Volume 37, Issue 2, February 1996
Index of content:

Path integrals for spinning particles, stationary phase and the Duistermaat–Heckmann theorem
View Description Hide DescriptionWe examine the problem of the evaluation of both the propagator and of the partition function of a spinning particle in an external field at the classical as well as the quantum level, in connection with the asserted exactness of the stationary phase approximation. At the classical level we argue that exactness of this approximation stems from the fact that the dynamics (on the two‐sphere S ^{2}) of a spinning particle in a magnetic field is the reduction from R ^{4} to S ^{2} of a linear dynamical system on R ^{4}. At the quantum level, however, and within the path integral approach, the restriction, inherent to the use of the stationary phase approximation, to regular paths clashes with the fact that no regulators are present in the action that enters the path integral. This is shown to lead to a prefactor for the path integral that is strictly divergent, except in the classical limit. A critical comparison is made with the various approaches that have been presented in the literature. The validity of a formula given in literature for the spin propagator is extended to the case of motion in an arbitrary magnetic field.

Semilocal Chern–Simons defects
View Description Hide DescriptionWe find both radially symmetric and more general solutions in a Chern–Simons–Higgs model with a Higgs doublet. They are of the kind of the so‐called semilocal defects, and show richer properties than those previously known in another type of Higgs models.

Coherent states over Grassmann manifolds and the WKB exactness in path integral
View Description Hide DescriptionU(N) coherent states over Grassmann manifold,G _{ N,n }≂U(N)/(U(n)×U(N−n)), are formulated to be able to argue the WKB exactness in the path integral representation of a character formula. The phenomena is the so‐called localization of Duistermaat–Heckman. The exponent in the path integral formula is proportional to an integer k labeling the U(N) representation. Thus, when k→∞ a usual semiclassical approximation, by regarding k∼1/ℏ, can be performed to yield a desired conclusion. The mechanism of the localization is uncovered by the help of the (generalized) Schwinger boson technique. The discussion on the Feynman kernel is also presented.

Berry’s phase on SU(n)/S(U(1)×U(n−1)) manifolds
View Description Hide DescriptionIt is shown that the Berry phase/matrix for Hamiltonians parametrized by the points of M≡SU(n)/S(U(1)×U(n−1)) and having no accidental degeneracy is expressible in terms of Riemannian connection on M and representations of S(U(1)×U(n−1)).

Quantization of the multifold Kepler system
View Description Hide DescriptionFor over a century, the Kepler problem and the harmonic oscillator have been known as the only central force dynamical systems, all of whose bounded motions are periodic. Two of the authors (T. I. and N. K.) have found an infinite number of dynamical systems possessing such a periodicity property, which have been called multi‐fold Kepler systems or ν‐fold Kepler systems, with ν a positive rational number. If ν is allowed to take the real positive numbers, say ν=α, then for the α‐fold Kepler system, all the bounded motions become periodic or not, according to whether the parameter α is a rational number or not. A purpose of this paper is to quantize the α‐fold Kepler system and thereby to figure out a quantum analog of the closed orbit property of the α‐fold Kepler system. It will turn out that the quantized α‐fold Kepler system admits accidental degeneracy in energy levels or not, according to whether α is a rational number or not.

The canonical connection in quantum mechanics
View Description Hide DescriptionIn this paper we investigate the form of induced gauge fields that arises in two types of quantum systems. In the first we consider quantum mechanics on coset spaces G/H, and argue that G invariance is central to the emergence of the H connection as induced gauge fields in the different quantum sectors. We then demonstrate why the same connection, now giving rise to the non‐Abelian generalization of Berry’s phase, can also be found in systems that have slow variables taking values in such a coset space.

Poisson brackets of Wilson loops and derivations of free algebras
View Description Hide DescriptionWe describe a finite analog of the Poisson algebra of Wilson loops in Yang–Mills theory. It is shown that this algebra arises in an apparently completely different context: as a Lie algebra of vector fields on a noncommutative space. This suggests that noncommutative geometry plays a fundamental role in the manifestly gauge invariant formulation of Yang–Mills theory. We also construct the deformation of the algebra of loops induced by quantization, in the large‐N _{ c } limit.

