Volume 47, Issue 12, December 2006
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Bayesian analog of Gleason’s theorem
View Description Hide DescriptionWe introduce a novel notion of probability within quantum history theories and give a Gleasonesque proof for these assignments. This involves introducing a tentative novel axiom of probability. We also discuss how we are to interpret these generalized probabilities as partially ordered notions of preference, and we introduce a tentative generalized notion of Shannon entropy. A Bayesian approach to probability theory is adopted throughout; thus the axioms we use will be minimal criteria of rationality rather than ad hoc mathematical axioms.

PauliHamiltonian in the presence of minimal lengths
View Description Hide DescriptionWe construct the PauliHamiltonian on a space where the position and momentum operators obey generalized commutation relations leading to the appearance of a minimal length. Using the momentum space representation we determine exactly the energy eigenvalues and eigenfunctions for a charged particle of spin half moving under the action of a constant magnetic field. The thermal properties of the system in the regime of high temperatures are also investigated, showing a magnetic behavior in terms of the minimal length.

On bipartite purestate entanglement structure in terms of disentanglement
View Description Hide DescriptionSchrödinger’s disentanglement [E. Schrödinger, Proc. Cambridge Philos. Soc.31, 555 (1935)], i.e., remote state decomposition, as a physical way to study entanglement, is carried one step further with respect to previous work in investigating the qualitative side of entanglement in any bipartite state vector. Remote measurement (or, equivalently, remote orthogonal state decomposition) from previous work is generalized to remote linearly independent complete state decomposition both in the nonselective and the selective versions. The results are displayed in terms of commutative square diagrams, which show the power and beauty of the physical meaning of the (antiunitary) correlation operator inherent in the given bipartite state vector. This operator, together with the subsystem states (reduced density operators), constitutes the socalled correlated subsystem picture. It is the central part of the antilinear representation of a bipartite state vector, and it is a kind of core of its entanglement structure. The generalization of previously elaborated disentanglement expounded in this article is a synthesis of the antilinear representation of bipartite state vectors, which is reviewed, and the relevant results of [Cassinelli et al., J. Math. Anal. Appl.210, 472 (1997)] in mathematical analysis, which are summed up. Linearly independent bases (finite or infinite) are shown to be almost as useful in some quantum mechanical studies as orthonormal ones. Finally, it is shown that linearly independent remote purestate preparation carries the highest probability of occurrence. This singles out linearly independent remote influence from all possible ones.

Shape invariance through Crum transformation
View Description Hide DescriptionWe show in a rigorous way that Crum’s result regarding the equal eigenvalue spectrum of SturmLiouville problems can be obtained iteratively by successive Darboux transformations. Furthermore, it can be shown that all neighboring Darbouxtransformed potentials of higher order, and , satisfy the condition of shape invariance provided the original potential does so. Based on this result, we prove that under the condition of shape invariance, the iteration of the original SturmLiouville problem defined solely through the shape invariance is equal to the Crum transformation.

Parametrictime coherent states for the generalized MICKepler system
View Description Hide DescriptionIn this study, we construct the parametrictime coherent states for the negative energy states of the generalized MICKepler system, in which a charged particle is in a monopole vector potential, a Coulomb potential, and a BohmAharonov potantial. We transform the system into four isotropic harmonic oscillators and construct the parametrictime coherent states for these oscillators. Finally, we compactify these states into the physical time coherent states for the generalized MICKepler system.

Timeofarrival probabilities and quantum measurements
View Description Hide DescriptionIn this paper we study the construction of probability densities for time of arrival in quantum mechanics. Our treatment is based upon the facts that (i) time appears in quantum theory as an external parameter to the system, and (ii) propositions about the time of arrival appear naturally when one considers histories. The definition of timeofarrival probabilities is straightforward in stochastic processes. The difficulties that arise in quantum theory are due to the fact that the time parameter of the Schrödinger’s equation does not naturally define a probability density at the continuum limit, but also because the procedure one follows is sensitive on the interpretation of the reduction procedure. We consider the issue in Copenhagen quantum mechanics and in historybased schemes like consistent histories. The benefit of the latter is that it allows a proper passage to the continuous limit—there are, however, problems related to the quantum Zeno effect and decoherence. We finally employ the historiesbased description to construct PositiveOperatorValuedMeasures (POVMs) for the timeofarrival, which are valid for a general Hamiltonian. These POVMs typically depend on the resolution of the measurement device; for a free particle, however, this dependence cancels in the physically relevant regime and the POVM coincides with that of Kijowski.

Hudson’s theorem for finitedimensional quantum systems
View Description Hide DescriptionWe show that, on a Hilbert space of odd dimension, the only pure states to possess a nonnegative Wigner function are stabilizer states. The Clifford group is identified as the set of unitary operations which preserve positivity. The result can be seen as a discrete version of Hudson’s theorem. Hudson established that for continuous variable systems, the Wigner function of a pure state has no negative values if and only if the state is Gaussian. Turning to mixed states, it might be surmised that only convex combinations of stabilizer states give rise to nonnegative Wigner distributions. We refute this conjecture by means of a counterexample. Further, we give an axiomatic characterization which completely fixes the definition of the Wigner function and compare two approaches to stabilizer states for Hilbert spaces of primepower dimensions. In the course of the discussion, we derive explicit formulas for the number of stabilizer codes defined on such systems.

