Volume 40, Issue 5, May 1999
 QUANTUM PHYSICS; PARTICLES AND FIELDS


PTsymmetric quantum mechanics
View Description Hide DescriptionThis paper proposes to broaden the canonical formulation of quantum mechanics. Ordinarily, one imposes the condition on the Hamiltonian, where † represents the mathematical operation of complex conjugation and matrix transposition. This conventional Hermiticity condition is sufficient to ensure that the Hamiltonian H has a real spectrum. However, replacing this mathematical condition by the weaker and more physical requirement where ‡ represents combined parity reflection and time reversal PT, one obtains new classes of complex Hamiltonians whose spectra are still real and positive. This generalization of Hermiticity is investigated using a complex deformation of the harmonic oscillator Hamiltonian, where ε is a real parameter. The system exhibits two phases: When the energy spectrum of H is real and positive as a consequence of PT symmetry. However, when the spectrum contains an infinite number of complex eigenvalues and a finite number of real, positive eigenvalues because PT symmetry is spontaneously broken. The phase transition that occurs at manifests itself in both the quantummechanical system and the underlying classical system. Similar qualitative features are exhibited by complex deformations of other standard real Hamiltonians with N integer and each of these complex Hamiltonians exhibits a phase transition at These PTsymmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space.

Hidden supersymmetry and Berezin quantization of spinning superparticles
View Description Hide DescriptionThe first quantized theory of massive superparticles with arbitrary fixed central charge and (half) integer or fractional superspin is constructed. The quantum states are realized on the fields carrying a finitedimensional, or a unitary infinitedimensional, representation of the supergroups OSp (22) or SU (1,12). The construction originates from quantization of a classical model of the superparticle we suggest. The physical phase space of the classical superparticle is embedded in a symplectic superspace where the inner Kähler supermanifold provides the particle with superspin degrees of freedom. We find the relationship between Hamiltonian generators of the global Poincaré supersymmetry and the “internal” SU(1,12) one. Quantization of the superparticle combines the Berezin quantization on and the conventional Dirac quantization with respect to space–time degrees of freedom. Surprisingly, to retain the supersymmetry, quantum corrections are required for the classical supercharges as compared to the conventional Berezin method. These corrections are derived and the Berezin correspondence principle for underlying their origin is verified.

An inversion inequality for potentials in quantum mechanics
View Description Hide DescriptionWe suppose: (1) that the groundstateeigenvalue of the Schrödinger Hamiltonian in one dimension is known for all values of the coupling and (2) that the potential shape can be expressed in the form where g is monotone increasing and convex. The inversion inequality is established, in which the “kinetic potential” is related to the energy function by the transformation { }. As an example, f is approximately reconstructed from the energy function F for the potential

Weyl anomaly in higher dimensions and Feynman rules in coordinate space
View Description Hide DescriptionAn algorithm to obtain the Weyl anomaly in higher dimensions is presented. It is based on the heatkernel method. Feynman rules, such as the vertex rule and the propagator rule, are given in (regularized) coordinate space. A graphical calculation is introduced. The sixdimensional scalargravity theory is taken as an example, and its explicit result is obtained.

Superintegrability on the twodimensional hyperboloid. II
View Description Hide DescriptionThis work is devoted to the investigation of the quantum mechanical systems on the twodimensional hyperboloid which admits separation of variables in at least two coordinate systems. Here we consider two potentials introduced in a paper of C. P. Boyer, E. G. Kalnins, and P. Winternitz [J. Math. Phys. 24, 2022 (1983)], which have not yet been studied. We give an example of an interbasis expansion and work out the structure of the quadratic algebra generated by the integrals of motion.

On the Coulomb Sturmian matrix elements of relativistic Coulomb Green’s operators
View Description Hide DescriptionThe Hamiltonian of the radial Coulomb Klein–Gordon and second order Dirac equations are shown to possess an infinite symmetric tridiagonal matrix structure on the relativistic Coulomb Sturmian basis. This allows us to give an analytic representation for the corresponding Coulomb Green’s operators in terms of continued fractions. The poles of the Green’s matrix reproduce the exact relativistic hydrogen spectrum.

