Volume 43, Issue 10, October 2002
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Behavior of the survival probability in some onedimensional problems
View Description Hide DescriptionWe study the behavior of the survival probability in potential barrier problems on the halfline. We consider two cases: when the initial state φ is a truncated outgoing solution corresponding to a scattering frequency and when it is an approximate bound state. Using techniques and methods from Spectral Theory, Fourier Analysis, and from Ordinary Differential Equations, we obtain approximate exponential decay in both cases.

Reduction of the Hilbert space in strongly correlated systems
View Description Hide DescriptionDefining commutation relations between symmetry operators and fundamental operators, we set up the symmetry group for a manyparticle Hamiltonian. Using the irreducible representations of the symmetry group, we decompose the Hilbert space. We discuss the advantage of this approach to find the dimensions of reduced Hilbert spaces in numerical exact diagonalization.

Hall effect in noncommutative coordinates
View Description Hide DescriptionWe consider electrons in uniform external magnetic and electric fields which move on a plane whose coordinates are noncommuting. Spectrum and eigenfunctions of the related Hamiltonian are obtained. We derive the electric current whose expectation value gives the Hall effect in terms of an effective magnetic field. We present a receipt to find the action which can be utilized in path integrals for noncommuting coordinates. In terms of this action we calculate the related Aharonov–Bohm phase and show that it also yields the same effective magnetic field. When magnetic field is strong enough this phase becomes independent of magnetic field. Measurement of it may give some hints on spatial noncommutativity. The noncommutativity parameter θ can be tuned such that electrons moving in noncommutative coordinates are interpreted as either leading to the fractional quantum Hall effect or composite fermions in the usual coordinates.

Wigner functions with boundaries
View Description Hide DescriptionWe consider the general Wigner function for a particle confined to a finite interval and subject to Dirichlet boundary conditions. We derive the boundary corrections to the “stargenvalue” equation and to the time evolution equation. These corrections can be cast in the form of a boundary potential contributing to the total Hamiltonian which together with a subsidiary boundary condition is responsible for the discretization of the energy levels. We show that a completely analogous formulation (in terms of boundary potentials) is also possible in standard operator quantum mechanics and that the Wigner and the operator formulations are also in onetoone correspondence in the confined case. In particular, we extend Baker’s converse construction to bounded systems. Finally, we elaborate on the applications of the formalism to the subject of Wigner trajectories, namely in the context of collision processes and quantum systems displaying chaotic behavior in the classical limit.

The dynamical elliptic quantum Gaudin models and their solutions
View Description Hide DescriptionIn this paper, we construct the Hamiltonians of both periodic and open elliptic quantum Gaudin models and show their relations with the elliptic quantum group, and the boundary elliptic quantum group, respectively. We define the eigenstates of these two models to be the Bethe vectors with of the elliptic quantum group and the boundary elliptic quantum group, respectively. Then, the Hamiltonians are exactly diagonalized by using the algebraic Bethe ansatz method.

Additivity for unital qubit channels
View Description Hide DescriptionAdditivity of the Holevo capacity is proved for product channels, under the condition that one of the channels is a unital qubit channel, with the other completely arbitrary. As a byproduct this proves that the Holevo bound is the classical information capacity of such qubit channels, and provides an explicit formula for this classical capacity. Additivity of minimal entropy and multiplicativity of norms are also proved under the same assumptions. The proof relies on a new bound for the norm of an output state from the halfnoisy phasedamping channel.

Method of solving the homogeneous Bloch equation
View Description Hide DescriptionA method of solving the Bloch equation with infinite relaxation times is given. Applying this method we have found unknown analytic solutions as well as the wellknown solutions.

Semiclassical generalization of the Darboux–Christoffel formula
View Description Hide DescriptionThe Darboux–Christoffel formula is a closedform expression for the kernel of the operator that projects onto the first of a system of onedimensional polynomials, orthonormal with respect to some weighting function. It is a key element in the theory of Gaussian integration and in the theory of discrete variable representation or Lagrangian mesh methods for diagonalizing quantum Hamiltonians of a few degrees of freedom. The onedimensional Darboux–Christoffel formula turns out to have a generalization that is valid in a semiclassical or asymptotic sense for a wider class of orthonormal functions than orthonormal polynomials. This class consists of the bound eigenfunctions of onedimensional Hamiltonians with timereversal invariance, such as kineticpluspotential Hamiltonians. It also has certain generalizations involving the unbound eigenfunctions of such Hamiltonians.