Fractional supersymmetry and hierarchy of shape invariant potentials
View Description Hide DescriptionFractional supersymmetric quantum mechanics is developed from a generalized WeylHeisenberg algebra. The Hamiltonian and the supercharges of fractional supersymmetric dynamical systems are built in terms of the generators of this algebra. The Hamiltonian gives rise to a hierarchy of isospectral Hamiltonians. Special cases of the algebra lead to dynamical systems for which the isospectral supersymmetric partner Hamiltonians are connected by a (translational or cyclic) shape invariance condition.

Propagator for finite range potentials
View Description Hide DescriptionThe Schrödinger equation in integral form is applied to the onedimensional scattering problem in the case of a general finite range, nonsingular potential. A simple expression for the Laplace transform of the transmission propagator is obtained in terms of the associated Fredholm determinant, by means of matrix methods; the particular form of the kernel and the peculiar aspects of the transmission problem play an important role. The application to an array of delta potentials is shown.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Relativistic quaternionic wave equation
View Description Hide DescriptionWe study a onecomponent quaternionic wave equation which is relativistically covariant. Bilinear forms include a conserved fourvector current and an antisymmetric second rank tensor. Waves propagate within the light cone and there is a conserved quantity which looks like helicity. The principle of superposition is retained in a slightly altered manner. External potentials can be introduced in a way that allows for gauge invariance. There are some results for scattering theory and for twoparticle wave functions as well as the beginnings of second quantization. However, we are unable to find a suitable Lagrangian or an energymomentum tensor.

Generating loop graphs via Hopf algebra in quantum field theory
View Description Hide DescriptionWe use the Hopf algebrastructure of the timeordered algebra of field operators to generate all connected weighted Feynman graphs in a recursive and efficient manner. The algebraic representation of the graphs is such that they can be evaluated directly as contributions to the connected point functions. The recursion proceeds by loop order and vertex number.

Endomorphisms on halfsided modular inclusions
View Description Hide DescriptionIn algebraic quantum field theory we consider nets of von Neumann algebras indexed over regions of the space time. Wiesbrock [“Conformal quantum field theory and halfsided modular inclusions of von Neumann algebras,” Commun. Math. Phys.158, 537–543 (Year: 1993)] has shown that strongly additive nets of von Neumann algebras on the circle are in correspondence with standard halfsided modular inclusions. We show that a finite index endomorphism on a halfsided modular inclusion extends to a finite index endomorphism on the corresponding net of von Neumann algebras on the circle. Moreover, we present another approach to encoding endomorphisms on nets of von Neumann algebras on the circle into halfsided modular inclusions. There is a natural way to associate a weight to a Möbius covariant endomorphism. The properties of this weight have been studied by Bertozzini et al. [“Covariant sectors with infinite dimension and positivity of the energy,” Commun. Math. Phys.193, 471–492 (Year: 1998)]. In this paper we show the converse, namely, how to associate a Möbius covariant endomorphism to a given weight under certain assumptions, thus obtaining a correspondence between a class of weights on a halfsided modular inclusion and a subclass of the Möbius covariant endomorphisms on the associated net of von Neumann algebras. This allows us to treat Möbius covariant endomorphisms in terms of weights on halfsided modular inclusions. As our aim is to provide a framework for treating endomorphisms on nets of von Neumann algebras in terms of the apparently simpler objects of weights on halfsided modular inclusions, we lastly give some basic results for manipulations with such weights.

Minimal redefinition of the OSV ensemble
View Description Hide DescriptionIn the interesting conjecture, , proposed by Ooguri, Strominger, and Vafa (OSV), the black hole ensemble is a mixed ensemble. So if working in the complex polarization, the black hole degeneracy of states as obtained from the ensemble inverseLaplace integration, generically receives prefactors that do not respect the electricmagnetic duality. One way to handle this, as claimed recently, is working instead of the complex polarization in the real polarization. The other idea would be imposing nontrivial measures for the ensemble sum in the complex polarization. We address this problem in the complex polarization, which is canonical, and upon a redefinition of the OSV ensemble with variables as numerous as the electric potentials, show that for restoring the symmetry no nonEuclidean measure is needed. In detail, applying the electricmagnetic duality as a constraint governing the proper definition of the ensemble variables, we rewrite the OSV free energy as a function of new variables that are combinations of the electric potentials and the black hole charges. Subsequently the Legendre transformation, which bridges between the entropy and the black holefree energy in terms of these variables, points to a generalized ensemble that is well behaved in the complex polarization. In this context, we will consider all the cases of relevance: small and large black holes, with or without charge. For the case of vanishing , the new ensemble is purely canonical and the electricmagnetic duality is restored exactly, leading to proper results for the black hole degeneracy of states to all orders in an asymptotic expansion. For more general cases as well, the construction does the job as far as the violation of the duality by the corresponding OSV result is restricted to a prefactor. In the case of black holes with nonvanishing charge, in a concrete example, we shall show that there are cases where the duality violation goes beyond this restriction and imposing nontrivial measures is incapable of restoring the duality.
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 GENERAL RELATIVITY AND GRAVITATION


