Volume 41, Issue 9, September 2000
Index of content:
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Non solutions to the Seiberg–Witten equations
View Description Hide DescriptionWe show that a previous paper of Freund describing a solution to the Seiberg–Witten equations has a sign error rendering it a solution to a related but different set of equations. The non nature of Freund’s solution is discussed and clarified and we also construct a whole class of solutions to the Seiberg–Witten equations.

Phase space observables and isotypic spaces
View Description Hide DescriptionWe give necessary and sufficient conditions for the set of Neumark projections of a countable set of phase space observables to constitute a resolution of the identity, and we give a criteria for a phase space observable to be informationally complete. The results will be applied to the phase space observables arising from an irreducible representation of the Heisenberg group.

Coupled harmonic oscillator systems: An elementary algebraic decoupling approach
View Description Hide DescriptionWe present simple explicit coordinate transformations which serve to decouple the Schrödinger equation for a pair of (not necessarily identical) harmonic oscillators in the presence of bilinear perturbing potentials. We derive general conditions for the decoupling, and give some examples of physical interest. These include the much studied example with just a static perturbation, the parallel problem with a dynamic coupling term, and the classic example of an isotropic twodimensional oscillator in a transverse magnetic field, first solved by Fock (1928) by separation of variables.

Geometric modular action, wedge duality, and Lorentz covariance are equivalent for generalized free fields
View Description Hide DescriptionThe Tomita–Takesaki modular groups and conjugations for the observable algebras of spacelike wedges and the vacuum state are computed for translationally covariant, but possibly not Lorentz covariant, generalized free quantum fields in arbitrary space–time dimension d. It is shown that for the condition of geometric modular action (CGMA) of Buchholz, Dreyer, Florig, and Summers [Rev. Math. Phys. (2000)], Lorentz covariance and wedge duality are all equivalent in these models. The same holds for if there is a mass gap. For massless fields in and for and arbitrary mass, CGMA does not imply Lorentz covariance of the field itself, but only of the maximal local net generated by the field.

“Massive” vector field in de Sitter space
View Description Hide DescriptionWe present in this paper a covariant quantization of the “massive” vector field on de Sitter (dS) space based on analyticity in the complexified pseudoRiemanian manifold. The correspondence between unitary irreducible representations of the de Sitter group and the field theory on de Sitter space–time is essential in our approach. We introduce the Wightman twopoint function for the case of generalized free vector fields on de Sitter space. This function satisfies the conditions of (a) positiveness, (b) locality, (c) covariance, (d) normal analyticity, (e) transversality and (f) divergencelessness. The Hilbert space structure and the unsmeared field operators are also defined. This work is in the direct continuation of previous ones concerning the scalar and spinor cases.

A perturbation of CHSH inequality induced by fluctuations of ensemble distributions
View Description Hide DescriptionWe reconsider the theory of hidden variables under the assumption that the conjecture on the ensemble (experiment run) independence of the distribution of hidden variables (which was indirectly used by J. Bell and his followers) is violated. Ensemble fluctuations imply perturbations of Bell’s inequality and its generalizations. We study (by experimental reasons) CHSH (Clauser, Horne, Shimony, Holt) inequality and obtain its modification. This modified inequality is not in disaccord with the predictions of quantum formalism. The deviation from the standard CHSH inequality depends on the magnitude of ensemble fluctuations. We find these magnitude for fluctuating families of Gaussian distributions. We found that if the dimension of the space of hidden variables is very high, then to obtain a contradiction between the local realism and quantum formalism, we must be sure there is no even negligibly small deviations in probability distributions of hidden variables corresponding to different runs of the experiment (in particular, the efficiency of detectors must be equal to one).

The role of infrared divergence for decoherence
View Description Hide DescriptionContinuous and discrete superselection rules induced by the interaction with the environment are investigated for a class of exactly soluble Hamiltonian models. The environment is given by a boson field. Stable superselection sectors can only emerge if the low frequences dominate and the ground state of the boson field disappears due to infrared divergence. The models allow uniform estimates of all transition matrix elements between different superselection sectors.