The twodimensional hydrogen atom revisited
View Description Hide DescriptionThe boundstate energyeigenvalues for the twodimensional Kepler problem are found to be degenerate. This “accidental” degeneracy is due to the existence of a twodimensional analog of the quantummechanical Runge–Lenz vector. Reformulating the problem in momentum space leads to an integral form of the Schrödinger equation. This equation is solved by projecting the twodimensional momentum space onto the surface of a threedimensional sphere. The eigenfunctions are then expanded in terms of spherical harmonics, and this leads to an integral relation in terms of special functions which has not previously been tabulated. The dynamical symmetry of the problem is also considered, and it is shown that the two components of the Runge–Lenz vector in real space correspond to the generators of infinitesimal rotations about the respective coordinate axes in momentum space.

Unification of the Jaynes–Cummings model and Planck’s radiation law
View Description Hide DescriptionBy combining iterative methods with Laplace transformation, we construct the solution of a dissipative Jaynes–Cummings model. The dissipative part of the model is based on the standard Markovian master equation for a harmonic oscillator that is coupled to a heat bath of nonzero temperature. Besides photon loss, we also take into account frequency detuning between atom and field. Before commencing the iteration, we subject the matrix elements of the density operator to a transformation that depends on temperature. As a result, the pole structure of all Laplace transformed matrix elements is improved. It becomes manifest which poles do not contribute to the asymptotic behavior of the density operator. In proving that our iterative process yields convergent results, we assume upper bounds on: the matrix elements of the density operator, the matrix elements of the initial density operator, the damping parameter of the heat bath, and the temperature of the heat bath. All of these bounds are physically acceptable. The photon field may start from a coherent state or a number state. For experiments in a microwavecavity, temperatures of the order of 0.1 [K] are allowed. As an application, the evolution of the atomic density matrix is studied. We propose a limit for which this matrix converges to the state of maximum von Neumann entropy. The time, the cubed initial energy density, and the inverse of the damping parameter must tend to infinity equally fast. The photon field is assumed to be in a number state at time zero, whereas the initial state of the atom can be chosen freely.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


On the invariance of residues of Feynman graphs
View Description Hide DescriptionWe use simple iterated oneloop graphs in massless Yukawa theory and QED to pose the following question: what are the symmetries of the residues of a graph under a permutation of places to insert subdivergences. The investigation confirms partial invariance of the residue under such permutations: the highest weight transcendental is invariant under such a permutation. For QED this result is gauge invariant, i.e., the permutation invariance holds for any gauge. Computations are done making use of the Hopf algebrastructure of graphs and employing GiNaC to automate the calculations.

Chronon corrections to the Dirac equation
View Description Hide DescriptionThe Dirac equation is not semisimple. We therefore regard it as a contraction of a simpler decontracted theory. The decontracted theory is necessarily purely algebraic and nonlocal. In one simple model the algebra is a Clifford algebra with generators. The quantum imaginary is the contraction of a dynamical variable whose backreaction provides the Dirac mass. The simplified Dirac equation is exactly Lorentz invariant but its symmetry group is SO(3, 3), a decontraction of the Poincaré group, and it has a slight but fundamental nonlocality beyond that of the usual Dirac equation. On operational grounds the nonlocality is in size and the associated mass is about the Higgs mass. There is a nonstandard small but unique spin–orbit coupling whose observation would be some evidence for the simpler theory. All the fields of the standard model call for similar nonlocal simplification.

Complex velocity transformations and the Bisognano–Wichmann theorem for quantum fields acting on Krein spaces
View Description Hide DescriptionIt is proven that in indefinite metric quantum field theory there exists a dense set of analytic vectors for the generator of the one parameter group of velocity transformations. This makes it possible to define complex velocity transformations also for the indefinite metric case. In combination with the results of Bros, Epstein, and Moschella [Commun. Math. Phys. 196, 535–570 (1998)], proving Bisognano–Wichmann (BW) analyticity within the linear program, one then obtains a suitable generalization of the BW theorem for local, relativistic quantum fields acting on Krein spaces (“quantum fields with indefinite metric”).

