of the functional trace of the heat kernel expressed in momentum space to a Gaussian integral. An explicit computation of the piece of the low energy effective action for the YangMills fields quadratic in derivatives of the field strength is carried out, extending previously known results on the form of the long wavelength limit of the low energy effective action.

Mathematical derivation of chiral anomaly in lattice gauge theory with Wilson’s action
View Description Hide DescriptionChiral U(1) anomaly is derived with mathematical rigor for a Euclidean fermion coupled to a smooth external U(1) gauge field on an even dimensional torus as a continuum limit of lattice regularized fermion field theory with the Wilson term in the action. The present work rigorously proves for the first time that the Wilson term correctly reproduces the chiral anomaly.

Feynman integrals for a class of exponentially growing potentials
View Description Hide DescriptionWe construct the Feynman integrands for a class of exponentially growing timedependent potentials as white noise functionals. We show that they solve the Schrödinger equation. The Morse potential is considered as a special case.

On simulating Liouvillian flow from quantum mechanics via Wigner functions
View Description Hide DescriptionThe interconnection between quantum mechanics and probabilistic classical mechanics for a free relativistic particle is derived in terms of Wigner functions (WF) for both Dirac and KleinGordon (KG) equations. Construction of WF is achieved by first defining a bilocal 4current and then taking its Fourier transform w.r.t. the relative 4coordinate. The KG and Proca cases also lend themselves to a closely parallel treatment provided the KemmerDuffin matrix formalism is employed for the former. Calculation of WF is carried out in a Lorentzcovariant fashion by standard “trace” techniques. The results are compared with a recent derivation due to Bosanac.

Twocomponent formulation of the Wheeler–DeWitt equation
View Description Hide DescriptionThe Wheeler–DeWitt equation for the minimally coupled Friedman–Robertson–Walkermassivescalarfield minisuperspace is written as a twocomponent Schrödinger equation with an explicitly “time”dependent Hamiltonian. This reduces the solution of the Wheeler–DeWitt equation to the eigenvalue problem for a nonrelativistic onedimensional harmonic oscillator and an infinite series of trivial algebraic equations whose iterative solution is easily found. The solution of these equations yields a mode expansion of the solution of the original Wheeler–DeWitt equation. Further analysis of the mode expansion shows that in general the solutions of the Wheeler–DeWitt equation for this model are doubly graded, i.e., every solution is a superposition of two definiteparity solutions. Moreover, it is shown that the mode expansion of both even and oddparity solutions is always infinite. It may be terminated artificially to construct approximate solutions. This is demonstrated by working out an explicit example which turns out to satisfy DeWitt’s boundary condition at initial singularity.

Quantum Anosov flows: A new family of examples
View Description Hide DescriptionA quantum version is presented for the Anosov system defined by the time evolution implemented by the geodesic coflow on the cotangent bundle of any compact quotient manifold obtained from the Poincaré halfplane. While the canonical Weyl algebra does not close under time evolution, the symplectic structure of these classical systems can be exploited to produce objects akin to the CCR algebras encountered in quantum field theory. This construction allows one to lift both the geodesic and the horocyclic flows to a Weyl algebra describing the quantum dynamics corresponding to the systems under consideration. The Anosov relations as proposed in Ref. Reference 1 are found to be valid for these models. A quantum version of the classical ergodicity of these systems is discussed in the last section.

Thermodynamic fermion loop in a constant magnetic field
View Description Hide DescriptionThe oneloop effective potential of a thermodynamic fermion loop in a constant magnetic field is studied. As expected, it can be interpreted literally as a discretized sum of dimensional energy density above the Dirac sea. Large/small mass expansions of the potential are also examined.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Amplitude equations for electrostatic waves: Multiple species
View Description Hide DescriptionThe amplitude equation for an unstable electrostatic wave is analyzed using an expansion in the mode amplitude In the limit of weak instability, i.e., where γ is the linear growth rate, the nonlinear coefficients are singular and their singularities predict the dependence of on γ. Generically the scaling as is required to cancel the coefficient singularities to all orders. This result predicts the electric field scaling will hold universally for these instabilities (including beamplasma and twostream configurations) throughout the dynamical evolution and in the timeasymptotic state. In exceptional cases, such as infinitely massive ions, the coefficients are less singular and the more familiar trapping scaling is recovered.

Multivector fields and connections: Setting Lagrangian equations in field theories
View Description Hide DescriptionThe integrability of multivector fields in a differentiable manifold is studied. Then, given a jet bundle it is shown that integrable multivector fields in E are equivalent to integrable connections in the bundle (that is, integrable jet fields in This result is applied to the particular case of multivector fields in the manifold and connections in the bundle (that is, jet fields in the repeated jet bundle in order to characterize integrable multivector fields and connections whose integral manifolds are canonical lifting of sections. These results allow us to set the Lagrangian evolution equations for firstorder classical field theories in three equivalent geometrical ways (in a form similar to that in which the Lagrangian dynamical equations of nonautonomous mechanical systems are usually given). Then, using multivector fields, we discuss several aspects of these evolution equations (both for the regular and singular cases); namely, the existence and nonuniqueness of solutions, the integrability problem and Noether’s theorem, giving insights into the differences between mechanics and field theories.

Limit analysis of the diffraction of a plane wave by a onedimensional periodic medium
View Description Hide DescriptionIn this paper we address the problem of the reflection coefficient limit in a periodic stratified structure when the number of periods tends to infinity. We show that the graph of the reflected energy admits a superior envelope and we exhibit an explicit formula.
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 STATISTICAL PHYSICS AND STOCHASTIC PROCESSES


Black hole thermodynamics and spectral analysis
View Description Hide DescriptionWe deal with some problems concerning quantum matter fields thermodynamics on static spherosymmetric black hole backgrounds. By means of variable separation, we study the selfadjointness of the Euclidean wave operators relevant in the calculation of the partition function and of the thermal Green functions for the case of scalar quantum fields in thermal equilibrium with a Schwarzschild black hole. In this respect, the role of the horizon surface is analyzed. An approach to this topic, related to completeness of the Euclidean manifold, is also discussed. It is found that, if the horizon surface is not included in the Euclidean manifold (or if it cannot be included, as in the case of a temperature different from the black hole one), essential selfadjointness is missing and boundary conditions on the horizon are required. We also analyze some qualitative spectral properties of the given wave operators after introducing an infrared regularization by means of a spherical box. We find that in the standard Euclidean path integral approach the spectrum is purely discrete. The same analysis is carried out for the noncovariant choice of the functional measure in the Euclidean path integral. An analogous study is carried for an extremal Reissner–Nordström black hole and it is sketched in the nonextremal one. The latter is completely analogous to the Schwarzschild case, whereas in the extremal case it is found that the corresponding Riemannian manifold is complete for every choice of the temperature and that at finite four volume there is a nonvoid continuous spectrum. Physical implications of our results are discussed.