Zero modes of rotationally symmetric generalized vortices and vortex scattering
View Description Hide DescriptionZero modes of rotationally symmetric vortices in a hierarchy of generalized Abelian Higgs models are studied. Under the finite energy and the smoothness condition, it is shown, that in all models, n self‐dual vortices superimposed at the origin have 2n modes. The relevance of these modes for vortex scattering is discussed: first, in the context of the slow‐motion approximation; second, a corresponding Cauchy problem for an all head‐on collision of nvortices is formulated. It is shown that the solution of this Cauchy problem has a π/n symmetry.

Time‐dependent plane‐wave spectrum representations for radiation from volume source distributions
View Description Hide DescriptionA new time‐domain spectral theory for radiation from a time‐dependent source distribution, is presented. The full spectral representation is based on a Radon transform of the source distribution in the four‐dimensional space‐time domain and consists of time‐dependent plane waves that propagate in all space directions and with all (spectral) propagation speeds v _{κ}. This operation, termed the slant stack transform, involves projection of the time‐dependent source distribution along planes normal to the spectral propagation direction and stacking them with a progressive delay corresponding to the spectral propagation speed v _{κ} along this direction. Outside the source domain, this three‐fold representation may be contracted into a two‐fold representation consisting of time‐dependent plane waves that satisfy the spectral constraint v _{κ}=c with c being the medium velocity. In the two‐fold representation, however, the complete spectral representation involves both propagating time‐dependent plane waves and evanescent time‐dependent plane waves. We explore the separate role of these spectral constituents in establishing the causal field, and determine the space‐time regions where the field is described only by the propagating spectrum. The spectral theory is presented here for scalar wave fields, but it may readily be extended to vector electromagnetic fields.

Time‐dependent multipoles and their application for radiation from volume source distributions
View Description Hide DescriptionThe radiation from a pulsed source distribution is expressed directly in the time‐domain using a sum of time‐dependent spherical (multipole) wave functions. Two alternative expressions for the time‐dependent multipole moments (the excitation pulses) are derived. It is shown how they are related to the time‐dependent plane‐wave spectrum of the source (obtained via a Radon transform of the source distribution in the four space‐time coordinates). Furthermore, the time‐dependent multipole moments, and thereby the total time‐dependent field outside the source region, are completely determined by the time‐dependent radiation pattern. The series convergence is addressed by showing that the high order multipole moments tend to the quasistatic extension of the static multipole moments. This also puts an upper limit on the spatial resolution that can be achieved by a source distribution with specified size and pulse length.

Topological Yang–Mills theory with scalar and vector fields in two and three dimensions
View Description Hide DescriptionWe study the topological Yang–Mills theory of scalar and vector fields in two and three dimensions in the superconnection framework. We modify the horizontality condition in such a way that we can take care of the topological symmetry of the δA _{μ}=β_{μ} type, as well as the usual gauge symmetry of the δA _{μ}=D _{μ} c type. We then obtain a complete set of BRST and anti‐BRST transformation rules of the component fields in a systematic way.

(2,0) superconformal anomaly
View Description Hide DescriptionThe (2,0) supersymmetric Wess–Zumino–Polyakov action is constructed and the (2,0) superconformal anomaly is given. The anomalous Ward‐identity in the right sector is derived and the known operator product expansion of the N=2 superstress energy tensor is recovered.

Imaging of velocity singularities with multiscale operators
View Description Hide DescriptionIn this paper, by using the multiscale operator and based on the Beylkin’s formula [J. Math. Phys. 26, 99–108 (1985)], the inversion method of velocity singularities is presented, and by applying the multiscale operator to upward scattered data represented by the Kirchhoff approximation, the multiscale imaging formula is shown. For the band‐limited inverse problem, based on the multiscale imaging formula in this paper, the behavior of the velocity singularities can be analyzed across the multiscale and the formula to reconstruct the velocity is obtained from the output of the multiscale operator interpreted in the terms of the Kirchhoff‐approximate data. Particularly, at wide offset and even postcritical case, for the full‐band input data, the same result with Beylkin’s in the above reference about the velocity discontinuities is obtained. Our method in this paper can suppress the effects of the noise in the real world data and analyze the effect produced by band‐limited input data.

