Volume 47, Issue 8, August 2006
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Coherent states expectation values as semiclassical trajectories
View Description Hide DescriptionWe study the time evolution of the expectation value of the anharmonic oscillator coordinate in a coherent state as a toy model for understanding the semiclassical solutions in quantum field theory. By using the deformation quantization techniques, we show that the coherent state expectation value can be expanded in powers of such that the zerothorder term is a classical solution while the firstorder correction is given as a phasespace Laplacian acting on the classical solution. This is then compared to the effective action solution for the onedimensional perturbative quantum field theory. We find an agreement up to the order , where is the coupling constant, while at the order there is a disagreement. Hence the coherent state expectation values define an alternative semiclassical dynamics to that of the effective action. The coherent statesemiclassical trajectories are exactly computable and they can coincide with the effective action trajectories in the case of twodimensional integrable field theories.

Minimally disturbing Heisenberg–Weyl symmetric measurements using hardcore collisions of Schrödinger particles
View Description Hide DescriptionIn a previous paper we have presented a general scheme for the implementation of symmetric generalized measurements (POVMs) on a quantum computer. This scheme is based on representation theory of groups and methods to decompose matrices that intertwine two representations. We extend this scheme in such a way that the measurement is minimally disturbing, i.e., it changes the state vector of the system to where is the positive operator corresponding to the measured result. Using this method, we construct quantum circuits for measurements with Heisenberg–Weyl symmetry. A continuous generalization leads to a scheme for optimal simultaneous measurements of position and momentum of a Schrödinger particle moving in one dimension such that the outcomes satisfy . The particle to be measured collides with two probe particles, one for the position and the other for the momentum measurement. The position and momentum resolution can be tuned by the entangled joint state of the probe particles which is also generated by a collision with hardcore potential. The parameters of the POVM can then be controlled by the initial widths of the wave functions of the probe particles. We point out some formal similarities and differences to simultaneous measurements of quadrature amplitudes in quantum optics.

Zero energy resonance and the logarithmically slow decay of unstable multilevel systems
View Description Hide DescriptionThe long time behavior of the reduced time evolution operator for unstable multilevel systems is studied based on the level Friedrichs model in the presence of a zero energy resonance. The latter means the divergence of the resolvent at zero energy. Resorting to the technique developed by Jensen and Kato [Duke Math. J.46, 583 (Year: 1979)], the zero energy resonance of this model is characterized by the zero energy eigenstate that does not belong to the Hilbert space. It is then shown that for some kinds of the rational form factors the logarithmically slow decay proportional to of the reduced time evolution operator can be realized.

Some physical applications of fractional Schrödinger equation
View Description Hide DescriptionThe fractional Schrödinger equation is solved for a free particle and for an infinite square potential well. The fundamental solution of the Cauchy problem for a free particle, the energy levels and the normalized wave functions of a particle in a potential well are obtained. In the barrier penetration problem, the reflection coefficient and transmission coefficient of a particle from a rectangular potential wall is determined. In the quantum scattering problem, according to the fractional Schrödinger equation, the Green’s function of the LippmannSchwinger integral equation is given.

Thin layer quantization in higher dimensions and codimensions
View Description Hide DescriptionWe consider the thin layer quantization with use of only the most elementary notions of differential geometry. We consider this method in higher dimensions and get an explicit formula for quantum potential. For codimension 1 surfaces the quantum potential is presented in terms of principal curvatures, and equivalence with Prokhorov quantization method is proved. It is shown that, in contrast with original da Costa method, Prokhorov quantization can be generalized directly to higher codimensions.

Analytic plane wave solutions for the quaternionic potential step
View Description Hide DescriptionBy using the recent mathematical tools developed in quaternionic differential operator theory, we solve the Schrödinger equation in the presence of a quaternionic step potential. The analytic solution for the stationary states allows one to explicitly show the qualitative and quantitative differences between this quaternionic quantum dynamical system and its complex counterpart. A brief discussion on reflected and transmitted times, performed by using the stationary phase method, and its implication on the experimental evidence for deviations of standard quantum mechanics is also presented. The analytic solution given in this paper represents a fundamental mathematical tool to find an analytic approximation to the quaternionic barrier problem (up to now solved by numerical method).

Toeplitz algebras and spectral results for the onedimensional Heisenberg model
View Description Hide DescriptionWe determine the structure of the spectrum and obtain nonpropagation estimates for a class of Toeplitz operators acting on a subset of the lattice . This class contains the Hamiltonian of the onedimensional Heisenberg model.

Grouptheoretical approach to reflectionless potentials
View Description Hide DescriptionWe examine the general form of potentials with zero reflection coefficient in onedimensional Hamiltonians connected with Casimir invariants of noncompact groups.

Decompositions of unitary evolutions and entanglement dynamics of bipartite quantum systems
View Description Hide DescriptionWe describe a decomposition of the Lie group of unitary evolutions for a bipartite quantum system of arbitrary dimensions. The decomposition is based on a recursive procedure that systematically uses the Cartan classification of the symmetric spaces of the Lie group. The resulting factorization of unitary evolutions clearly displays the local and entangling character of each factor.