Twodimensional theory of chirality. I. Absolute chirality
View Description Hide DescriptionChirality is a notion already familiar to undergraduates as the fact that an object is not superposable to its mirror image. Its dichotomous or “yes/no” character, implicit in this definition due to Lord Kelvin, has always been considered selfevident. We prove here and in the companion article that in two dimensions chirality—the very concept of chirality—is a continuous phenomenon in the special case of squareintegrable wave functions. This conception of chirality is to Kelvin’s definition what the continuous conception of door opening is to the closed/nonclosed dichotomy; hence it provides a continuous description of the discrete achiral symmetry breaking. This result is first extended to three and higher dimensions, then to the whole nonrelativistic quantum description of matter. Thus molecules are more or less chiral just as doors are more or less open, and molecular chirality changes continuously during chemical reactions. Chirality splits into two complementary forms—absolute and relative chirality. We present here the theory of absolute chirality. More generally, this unexpected and paradoxical breakthrough in symmetry theory is based on a geometrical description of wave functions that should find broad applications in molecular physics and in stereochemistry, where the notion of chirality has an overwhelming importance since long ago.

Twodimensional theory of chirality. II. Relative chirality and the chirality of complex fields
View Description Hide DescriptionChirality can equally be correctly viewed as a dichotomous symmetry property, as in Kelvin’s historical conception, and as a continuous phenomenon. This highly paradoxical result is proved, in this and the previous article (I) [Le Guennec, J. Math. Phys. 41, 5954 (2000)], in the case of the basic ingredient of quantum mechanics, squareintegrable wave functions. In the continuous conception, chirality appears as the combination of two complementary forms—absolute and relative chirality. Accordingly, while (I) focused on the conceptual issue and on the 2D theory of absolute chirality, this article focuses on 2D relative chirality. We show that relative chirality is a continuous phenomenon described by relative radial functions and relative chiral loops whose features are surprisingly close to those of their absolute counterparts. This is illustrated on a 2D model of cistrans isomerism. We then show that chirality as such is the “addition” of absolute and relative chirality just as a vector is the addition of its projection on a basis. As a test of versatility, the continuous conception of chirality is extended to 2D complex fields and Fourier transforms. Why this conception of chirality is possible in spaces is tentatively discussed.

The uniqueness and approximation of a positive solution of the Bardeen–Cooper–Schrieffer gap equation
View Description Hide DescriptionIn this paper we study the Bardeen–Cooper–Schrieffer energy gap equation at finite temperatures. When the kernel is positive representing a phonondominant phase in a superconductor, the existence and uniqueness of a gap solution is established in a class which contains solutions obtainable from bounded domain approximations. The critical temperatures that characterize superconducting–normal phase transitions realized by bounded domain approximations and full space solutions are also investigated. It is shown under some sufficient conditions that these temperatures are identical. In this case the uniqueness of a full space solution follows directly. We will also present some examples for the nonuniqueness of solutions. The case of a kernel function with varying signs is also considered. It is shown that, at low temperatures, there exist nonzero gap solutions indicating a superconducting phase, while at high temperatures, the only solution is the zero solution, representing the dominance of the normal phase, which establishes again the existence of a transition temperature.

Hyperspherical theory of anisotropic exciton
View Description Hide DescriptionA new approach to the theory of anisotropicexciton based on Fock transformation, i.e., on a stereographic projection of the momentum to the unit fourdimensional (4D) sphere, is developed. Hyperspherical functions are used as a basis of the perturbation theory. The binding energies, wave functions and oscillator strengths of elongated as well as flattened excitons are obtained numerically. It is shown that with an increase of the anisotropy degree the oscillator strengths are markedly redistributed between optically active and formerly inactive states, making the latter optically active. An approximate analytical solution of the anisotropicexciton problem taking into account the angular momentumconserving terms is obtained. This solution gives the binding energies of moderately anisotropicexciton with a good accuracy and provides a useful qualitative description of the energy level evolution.

Dual WDVV equations in supersymmetric Yang–Mills theory
View Description Hide DescriptionThis paper studies the dual form of Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations in supersymmetric Yang–Mills theory by applying a duality transformation to WDVV equations. The dual WDVV equations called in this paper are nonlinear differential equations satisfied by dual prepotential and are found to have the same form with the original WDVV equations. However, in contrast with the case of weak coupling calculus, the perturbative part of dual prepotential itself does not satisfy the dual WDVV equations. Nevertheless, it is possible to show that the nonperturbative part of dual prepotential can be determined from dual WDVV equations, provided the perturbative part is given. As an example, the SU(4) case is presented. The nonperturbative dual prepotential derived in this way is consistent to the dual prepotential obtained by D’Hoker and Phong.

Reformulation of the standard model in a generalized differential geometry on the discrete space
View Description Hide DescriptionThe standard model is reconstructed in a generalized differential geometry (GDG) on the product space by reformulating the work of Coquereaux et al. that dealt with the same theme based on the noncommutative geometry (NCG). A GDG on is constructed by adding the basis of a differential form on the discrete space to the ordinary basis on Minkowski space and so it is a direct generalization of the differential geometry on the continuous manifold. A GDG is a version of NCG. The Yang–Mills–Higgs Lagrangian and the Dirac Lagrangian are reconstructed by using the fermion representation similar to that in a grand unified model.