Feynman–Kac kernels in Markovian representations of the Schrödinger interpolating dynamics
View Description Hide DescriptionProbabilistic solutions of the so‐called Schrödinger boundary data problem provide for a unique Markovianinterpolation between any two strictly positive probability densities designed to form the input–output statistics data for the process taking place in a finite‐time interval. The key issue is to select the jointly continuous in all variables positive Feynman–Kac kernel, appropriate for the phenomenological (physical) situation. We extend the existing formulations of the problem to cases when the kernel is not a fundamental solution of a parabolic equation, and prove the existence of a continuous Markovianinterpolation in this case. Next, we analyze the compatibility of this stochastic evolution with the original parabolic dynamics, which is assumed to be governed by the temporally adjoint pair of (parabolic) partial differential equations, and prove that the pertinent random motion is a diffusion process. In particular, in conjunction with Born’s statistical interpretation postulate in quantum theory, we consider stochastic processes which are compatible with the Schrödinger picture quantum evolution.

A uniform diffusion limit for random wave propagation with turning point
View Description Hide DescriptionA random wave propagation problem with turning point is considered for a refractive, layered random medium. The variations of the medium structure are assumed to have two spatial scales; microscopic random fluctuations are superposed upon slowly varying macroscopic variations. An extension of a limit theorem for stochastic differential equations with multiple spatial scales is derived and proved to obtain a uniformly valid diffusion limit for random multiple scattering up to the turning point region. The scale dependence of the infinitesimal generator of the backward Kolmogorov equation provides an insight into the interplay of internal refraction and random scattering as one approaches the turning point.

Stochastic mechanics of reciprocal diffusions
View Description Hide DescriptionThe dynamics and kinematics of reciprocal diffusions were examined in a previous paper [J. Math. Phys. 34, 1846 (1993)], where it was shown that reciprocal diffusions admit a chain of conservation laws, which close after the first two laws for two disjoint subclasses of reciprocal diffusions, the Markov and quantum diffusions. For the case of quantum diffusions, the conservation laws are equivalent to Schrödinger’s equation. The Markov diffusions were employed by Schrödinger [Sitzungsber. Preuss. Akad. Wiss. Phys. Math Kl. 144 (1931); Ann. Inst. H. Poincaré 2, 269 (1932)], Nelson [Dynamical Theories of Brownian Motion (Princeton University, Princeton, NJ, 1967); Quantum Fluctuations (Princeton University, Princeton, NJ, 1985)], and other researchers to develop stochastic formulations of quantum mechanics, called stochastic mechanics. We propose here an alternative version of stochastic mechanics based on quantum diffusions. A procedure is presented for constructing the quantum diffusion associated to a given wave function. It is shown that quantum diffusions satisfy the uncertainty principle, and have a locality property, whereby given two dynamically uncoupled but statistically correlated particles, the marginal statistics of each particle depend only on the local fields to which the particle is subjected. However, like Wigner’s joint probability distribution for the position and momentum of a particle, the finite joint probability densities of quantum diffusions may take negative values.