Quantum geometry of field extensions
View Description Hide DescriptionWe introduce a new kind of topological gauge configuration or “soliton” associated to the extension of the real numbers to the complex ones. These configurations describe zerocurvature gauge fields and nontrivial cohomology over the real line, but with a quantum choice of differential calculus. In general, the quantum differential 1forms on the line with coordinate algebra are in correspondence with field extensions of k, and the quantum cohomology detects the nontriviality of the extension.

Nonunitary representations of the SU(2) algebra in the Dirac equation with a Coulomb potential
View Description Hide DescriptionA novel realization of the classical SU(2) algebra is introduced for the Dirac relativistic hydrogen atom defining a set of operators that allow the factorization of the problem. An extra phase is needed as a new variable in order to define the algebra. We take advantage of the operators to solve the Dirac equation using algebraic methods. A similar path to the one used in the angular momentum case is used; hence, the radial eigenfunctions so calculated comprise nonunitary representations of the algebra. One of the interesting properties of such nonunitary representations is that they are not labeled by integer nor by halfinteger numbers, as occurs in the usual angular momentum representation.

Physical properties of quantum field theory measures
View Description Hide DescriptionWell known methods of measure theory on infinite dimensional spaces are used to study physical properties of measures relevant to quantum field theory. The difference of typical configurations of free massive scalar field theories with different masses is studied. We apply the same methods to study the Ashtekar–Lewandowski (AL) measure on spaces of connections. In particular we prove that the diffeomorphism group acts ergodically, with respect to the AL measure, on the Ashtekar–Isham space of quantum connections modulo gauge transformations. We also prove that a typical, with respect to the AL measure, quantum connection restricted to a (piecewise analytic) curve leads to a parallel transport discontinuous at every point of the curve.

New generalized coherent states
View Description Hide DescriptionWe construct a new family of boson coherent states using a specially designed function which is a solution of a functional equation with and . We use this function in place of the usual exponential to generate new coherent states from the vacuum, which are normalized and continuous in their label . These states allow the resolution of unity, and a corresponding weight function is furnished by the exact solution of the associated Stieltjes moment problem. They also permit exact evaluation of matrix elements of an arbitrary polynomial given as a normallyordered function of boson operators. We exemplify this by showing that the photon number statistics for these states is subPoissonian. For any the states are squeezed; we obtain and discuss their signal to quantum noise ratio. The function allows a natural generation of multiboson coherent states of arbitrary multiplicity, which is impossible for the usual coherent states. For all the above results reduce to those for conventional coherent states. Finally, we establish a link with qdeformed bosons.

Waves and particles in Kaluza–Klein theory
View Description Hide DescriptionWe examine three overlapping problems in the application of fivedimensional (5D) manifolds to physics. First, we linearize the 5D theory along the lines of the fourdimensional (4D) theory, using the harmonic gauge condition. The resulting wave equations have sources, and can in principle describe gravitons and scalar particles with finite masses, but the natural choice of gauge parameters makes both massless. Second, we generalize the 5D metric by including separate conformal factors on its 4D and extra parts. Then the 5D harmonic gauge gives back the 4D harmonic gauge of gravitational waves and the Lorentz gauge of electromagnetic waves, but both particles are massless. Third, we again take a conformally rescaled metric, but rewrite the field equations in a novel manner. This allows us to interpret the finite masses of ordinary particles in terms of the wavelengths associated with 4D spaces embedded in a 5D space.
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 DYNAMICAL SYSTEMS AND FLUID DYNAMICS