Fractional Israel layers
View Description Hide DescriptionA fractional Lie derivative, valid in the thin shell limit, is developed. The nonlocal nature of the fractional derivative allows the inclusion of shell thickness in the stress energy description of zero thickness Israel layers. The method is applied to several examples.
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 DYNAMICAL SYSTEMS


Infinitely many periodic orbits for the rhomboidal fivebody problem
View Description Hide DescriptionWe prove the existence of infinitely many symmetric periodic orbits for a regularized rhomboidal fivebody problem with four small masses placed at the vertices of a rhombus centered in the fifth mass. The main tool for proving the existence of such periodic orbits is the analytic continuation method of Poincaré together with the symmetries of the problem.

Perturbed block circulant matrices and their application to the wavelet method of chaotic control
View Description Hide DescriptionControlling chaos via wavelet transform was proposed by Wei et al. [Phys. Rev. Lett.89, 284103–1284103–4 (2002)]. It was reported there that by modifying a tiny fraction of the waveletsubspace of a coupling matrix, the transverse stability of the synchronous manifold of a coupled chaotic system could be dramatically enhanced. The stability of chaotic synchronization is actually controlled by the second largest eigenvalue of the (wavelet) transformed coupling matrix for each and . Here is a mixed boundary constant and is a scalar factor. In particular, (0) gives the nearest neighbor coupling with periodic (Neumann) boundary conditions. In this paper, we obtain two main results. First, the reduced eigenvalue problem for is completely solved. Some partial results for the reduced eigenvalue problem of are also obtained. Second, we are then able to understand behavior of and for any wavelet dimension and block dimension . Our results complete and strengthen the work of Shieh et al. [J. Math. Phys.47, 082701–1082701–10 (2006)] and Juang and Li [J. Math. Phys.47, 072704–1072704–16 (2006)].
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Existence and uniqueness for the reflection and transmission problem in stratified electromagnetic media
View Description Hide DescriptionThe reflectiontransmission problem of timeharmonic waves in a stratified electromagnetic medium is investigated. The waves are sent from upward or downward with oblique incidence. By means of the energy flux, upgoing and downgoing waves are distinguished and the reflection and transmission matrices are introduced. When the solid occupies a strip between two homogeneous media, the existence and uniqueness of the reflected and transmitted waves are proved. The same conclusions are obtained for a dielectric without memory extended in the whole space.

Green functions for wave propagation on a fivedimensional manifold and the associated gauge fields generated by a uniformly moving point source
View Description Hide DescriptionGauge fields associated with the manifestly covariant dynamics of particles in (3,1) space time are five dimensional (5D). We provide solutions of the classical 5D gauge fieldequations in both (4,1) and (3,2) flat spacetime metrics for the simple example of a uniformly moving point source. Green functions for the 5D field equations are obtained, which are consistent with the solutions for uniform motion obtained directly from the field equations with free asymptotic conditions.
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 STATISTICAL PHYSICS


New class of statistical models: Transfer matrix eigenstates, chain Hamiltonians, factorizable matrix
View Description Hide DescriptionStatistical models corresponding to a new class of braid matrices presented in a previous paper are studied. Indices labeling states spanning the dimensional base space of , the th order transfer matrix are so chosen that the operators (the sum of the state labels) and (CP) (the circular permutation of state labels) commute with . This drastically simplifies the construction of eigenstates, reducing it to solutions of relatively small number of simultaneous linear equations. Roots of unity play a crucial role. Thus for diagonalizing the 81 dimensional space for , , one has to solve a maximal set of five linear equations. A supplementary symmetry relates invariant subspaces pairwise [ and so on] so that only one of each pair needs study. The case is studied fully for . Basic aspects for all are discussed. Full exploitation of such symmetries lead to a formalism quite different from, possibly generalized, algebraic Bethe ansatz. Chain Hamiltonians are studied. The specific types of spin flips they induce and propagate are pointed out. The inverse Cayley transform of the YB matrix giving the potential leading to factorizable matrix is constructed explicitly for as also the full set of relations. Perspectives are discussed in a final section.

Localization at low temperature and infrared bounds
View Description Hide DescriptionWe consider a class of classical lattice spin systems, with valued spins and twobody interactions. Our main result states that the associated Gibbs measure localizes in certain cylindrical neighborhoods of the global minima of the unperturbed Hamiltonian. As an application we establish existence of a first order phase transition at low temperature, for a reflection positive mexican hat model on , , with a nonferromagnetic interaction.