Nonequilibrium fluctuation–dissipation relations for independent random rate processes with dynamical disorder
View Description Hide DescriptionA class of rate processes with dynamical disorder is investigated based on the two following assumptions: (a) the system is composed of a random number of particles (or quasiparticles) which decay according to a first‐order kinetic law; (b) the rate coefficient of the process is a random function of time with known stochastic properties. The formalism of characteristic functionals is used for the direct computation of the dynamical averages. The suggested approach is more general than the other approaches used in the literature: it is not limited to a particular type of stochastic process and can be applied to any type of random evolution of the rate coefficient. We derive an infinity of exact fluctuation–dissipation relations which establish connections among the moments of the survival function and the moments of the number of surviving particles.
The analysis of these fluctuation–dissipation relations leads to the unexpected result that in the thermodynamic limit the fluctuations of the number of particles have an intermittent behavior. The moments are explicitly evaluated in two particular cases: (a) the random behavior of the rate coefficient is given by a non‐Markovian process which can be embedded in a Markovian process by increasing the number of state variables and (b) the stochastic behavior of the rate coefficient is described by a stationary Gaussian random process which is generally non‐Markovian. The method of curtailed characteristic functionals is used to recover the conventional description of dynamical disorder in terms of the Kubo–Zwanzig stochastic Liouville equations as a particular case of our general approach. The fluctuation–dissipation relations can be used for the study of fluctuations without making use of the whole mathematical formalism.
To illustrate the efficiency of our method for the analysis of fluctuations we discuss three different physicochemical and biochemical problems. A first application is the kinetic study of the decay of positrons or positronium atoms thermalized in dense fluids: in this case the time dependence of the rate coefficient is described by a stationary Gaussian random function with an exponentially decaying correlation coefficient. A second application is an extension of Zwanzig’s model of ligand–protein interactions described in terms of the passage through a fluctuating bottle neck; we complete the Zwanzig’s analysis by studying the concentration fluctuations. The last example deals with jump rate processes described in terms of two independent random frequencies; this model is of interest in the study of dielectric or conformational relaxation in condensed matter and on the other hand gives an alternative approach to the problem of protein–ligand interactions. We evaluate the average survival function in several particular cases for which the jump dynamics is described by two activated processes with random energy barriers. Depending on the distributions of the energy barriers the average survival function is a simple exponential, a stretched exponential, or a statistical fractal of the inverse power law type. The possible applications of the method in the field of biological population dynamics are also investigated.

Gauge invariant perturbations of black holes. I. Schwarzschild space–time
View Description Hide DescriptionWe cast the perturbed Bianchi identities, in the Schwarzschild background, into a form involving only tetrad and coordinate gauge invariant Newman–Penrose field quantities. These quantities, which arise naturally in our approach, are gauge invariant quantities of spin‐weight ±2, ±1, and 0. Some of the integrability conditions for the Bianchi identities then provide a system of six gauge invariant perturbation wave equations for the spin‐weighted quantities. These wave equations are, respectively, the (spin‐weight ±2) Bardeen–Press equations, two new (spin‐weight ±1) gravitational waveequations, and two (spin‐weight 0) Regge–Wheeler equations. Other integrability conditions provide the transformation identities that relate the field quantities to each other, and hence relate the various perturbation wave equations to one another. In particular, this method provides an alternative derivation of the transformations between the Bardeen–Press and Regge–Wheeler equations. The integrability conditions also allow us to relate the Bardeen–Press quantities of opposite spin‐weight, and we investigate how this relationship compares with the Teukolsky–Starobinsky identities. Finally, we give a derivation of the gauge invariant Zerilli equation, and show how it is related to the fundamental equations mentioned above.

Cosmological models expressible as gradient vector fields
View Description Hide DescriptionClasses of cosmological models, for which Einstein’s equations reduce to two‐dimensional dynamical systems, are studied in the presence of stochastic perturbations using the steady state of the associated Fokker–Planck equations and Zeeman’s notion of ε‐stability. In all cases, a set of variables is found for which the dynamical systems are expressible as gradient vector flows, showing that the associated cosmologies are stochastically stable. Such models are also important in connection with application of catastrophy theory to cosmology.

The Dirac equation in the Robertson–Walker space–time
View Description Hide DescriptionThe Dirac equation is considered, via the Newman–Penrose formalism, in the context of the Robertson–Walker geometry. The solution of the equation, which contrary to the neutrino case is not directly separable, is reduced to the study of decoupled spatial and temporal equations. The spatial equations are explicitly integrated and show the existence of discrete energy levels in case of closed universe. Besides the neutrino, the time equation is discussed in limiting situations of the standard cosmology.