Volume 48, Issue 9, September 2007
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Incomplete KochenSpecker coloring
View Description Hide DescriptionA particular incomplete KochenSpecker coloring, suggested by Appleby [Stud. Hist. Philos. Mod. Phys.36, 1 (2005)] in dimension three, is generalized to arbitrary dimension. We investigate its effectivity as a function of dimension, using two different measures. A limit is derived for the fraction of the sphere that can be colored using the generalized Appleby construction as the number of dimensions approaches infinity. The second, and physically more relevant measure of effectivity, is to look at the fraction of properly colored bases. Using this measure, we derive a “lower bound for the upper bound” in three and four real dimensions.

Limit relation for quantum entropy and channel capacity per unit cost
View Description Hide DescriptionIn a thermodynamic model, Diósi et al. [Int. J. Quantum Inf.4, 99–104 (2006)] arrived at a conjecture stating that certain differences of von Neumann entropies converge to relative entropy as system size goes to infinity. The conjecture is proven in this paper for density matrices. The analytic proof uses the quantum law of large numbers and the inequality between the BelavkinStaszewski and Umegaki relative entropies. Moreover, the concept of channel capacity per unit cost is introduced for classicalquantum channels. For channels with binary input alphabet, this capacity is shown to equal the relative entropy. The result provides a second proof of the conjecture and a new interpretation. Both approaches lead to generalizations of the conjecture.

Notes on coherent backscattering from a random potential
View Description Hide DescriptionWe consider the quantum scattering from a random potential of strength and with a support on the scale of the mean free path, which is of order . On the basis of maximally crossed diagrams, we provide a concise formula for the backscattering rate in terms of Green’s function for the kinetic Boltzmann equation. We briefly discuss the extension to wave scattering.

Convergence of resonances on thin branched quantum waveguides
View Description Hide DescriptionWe prove an abstract criterion stating resolvent convergence in the case of operators acting in different Hilbert spaces. This result is then applied to the case of Laplacians on a family of branched quantum waveguides. Combining it with an exterior complex scaling we show, in particular, that the resonances on approximate those of the Laplacian with “free” boundary conditions on , the skeleton graph of .

A nonstandard geometric quantization of the harmonic oscillator
View Description Hide DescriptionIn the standard geometric quantization of the harmonic oscillator in , using standard holomorphic coordinate induces a standard complex structure which Kähler polarizes the prequantum line bundle. Hence, the quantum Hilbert space maybe identified with holomorphic functions which are integrable with respect to Gaussian measure, denoted by . The corresponding quantized Hamiltonian is then given by . We propose using a different almost complex structure (under some restrictions) compatible with and construct a quantum Hilbert space containing nontrivial sections and a quantized Hamiltonian . The main result is that is unitarily equivalent to .

Biorthogonal quantum systems
View Description Hide DescriptionModels of PT symmetric quantum mechanics provide examples of biorthogonal quantum systems. The latter incorporate all the structure of PT symmetric models, and allow for generalizations, especially in situations where the PT construction of the dual space fails. The formalism is illustrated by a few exact results for models of the form . In some nontrivial cases, equivalent Hermitian theories are obtained and shown to be very simple: They are just free (chiral) particles. Field theory extensions are briefly considered.

Linear Wegner estimate for alloytype Schrödinger operators on metric graphs
View Description Hide DescriptionWe study spectra of alloytype random Schrödinger operators on metric graphs. For finite edge subsets we prove a Wegner estimate which is linear in the volume (i.e., the total length of the edges) and the length of the energy interval. The single site potential needs to have fixed sign; the metric graph does not need to have a periodic structure. A further result is the existence of the integrated density of states for ergodic random Hamiltonians on metric graphs with a structure. For certain models the two above results together imply the Lipschitz continuity of the integrated density of states.

Supersymmetric biorthogonal quantum systems
View Description Hide DescriptionWe discuss supersymmetric biorthogonal systems, with emphasis given to the periodic solutions that occur at spectral singularities of PT symmetric models. For these periodic solutions, the dual functions are associated polynomials that obey inhomogeneous equations. We construct in detail some explicit examples for the supersymmetric pairs of potentials where . In particular, we consider the cases generated by and . We also briefly consider the effects of magnetic vector potentials on the partition functions of these systems.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


Dirac equation in the background of the Nutku helicoid metric
View Description Hide DescriptionWe study the solutions of the Dirac equation in the background of the Nutku helicoid metric. This metric has curvature singularities, which necessitates imposing a boundary to exclude this point. We use the AtiyahPatodiSinger [Math. Proc. Cambridge Philos. Soc.77, 43 (1975)] nonlocal spectral boundary conditions for both the four and the five dimensional manifolds.

CPT and Lorentz violation effects in hydrogenlike atoms
View Description Hide DescriptionWithin the framework of Lorentzviolating extended electrodynamics, the Dirac equation for a bound electron in an external electromagnetic field is considered assuming the interaction with a CPTodd axial vector background . The quasirelativistic Hamiltonian is obtained using a series expansion. Relativistic Dirac eigenstates in a spherically symmetric potential are found accurate up to the second order in . induced CPTodd corrections to the electromagnetic dipole moment operators of a bound electron are calculated that contribute to the anapole moment of the atomic orbital and may cause a specific asymmetry of the angular distribution of the radiation of a hydrogen atom.
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 GENERAL RELATIVITY AND GRAVITATION


Quantum singularities in spacetimes with spherical and cylindrical topological defects
View Description Hide DescriptionExact solutions of Einstein equations with null RiemmanChristoffel curvature tensor everywhere, except on a hypersurface, are studied using quantum particles obeying the KleinGordon equation. We consider the particular cases when the curvature is represented by a Dirac delta function with support either on a sphere or on a cylinder (spherical and cylindrical shells). In particular, we analyze the necessity of extra boundary conditions on the shells.