Classical and quantum mechanics of jointed rigid bodies with vanishing total angular momentum
View Description Hide DescriptionA gauge theoretical treatment proves to have been successful in the study of systems of point particles; the centerofmass system is made into a principal fiber bundle, on which is defined a natural connection. The gauge theoretical approach may be generalized to be applicable to a system of rigid bodies. The present article deals with a system of two identical axially symmetric cylinders jointed together by a special type of joint. This system is the model made by Kane and Scher and reformulated later by Montgomery, in order to study the falling cats who can land on their legs when released upside down. With the notwist condition, the system turns out to have the configuration space diffeomorphic with SO(3), which is made into a principal O(2) bundle over the real projective space of dimension two, and endowed with a natural connection. An optimal control problem for this system with the vanishing total angular momentum is satisfactorily treated in this bundle picture. Along with a certain performance index, the Maximum Principle gives rise to a Hamiltonian system on the cotangent bundle of SO(3). This Hamiltonian system is shown to admit a symmetry group O(2), which is not the structure group, but comes from the material symmetry of the respective cylinders. Moreover, quantization of this “classical” system is carried out, giving rise to a quantum system with the constraints of the vanishing total angular momentum. Through the symmetry by the structure group O(2), the reduction procedure is performed for both the classical and the quantum systems. It then turns out that the respective reduced systems, classical and quantum, admit the material symmetry group O(2), in general.

Algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations
View Description Hide DescriptionAn algebraic structure related to discrete zero curvature equations is established. It is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries. The key of the theory is to generate nonisospectral flows from the discrete spectral problem associated with a given system of discrete evolution equations. Three examples are given.

On the Fay identity for Korteweg–de Vries tau functions and the identity for the Wronskian of squared solutions of Sturm–Liouville equation
View Description Hide DescriptionWe show that the wellknown identity for the Wronskian of squared solutions of a Sturm–Liouville equation follows from the Fay identity. We also study some oddorder order, identities which are specific for tau functions, related to the KdV hierarchy.

Differential polynomial expressions related to the Kadomtsev–Petviashvili and Korteweg–de Vries hierarchies
View Description Hide DescriptionAn integrable nonlinear partial differential equation typically extends to a hierarchy of integrable equations. There exist several recursive schemes for obtaining these hierarchies. Recently, explicit expressions for the KdV hierarchy have been found. We derive explicit expressions for the hierarchy associated with the KP equation. The main tools are Sato’s theory, Hirota’s formalism, and Bell’s polynomials.

Attractors for the Klein–Gordon–Schrödinger equation
View Description Hide DescriptionIn this paper we deal with the asymptotic behavior of solutions for the Klein–Gordon–Schrödinger equation. We prove the existence of compact global attractors for this model in the space for each integer
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 RELATIVITY AND GRAVITATION


Anisotropic fluid spherically symmetric space–times admitting a kinematic selfsimilarity
View Description Hide DescriptionAnisotropic fluid spherically symmetric space–times admitting a kinematic selfsimilar vector are investigated. The geodesic case is considered, and some special subcases in which the anisotropic fluid satisfies additional physical conditions are investigated in detail. A number of other special cases are studied. Particular attention is focused on the possible asymptotic behavior of the models, and it is shown that the models considered always asymptote towards an exact homothetic solution, which is in general either a perfect fluid model or a static solution.

The generalized thinsandwich problem and its local solvability
View Description Hide DescriptionWe consider Einstein gravity coupled to matter consisting of a gauge field with any compact gauge group and minimally coupled scalar fields. We investigate under what conditions a free specification of a spatial field configuration and its time derivative allows us to solve the constraints for lapse, shift, and other gauge parameters and hence determine a solution to the field equations (thinsandwich problem). We establish sufficient conditions under which the thinsandwich problem can be solved locally in field space.

Null cones from and Legendre submanifolds
View Description Hide DescriptionIt is shown that the main variable Z of the null surface formulation of GR is the generating family of a constrained Lagrange submanifold that lives on the energy surface and that its level surfaces yield Legendre submanifolds on that energy surface. Thus, the singularity structure of past null cones with apex at is obtained by studying the projection map of the Legendre submanifolds to the configuration space. The behavior of the coordinate system defined by the variable Z at the caustic points is analyzed. It is shown that a single function cannot generate the conformal structure of an asymptotically flat space–time that satisfies the generic and weak energy condition.
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 MISCELLANEOUS TOPICS IN MATHEMATICAL METHODS


Hopf algebra structure of related to
View Description Hide DescriptionWe show that the algebra of functions on the Grassmann supergroup has a (graded) Hopf algebrastructure related to
