Volume 49, Issue 5, May 2008
Index of content:
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Comments on “Gazeau–Klauder coherent states for trigonometric Rosen–Morse potential” [J. Math. Phys.49, 022104 (2008)]
View Description Hide DescriptionIn a recently published paper in this journal [A. Cheaghlou and O. Faizy, J. Math. Phys.49, 022104 (2008)], the authors introduce the Gazeau–Klauder coherent states for the trigonometric Rosen–Morse potential as an infinite superposition of the wavefunctions. It is shown that their proposed measure to realize the resolution of the identity condition is not positive definite. Consequently, the claimed coherencies for the trigonometric Rosen–Morse wavefunctions cannot actually exist.

The role of von Neumann and Lüders postulates in the Einstein, Podolsky, and Rosen considerations: Comparing measurements with degenerate and nondegenerate spectra
View Description Hide DescriptionWe show that the projection postulate plays a crucial role in the discussion on the socalled quantum nonlocality, in particular, in the Einstein, Podolsky, and Rosen argument. We stress that the original von Neumann projection postulate was crucially modified by extending it to observables with degenerate spectra (the Lüders postulate) and we show that this modification is highly questionable from a physical point of view and is the real source of quantum nonlocality. The use of the original von Neumann postulate eliminates this problem: instead of an action at a distance nonlocality we obtain a classical measurement nonlocality, which is related to the synchronization of two measurements (on the two parts of a composite system).

Decoherence rates for Galilean covariant dynamics
View Description Hide DescriptionWe introduce a measure of decoherence for a class of density operators. For Gaussian density operators in dimension one, it coincides with an index used by Morikawa [“Quantum decoherence and classical correlation in quantum mechanics,” Phys. Rev. D42, 2929–2931 (1990)]. Spatial decoherence rates are derived for three large classes of the Galilean covariant quantum semigroups introduced by Holevo [“On conservativity of covariant dynamical semigroups,” Rep. Math. Phys.33, 95–110 (1993)]. We also characterize the relaxation to a Gaussian state for these dynamics and give a theorem for the convergence of the Wigner function to the probability distribution of the classical analog of the process.

Optimal covariant measurement of momentum on a half line in quantum mechanics
View Description Hide DescriptionWe cannot perform the projective measurement of a momentum on a half line since it is not an observable. Nevertheless, we would like to obtain some physical information of the momentum on a half line. We define an optimality for measurement as minimizing the variance between an inferred outcome of the measured system before a measuring process and a measurement outcome of the probe system after the measuring process, restricting our attention to the covariant measurement studied by Holevo [Rep. Math. Phys.13, 379 (1978)]. Extending the domain of the momentum operator on a half line by introducing a two dimensional Hilbert space to be tensored, we make it selfadjoint and explicitly construct a model Hamiltonian for the measured and probe systems. By taking the partial trace over the newly introduced Hilbert space, the optimal covariant positive operator valued measure of a momentum on a half line is reproduced. We physically describe the measuring process to optimally evaluate the momentum of a particle on a half line.

Applications of continuity and discontinuity of a fractional derivative of the wave functions to fractional quantum mechanics
View Description Hide DescriptionThe space fractional Schrödinger equation with a finite square potential, periodic potential, and deltafunction potential is studied in this paper. We find that the continuity or discontinuity condition of a fractional derivative of the wave functions should be considered to solve the fractional Schrödinger equation in fractional quantum mechanics. More parity states than those given by standard quantum mechanics for the finite square potential well are obtained. The corresponding energy equations are derived and then solved by graphical methods. We show the validity of Bloch’s theorem and reveal the energy band structure for the periodic potential. The jump (discontinuity) condition for the fractional derivative of the wave function of the deltafunction potential is given. With the help of the jump condition, we study some deltafunction potential fields. For the deltafunction potential well, an alternate expression of the wave function (the H function form of it was given by Dong and Xu [J. Math. Phys.48, 072105 (2007)]) is obtained. The problems of a particle penetrating through a deltafunction potential barrier and the fractional probability current density of the particle are also discussed. We study the Dirac comb and show the energy band structure at the end of the paper.

Characterization of the sequential product on quantum effects
View Description Hide DescriptionWe present a characterization of the standard sequential product of quantum effects. The characterization is in terms of algebraic, continuity and duality conditions that can be physically motivated.