Stochastic mechanics and the Feynman integral
View Description Hide DescriptionThe Feynman integral is given a stochastic interpretation in the framework of Nelson’s stochastic mechanics employing a timesymmetric variant of Nelson’s kinematics recently developed by the author.

Modular localization of elementary systems in the theory of Wigner
View Description Hide DescriptionStarting from Wigner’s theory of elementary systems and following a recent approach of Brunetti, Guido, and Longo also taken up by Schroer we define certain subspaces of localized wave functions in the underlying Hilbert space with the help of the theory of modular vonNeumann algebras of Tomita and Takesaki. We characterize the elements of these subspaces as boundary values of holomorphic functions in the sense of distribution theory and show that the corresponding holomorphic functions satisfy the sufficient conditions of the theorems of Paley–Wiener–Schwartz and Hörmander.

Effect of couplings weakening and reversing in ferromagnetic Ising systems— Rigorous inequalities
View Description Hide DescriptionWe consider Ising systems where all the manyspin couplings are positive. We show that the absolute value of all the manyspin correlations does not increase when the value of any of the couplings is reduced, taking any value in the interval Results of this type are motivated by work in systems such as random field Ising models.

Quantum time and spatial localization: An analysis of the Hegerfeldt paradox
View Description Hide DescriptionTwo related problems in relativistic quantum mechanics, the apparent superluminal propagation of initially localized particles and dependence of spatial localization on the motion of the observer, are analyzed in the context of Dirac’s theory of constraints. A parametrization invariant formulation is obtained by introducing time and energy operators for the relativistic particle and then treating the Klein–Gordon equation as a constraint. The standard, physical Hilbert space is recovered, via integration over proper time, from an augmented Hilbert space wherein time and energy are dynamical variables. It is shown that the Newton–Wigner position operator, being in this description a constant of motion, acts on states in the augmented space. States with strictly positive energy are nonlocal in time; consequently, position measurements receive contributions from states representing the particle’s position at many times. Apparent superluminal propagation is explained by noting that, as the particle is potentially in the past (or future) of the assumed initial place and time of localization, it has time to propagate to distant regions without exceeding the speed of light. An inequality is proven showing the Hegerfeldt paradox to be completely accounted for by the hypotheses of subluminal propagation from a set of initial space–time points determined by the quantum time distribution arising from the positivity of the system’s energy. Spatial localization can nevertheless occur through quantum interference between states representing the particle at different times. The nonlocality of the same system to a moving observer is due to Lorentz rotation of spatial axes out of the interference minimum.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


A frequency–domain inverse problem for a dispersive stratified chiral medium
View Description Hide DescriptionWe study the inverse problem for a dispersive stratified chiral slab. The problem is treated as an analytic factorization problem in the complex plane of frequencies. Emphasis is made on the reconstruction of the spatial dependence of the medium parameters, whereas the frequency dependence is supposed to be a singleresonance Lorentz model. It is shown that, under the normal incidence of exciting plane waves, the scattering data as functions of alternating frequency allows reconstructing three independent combinations of four spacevarying medium parameters. If one parameter is known, all other parameters are uniquely reconstructed.

Twodimensional, highly directive currents on large circular loops
View Description Hide DescriptionProperties of idealized, twodimensional current distributions on circular loops are investigated analytically via the solution of a constrained optimization problem. The directivity in the far field is maximized under a fixed where is the integral of the square of the current magnitude and is the total radiated power. enters the ensuing Fourier series for the current implicitly through a Lagrange multiplier α. For nonnegative α and large electrical radius the directivity and the current are evaluated approximately via combined use of the Poisson summation formula and the Mellin transform technique. As a result, a geometricalray representation for the current is derived for the case of directivities that are slightly larger than that of the uniform distribution. The analysis indicates certain advantages of large radiating structures for moderate values of the constraint In the limit of Oseen’s “Einstein needle radiation,” an asymptotic formula for the directivity is obtained. Possible extensions of these results to classes of smooth convex loops are briefly discussed.

 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


Classification of certain integrable coupled potential KdV and modified KdVtype equations
View Description Hide DescriptionIn this article we describe the classification of integrable symmetrically coupled potential KdV and modified KdVtype equations that possess higher symmetries. Restricting our attention to the systems that cannot be decoupled by a change of dependent variables, we obtain 11 previously unknown classes of integrable equations. In some cases we present Hamiltonian or biHamiltonian formulations.