Finitely many Diracdelta interactions on Riemannian manifolds
View Description Hide DescriptionThis work is intended as an attempt to study the nonperturbative renormalization of bound state problem of finitely many Diracdelta interactions on Riemannian manifolds,, , and . We formulate the problem in terms of a finite dimensional matrix, called the characteristic matrix . The bound state energies can be found from the characteristicequation. The characteristic matrix can be found after a regularization and renormalization by using a sharp cutoff in the eigenvalue spectrum of the Laplacian, as it is done in the flat space, or using the heat kernel method. These two approaches are equivalent in the case of compact manifolds. The heat kernel method has a general advantage to find lower bounds on the spectrum even for compact manifolds as shown in the case of . The heat kernels for and are known explicitly, thus we can calculate the characteristic matrix . Using the result, we give lower bound estimates of the discrete spectrum.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Anomalyfree sets of fermions
View Description Hide DescriptionWe present new techniques for finding anomalyfree sets of fermions. Although the anomaly cancellation conditions typically include cubic equations with integer variables that cannot be solved in general, we prove by construction that any chiral set of fermions can be embedded in a larger set of fermions which is chiral and anomalyfree. Applying these techniques to extensions of the standard model, we find anomalyfree models that have arbitrary quark and lepton charges under an additional gauge group.

Exact analytical solution to the relativistic KleinGordon equation with noncentral equal scalar and vector potentials
View Description Hide DescriptionWe present an alternative and simple method for the exact solution of the KleinGordon equation in the presence of the noncentral equal scalar and vector potentials by using NikiforovUvarov method. The exact bound state energy eigenvalues and corresponding eigenfunctions are obtained for a particle bound in a potential of type.

Quantum energy inequalities and local covariance. I. Globally hyperbolic spacetimes
View Description Hide DescriptionWe begin a systematic study of quantum energy inequalities (QEIs) in relation to local covariance. We define notions of locally covariant QEIs of both “absolute” and “difference” types and show that existing QEIs satisfy these conditions. Local covariance permits us to place constraints on the renormalized stressenergy tensor in one spacetime using QEIs derived in another, in subregions where the two spacetimes are isometric. This is of particular utility where one of the two spacetimes exhibits a high degree of symmetry and the QEIs are available in simple closed form. Various general applications are presented, including a priori constraints (depending only on geometric quantities) on the groundstate energy density in a static spacetime containing locally Minkowskian regions. In addition, we present a number of concrete calculations in both two and four dimensions that demonstrate the consistency of our bounds with various known ground and thermalstate energy densities. Examples considered include the Rindler and Misner spacetimes, and spacetimes with toroidal spatial sections. In this paper we confine the discussion to globally hyperbolic spacetimes; subsequent papers will also discuss spacetimes with boundary and other related issues.

Liouville theory and uniformization of fourpunctured sphere
View Description Hide DescriptionA few years ago Zamolodchikov and Zamolodchikov proposed an expression for the fourpoint classical Liouville action in terms of the threepoint actions and the classical conformal block [Nucl. Phys. B477, 577 (1996)]. In this paper we develop a method of calculating the uniformizing map and the uniformizing group from the classical Liouville action on punctured sphere and discuss the consequences of Zamolodchikovs conjecture for an explicit construction of the uniformizing map and the uniformizing group for the sphere with four punctures.

Gauge theories of Dirac type
View Description Hide DescriptionA specific class of gauge theories is geometrically described in terms of fermions. In particular, it is shown how the geometrical frame presented naturally includes spontaneous symmetry breaking of YangMillsgauge theories without making use of a Higgs potential. In more physical terms, it is shown that the Yukawa coupling of fermions, together with gravity, necessarily yields a symmetry reduction provided the fermionic mass is considered as a globally welldefined concept. The structure of this symmetry breaking is shown to be compatible with the symmetry breaking that is induced by the Higgs potential of the minimal Standard Model. As a consequence, it is shown that the fermionic mass has a simple geometrical interpretation in terms of curvature and that the (semiclassical) “fermionic vacuum” determines the intrinsic geometry of spacetime. We also discuss the issue of “fermion doubling” in some detail and introduce a specific projection onto the “physical subspace” that is motivated by the Standard Model.
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 GENERAL RELATIVITY AND GRAVITATION


stars in scalartensor gravitational theories in extra dimensions
View Description Hide DescriptionWe present JordanBransDicke and general scalartensor gravitational theory in extra dimensions in an asymptotically flat or anti de Sitter spacetime. We consider a special gravitating, boson field configuration, a star, in three, four, five, and six dimensions, within the framework of the above gravitational theory, and find that the parameters of the stable stars are a few percent different from the case of General Relativity.
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 DYNAMICAL SYSTEMS


Mathematical analysis of the wavelet method of chaos control
View Description Hide DescriptionIn this paper, we provide mathematical analysis for the controllability of chaos in waveletsubspaces. We prove that depending on the scale of the wavelet operation and the number of the coupled oscillators, the critical coupling strength for the occurrence of chaossynchronization becomes many times smaller if the original coupling matrix is appropriately treated with the wavelet transform. Moreover, we obtain rigorous relations connecting the critical values and the waveletsubspace operations. Our mathematical results are completely consistent with early numerical simulations.

Goldfishing by gauge theory
View Description Hide DescriptionA new solvablemanybody problem of goldfish type is identified and used to revisit the connection between two different approaches to solvable dynamical systems. An isochronous variant of this model is identified and investigated. Alternative versions of these models are presented. The behavior of the alternative isochronous model near its equilibrium configurations is investigated, and a remarkable Diophantine result, as well as related Diophantine conjectures, are thereby obtained.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Optimization problems for an elastic plate
View Description Hide DescriptionThis paper concerns optimization problems related to biharmonic equations subject to either Navier or Dirichlet homogeneous boundary conditions. Physically, in dimension two, our equation models the deformation of an elastic plate which is either hinged or clamped along the boundary, under load. We discuss existence, uniqueness, and properties of the optimizers.

Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives
View Description Hide DescriptionTwo nonlinear anelastic models with fractional derivatives, describing the properties of a series of materials as polymers, and polycrystallinematerials are presented in this paper. These models are studied analytically, using a variational iteration method. The paper clarifies the different ways in which the fractional differentiation operator can be defined. A Volterra series method of model parameters identification from the experimental data is also presented.