 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Reflection positivity and monotonicity
View Description Hide DescriptionWe prove general reflection positivity results on Riemannian manifolds for both scalar fields and Dirac fields, and comment on applications to quantum field theory. As another application, we prove the inequality between Dirichlet and Neumann covariance operators on a manifold with a reflection.

Abelian gauge theories on compact manifolds and the Gribov ambiguity
View Description Hide DescriptionWe study the quantization of Abelian gauge theories of principal torus bundles over compact manifolds with and without boundary. It is shown that these gauge theories suffer from a Gribov ambiguity originating in the nontriviality of the bundle of connections whose geometrical structure will be analyzed in detail. Motivated by the stochastic quantization approach, we propose a modified functional integral measure on the space of connections that takes the Gribov problem into account. This functional integral measure is used to calculate the partition function, Green’s functions, and the field strength correlating functions in any dimension by using the fact that the space of inequivalent connections itself admits the structure of a bundle over a finite dimensional torus. Green’s functions are shown to be affected by the nontrivial topology, giving rise to nonvanishing vacuum expectation values for the gauge fields.

Relating onshell and offshell formalisms in perturbative quantum field theory
View Description Hide DescriptionIn the onshell formalism (mostly used in perturbative quantum field theory), the entries of the timeordered product are onshell fields (i.e., the basic fields satisfy the free field equations). With that, (multi)linearity of is incompatible with the action Ward identity. This can be circumvented by using the offshell formalism in which the entries of are offshell fields. To relate on and offshell formalisms correctly, a map from onshell fields to offshell fields was axiomatically introduced by Dütsch and Fredenhagen [Common. Math. Phys.243, 275 (2003)]. In that paper it is shown that, in the case of one real scalar field in dimensional Minkowski space, these axioms have a unique solution. However, this solution is only recursively given there. We solve this recurrence relation and give a fully explicit expression for in the cases of the scalar, Dirac, and gauge fields for arbitrary values of the dimension .

Solution of a linearized model of Heisenberg’s fundamental equation II
View Description Hide DescriptionWe propose to look at (a simplified version of) Heisenberg’s fundamental field equation [see Heisenberg, W., Introduction to the Unified Field Theory of Elementary Particles (Wiley, New York, 1966)] as a relativistic quantum field theory with a fundamental length, as introduced by Brüning and Nagamachi [J. Math. Phys.45, 2199 (2004)], and give a solution in terms of Wick power series of free fields which converge in the sense of ultrahyperfunctions but not in the sense of distributions. The solution of this model has been prepared by Nagamachi and Brüning [arXiv:0804.1663] by calculating all point functions by using path integral quantization. The functional representation derived in this part is essential for the verification of our condition of extended causality. The verification of the remaining defining conditions of a relativistic quantum field theory is much simpler through the use of Wick power series. Accordingly in this second part, we use Wick power series techniques to define our basic fields and derive their properties.

 GENERAL RELATIVITY AND GRAVITATION


Solution of the Dirac equation in the rotating Bertotti–Robinson spacetime
View Description Hide DescriptionThe Dirac equation is solved in the rotating Bertotti–Robinson spacetime. The set of equations representing the Dirac equation in the Newman–Penrose formalism is decoupled into an axial and an angular part. The axial equation, which is independent of mass, is exactly solved in terms of hypergeometric functions. The angular equation is considered both for massless (neutrino) and massive spin particles. For the neutrinos, it is shown that the angular equation admits an exact solution in terms of the confluent Heun equation. In the existence of mass, the angular equation does not allow an analytical solution, however, it is expressible as a set of first order differential equations apt for a numerical study.

On global properties of certain homogeneous exact solutions of Einstein field equations
View Description Hide DescriptionThe aim of this paper is an application of some general results on homogeneous Lorentz manifolds to exact solutions of Einstein field equations. We study the relation “<” of a chronological sequence of events. We examine a number of known solutions and show that in all but one, the relation < is maximal, i.e., for every and from , one has both and .