Twistor representation of null twosurfaces
View Description Hide DescriptionWe present a twistor description for null twosurfaces (null strings) in fourdimensional Minkowski space–time. The Lagrangian density for a variational principle is taken as a surfaceforming null bivector. The proposed formulation is reparametrization invariant and free of any algebraic and differential constraints. The spinor formalism of Cartan–Penrose allows us to derive a nonlinear evolution equation for the worldsheet coordinate An example of null twosurface given by the twodimensional selfintersection (caustic) of a null hypersurface is studied.

The peculiarity of a negative coordinate axis in dyonic solutions of noncommutative super Yang–Mills
View Description Hide DescriptionWe show that in the neighborhood of a negative coordinate axis, the U(1) sector of the static dyonic solutions to the noncommutative U(4) super Yang–Mills (SYM) can be consistently decoupled from the SU(4) to all orders in the noncommutativity parameter. We show the above decoupling in two ways. First, we show the noncommutative dyon being the same as the commutative dyon, is a consistent solution to noncommutative equations of motion in the abovementioned region of noncommutative space. Second, as an example of decoupling of a nonnull U(1) sector, we also obtain a family of solutions with nontrivial U(1) components for all components of the gauge field in the same region of noncommutative space. In both cases, the SU(4) and U(1) components separately satisfy the equations of motion.

Becchi–Rouet–Stora–Tyutin quantization of histories electrodynamics
View Description Hide DescriptionThis article is a continuation of earlier work where a classical history theory of pure electrodynamics was developed in which the history fields have five components. The extra component is associated with an extra constraint, thus enlarging the gauge group of histories electrodynamics. In this article we quantize the classical theory developed previously by two methods. First we quantize the reduced classical history space to obtain a reduced quantum history theory. Second we quantize the classical BRSTextended history space, and use the Becchi–Rouet–Stora–Tyutin charge to define a “cohomological” quantum history theory. Finally, we show that the reduced history theory is isomorphic (as a history theory) to the cohomological history theory.

Chiral superconducting strings and Nambu–Goto strings in arbitrary dimensions
View Description Hide DescriptionWe present general solutions to the equations of motion for a superconducting relativistic chiral string that satisfy the unit magnitude constraint in terms of products of rotations. From this result we show how to construct a general family of odd harmonic superconductingchiral loops. We further generalize the product of rotations to an arbitrary number of dimensions.

On threedimensional coupled bosons
View Description Hide DescriptionIn this work we study two complex scalar fields coupled through a quadratic interaction in dimensions using the method of bilinears as suggested by Rajeev [Int. J. Mod. Phys. A 9, 5583 (1994)]. The resulting theory can be formulated as a classical theory. We study the linear approximation, and show that there is a possible bound state in a range of coupling constants.
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 GENERAL RELATIVITY AND GRAVITATION


potential without charge
View Description Hide DescriptionIn order to get geodesically complete Reißner–Nordstrøm space–times, it is necessary to identify pairs of singular points. This can be done in such a way that “wormholes” are created which generate electric field lines without any charge. Finally, it is shown that it is possible to glue this space–time not in the singularities but at some The surface energy generated by this gluing is exotic, but tends to zero in the limit

Inhomogeneous Mtheory cosmologies
View Description Hide DescriptionWe study a class of inhomogeneous and anisotropic string cosmological models. In the case of separable models we show that the governing equations reduce to a system of ordinary differential equations. We focus on a class of separable Mtheory cosmological models, and study their qualitative behavior (a class of models with timereversed dynamics is also possible). We find that generically these inhomogeneous Mtheorycosmologies evolve from a spatially inhomogeneous and negatively curved model with a nontrivial form field toward spatially flat and spatially homogeneous dilatonmodulivacuum solutions with trivial form fields. The late time behavior is the same as that of spatially homogeneous models previously studied. However, the inhomogeneities are not dynamically insignificant at early times in these models.