Simplified mathematical model for the formation of null singularities inside black holes. I. Basic formulation and a conjecture
View Description Hide DescriptionEinstein’s equations are known to lead to the formation of black holes and space timesingularities. This appears to be a manifestation of the mathematical phenomenon of finitetime blowup: A formation of singularities from regular initial data. We present a simple hyperbolic system of two semilinear equations inspired by the Einstein equations. We explore a class of solutions to this system which are analogous to static blackhole models. These solutions exhibit a blackhole structure with a finitetime blowup on a characteristic line mimicking the null inner horizon of spinning or charged black holes. We conjecture that this behavior—namely, blackhole formation with blowup on a characteristic line—is a generic feature of our semilinear system. Our simple system may provide insight into the formation of null singularities inside spinning or charged black holes in the full system of Einstein equations.

Simplified mathematical model for the formation of null singularities inside black holes. II. Proof for a particular case
View Description Hide DescriptionWe study a simple system of two hyperbolic semilinear equations inspired by the Einstein equations. The system, which was introduced by Ori and Gorbonos [J. Math. Phys., 48, 092502 (2007)], is a model for singularity formation inside black holes. We show for a particular case of the equations that the system demonstrates a finite time blowup. The singularity that is formed is a null singularity. Then we show that in this particular case the singularity has features that are analogous to known features of models of blackhole interiors—which describe the innerhorizon instability. Our simple system may provide insight into the formation of null singularities inside spinning or charged black holes.

Anatomy of malicious singularities
View Description Hide DescriptionAs well known, the boundaries of the closed Friedman world model and of Schwarzschild solution consist of a single point. We study this phenomenon in a broader context of differential and structured spaces. We show that it is an equivalence relation , defined on the Cauchy completed total space of the frame bundle over a given spacetime, that is responsible for this pathology. A singularity is called malicious if the equivalence class related to the singularity remains in close contact with all other equivalence classes, i.e., if for every . We formulate conditions for which such a situation occurs. The differential structure of any spacetime with malicious singularities consists only of constant functions which means that, from the topological point of view, everything collapses to a single point. It was noncommutative geometry that was especially devised to deal with such situations. A noncommutative algebra on , which turns out to be a von Neumann algebra of random operators, allows us to study probabilistic properties (in a generalized sense) of malicious singularities. Our main result is that, in the noncommutative regime, even the strongest singularities are probabilistically irrelevant.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Radiation reaction in electrodynamics
View Description Hide DescriptionA selfaction problem for a pointlike charged particle arbitrarily moving in flat spacetime of three dimensions is considered. Outgoing waves carry energymomentum and angular momentum; the radiation removes energy, momentum, and angular momentum from the source which then undergoes a radiation reaction. We decompose Noether quantities carried by electromagnetic field into bound and radiative components. The bound terms are absorbed by individual particle’s characteristics within the renormalization procedure. Radiative terms together with already renormalized 3momentum and angular momentum of pointlike charge constitute the total conserved quantities of our particle plus field system. Their differential consequences yield the effective equation of motion of radiating charge in an external electromagnetic field. In this integrodifferential equation the radiation reaction is determined by Lorentz force of pointlike charge acting upon itself plus nonlocal term which provides finiteness of the selfaction.

Integrable almostsymplectic Hamiltonian systems
View Description Hide DescriptionWe extend the notion of Liouville integrability, which is peculiar to Hamiltonian systems on symplectic manifolds, to Hamiltonian systems on almostsymplectic manifolds, namely, manifolds equipped with a nondegenerate (but not closed) 2form. The key ingredient is to require that the Hamiltonian vector fields of the integrals of motion in involution (or equivalently, the generators of the invariant tori) are symmetries of the almostsymplectic form. We show that, under this hypothesis, essentially all of the structure of the symplectic case (from quasiperiodicity of motions to an analog of the actionangle coordinates and of the isotropiccoisotropic dual pair structure characteristic of the fibration by the invariant tori) carries over to the almostsymplectic case.

Isochronous extension of the Hamiltonian describing free motion in the Poincaré halfplane: Classical and quantum treatments
View Description Hide DescriptionWe modify (in two different manners) the Hamiltonian describing motions in the Poincaré halfplane so that the modified Hamiltonians thereby obtained are entirely isochronous: indeed, in the classical context, all the motions they entail are periodic with the same period. We then investigate suitably quantized versions of these systems and show that their spectra are equispaced.
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 STATISTICAL PHYSICS


Asymptotic expansion of the logpartition function for a gas of interacting Brownian loops
View Description Hide DescriptionThe asymptotic behavior of the logarithm of the grand partition function of a quantum gas in the FeynmanKac representation is studied. Under suitable restrictions on the stable pair potential the logpartition function is expanded at low activity as the domain is dilated to the infinity. The volume and the boundary terms are given explicitly as functional integrals. The proof is based on the cluster expansion method.
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 METHODS OF MATHEMATICAL PHYSICS


Reflection equation and twisted Yangians
View Description Hide DescriptionWith any involutive antialgebra and coalgebra automorphism of a quasitriangular bialgebra we associate a reflection equation algebra. A Hopf algebraic treatment of the reflection equation of this type and its universal solution is given. Applications to the twisted Yangians are considered.

Multidimensional integrable systems and deformations of Lie algebra homomorphisms
View Description Hide DescriptionWe use deformations of Lie algebra homomorphisms to construct deformations of dispersionless integrable systems arising as symmetry reductions of antiselfdual YangMillsequations with a gauge group .