 DYNAMICAL SYSTEMS


A new class of solvable dynamical systems
View Description Hide DescriptionA new class of dynamical systems are presented, together with their solutions. Some of these models are isochronous, namely, their generic solutions are all completely periodic with the same period; others are characterized by friction, all solutions vanishing in the remote future; and still others are “asymptotically isochronous,” approaching an isochronous behavior in the remote future.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Central configurations with a quasihomogeneous potential function
View Description Hide DescriptionFor the classical body problem, if the masses are paced at rest in the configuration that is central, then the masses will tend to the origin and result in a simultaneous collision [A. Winter, The Analytical Foundations of Celestial Mechanics, 1st ed. (Princeton University Press, Princeton, 1941)]. However, if the potential is quasihomogeneous in nature, that is, the potential is of the form , where and , then we will dramatically get different results. This paper determines the orbits that will result when the potential function is quasihomogeneous. It also shows when there existences a relative equilibrium solution for the regular gon configuration. This is onehalf of Perko’s theorem for a homogeneous potential. Finally, we show the existence of a central configuration which is not central for either or . This configuration is the placement of two mutual equal pairs of masses placed at the vertices of a square. This shows that the reverse implication of Perko’s theorem for the regular gon does not hold for quasihomogeneous potentials.

 STATISTICAL PHYSICS


A new eight vertex model and higher dimensional, multiparameter generalizations
View Description Hide DescriptionWe study statistical models, specifically transfer matrices corresponding to a multiparameter hierarchy of braid matrices of dimensions with free parameters . The simplest, case, is treated in detail. Powerful recursion relations are constructed by explicitly giving the dependence on the spectral parameter of the eigenvalues of the transfer matrix at each level of coproduct sequence. A brief study of higher dimensional cases is presented, pointing out features of particular interest. Spin chain Hamiltonians are also briefly presented for the hierarchy. In a long final section, basic results are recapitulated with systematic analysis of their contents. Our eight vertex case is compared to standard six vertex and eight vertex models.

Correlation function of the Schur process with a fixed final partition
View Description Hide DescriptionWe consider a generalization of the Schur process in which a partition evolves from the empty partition into an arbitrary fixed final partition. We obtain a double integral representation of the correlation kernel. For a special final partition with only one row, the edge scaling limit is also discussed by the use of the saddle point analysis. If we appropriately scale the length of the row, the limiting correlation kernel changes from the extended Airy kernel.

On the Kertész line: Some rigorous bounds
View Description Hide DescriptionWe study the Kertész line of the state Potts model at (inverse) temperature in the presence of an external magnetic field . This line separates the two regions of the phase diagram according to the existence or not of an infinite cluster in the Fortuin–Kasteleyn representation of the model. It is known that the Kertész line coincides with the line of first order phase transition for small fields when is large enough. Here, we prove that the first order phase transition implies a jump in the density of the infinite cluster; hence, the Kertész line remains below the line of first order phase transition. We also analyze the region of large fields and prove, using techniques of stochastic comparisons, that equals to the leading order, as goes to , where is the threshold for bond percolation.

 METHODS OF MATHEMATICAL PHYSICS


Lotka–Volterra systems integrable in quadratures
View Description Hide DescriptionHamiltonian dimensional Lotka–Volterra systems are introduced that have conserved quantities. The explicit integrability in quadratures is demonstrated.

Representation theory of algebras for a higherorder class of spheres and tori
View Description Hide DescriptionWe construct algebras for a class of surfaces that are inverse images of certain polynomials of arbitrary degree. By using the directed graph associated with a matrix, the representation theory can be understood in terms of “loop” and “string” representations, which are closely related to the dynamics of an iterated map in the plane. As a particular class of algebras, we introduce the “Hénon algebras,” for which the dynamical map is a generalized Hénon map, and give an example where irreducible representations of all dimensions exist.

Orthogonal polynomials from Hermitian matrices
View Description Hide DescriptionA unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of Hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schrödinger equations. The Hermitian matrices (factorizable Hamiltonians) are real symmetric tridiagonal (Jacobi) matrices corresponding to second order difference equations. By solving the eigenvalue problem in two different ways, the duality relation of the eigenpolynomials and their dual polynomials is explicitly established. Through the techniques of exact Heisenberg operator solution and shape invariance, various quantities, the two types of eigenvalues (the eigenvalues and the sinusoidal coordinates), the coefficients of the three term recurrence, the normalization measures and the normalisation constants, etc., are determined explicitly.
