Volume 48, Issue 1, January 2007
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Strongcoupling limit for the ground state of a particle harmonic oscillator interaction
View Description Hide DescriptionWe consider the quantum Hamiltonian for a particle interacting with a harmonic oscillator,, for various choices of coupling potential , where and are creation and annihilation operators, and is a parameter to be thought of as large. This operator is a caricature of the polaron Hamiltonian where the quantum field is approximated by a single mode. The large corresponds to large coupling between the electron and the field. Let be the infimum of the spectrum of , and let where the infimum is taken over product states for the electron and oscillator, the oscillator function taken as a coherent state. It is a remarkable fact that is a “good” approximation of . More specifically, for all , as observed by Lieb. In this paper we examine this gap for various choices of , showing cases where it closes and cases where it does not.

Canonical coset parametrization and the Bures metric of the threelevel quantum systems
View Description Hide DescriptionAn explicit parametrization for the state space of an level density matrix is given. The parametrization is based on the canonical coset decomposition of unitary matrices. We also compute, explicitly, the Bures metric tensor over the state space of two and threelevel quantum systems.

Extended weak coupling limit for Friedrichs Hamiltonians
View Description Hide DescriptionWe study a class of selfadjoint operators defined on the direct sum of two Hilbert spaces: a finite dimensional one called sometimes a “small subsystem” and an infinite dimensional one called a “reservoir.” The operator, which we call a “Friedrichs Hamiltonian,” has a small coupling constant in front of its offdiagonal term. It is well known that under some conditions in the weak coupling limit the appropriately rescaled evolution in the interaction picture converges to a contractive semigroup when restricted to the subsystem. We show that in this model, the properly renormalized and rescaled evolution converges on the whole space to a new unitary evolution, which is a dilation of the above mentioned semigroup. Similar results have been studied before (Accardi et al., 1990) in more complicated models under the name of “stochastic limit.”

On Weyl channels being covariant with respect to the maximum commutative group of unitaries
View Description Hide DescriptionWe investigate the Weyl channels being covariant with respect to the maximum commutative group of unitary operators. This class includes the quantum depolarizing channel and the “twoPauli” channel as well. Then, we show that our estimation of the output entropy for a tensor product of the phase damping channel and the identity channel based upon the decreasing property of the relative entropy allows to prove the additivity conjecture for the minimal output entropy for the quantum depolarizing channel in any prime dimension and for the twoPauli channel in the qubit case.

Quasiseparation of variables in the Schrödinger equation with a magnetic field
View Description Hide DescriptionWe consider a twodimensional integrable Hamiltonian system with a vector and scalar potential in quantum mechanics. Contrary to the case of a pure scalar potential, the existence of a second order integral of motion does not guarantee the separation of variables in the Schrödinger equation. We introduce the concept of “quasiseparation of variables” and show that in many cases it allows us to reduce the calculation of the energy spectrum and wave functions to linear algebra.

Quantum dynamical semigroups for finite and infinite Bose systems
View Description Hide DescriptionA new class of quasifree quantum Markov semigroups on algebras of canonical commutation relations is introduced and discussed. Two applications to decoherence in the Heisenberg representation are given. In the first one the dynamical semigroup which leads to the appearance of decoherence induced superselection rules corresponding to the boundary conditions of a quantum particle in a finite interval is considered. The second example analyzes the possibility of the transition from infinite systems to systems with a finite number of degrees of freedom.

Complementary reductions for two qubits
View Description Hide DescriptionReduction of a state of a quantum system to a subsystem gives partial quantum information about the true state of the total system. In connection with optimal state determination for two qubits, the question was raised about the maximum number of pairwise complementary reductions. The main result of the paper tells that the maximum number is 4, that is, if are pairwise complementary (or quasiorthogonal) subalgebras of the algebra of all matrices and they are isomorphic to , then . The proof is based on a Cartan decomposition of SU(4). In the way to the main result, contributions are made to the understanding of the structure of complementary reductions.

Dual monogamy inequality for entanglement
View Description Hide DescriptionWe establish duality for monogamy of entanglement: whereas monogamy of entanglementinequalities provide an upper bound for bipartite sharability of entanglement in a multipartite system, as quantified by linear entropy, we prove that the same quantity (namely, linear entropy) provides a lower bound for distribution of bipartite entanglement in a multipartite system. Our theorem for monogamy of entanglement is used to establish relations between bipartite entanglement that separate one qubit from the rest versus separating two qubits from the rest.

Features of Moyal trajectories
View Description Hide DescriptionWe study the Moyal evolution of the canonical position and momentum variables. We compare it with the classical evolution and show that, contrary to what is commonly found in the literature, the two dynamics do not coincide. We prove that this divergence is quite general by studying Hamiltonians of the form . Several alternative formulations of Moyal dynamics are then suggested. We introduce the concept of star function and use it to reformulate the Moyal equations in terms of a system of ordinary differential equations on the noncommutative Moyal plane. We then use this formulation to study the semiclassical expansion of Moyal trajectories, which is cast in terms of a (order by order in ) recursive hierarchy of (i) first order partial differential equations as well as (ii) systems of first order ordinary differential equations. The latter formulation is derived independently for analytic Hamiltonians as well as for the more general case of locally integrable ones. We present various examples illustrating these results.

Degenerate discrete energy spectra and associated coherent states
View Description Hide DescriptionGeneralized and Gaussian coherent states constructed for quantum system with degeneracies in the energy spectrum are compared with respect to some minimal definitions and fundamental properties they have to satisfy. The generalized coherent states must be eigenstates of a certain annihilation operator that has to be properly defined in the presence of degeneracies. The Gaussian coherent states are, in the particular harmonic oscillator case, an approximation of the generalized coherent states and so the localizability in phase space of the particle in those states is very good. For other quantum systems, this last property serves as a definition of those Gaussian coherent states. The example of a particle in a twodimensional square box is thus revisited having in mind the preceding definitions of generalized and Gaussian coherent states and also the preservation of the important property known as the resolution of the identity operator.

Finite size effects in bistable models
View Description Hide DescriptionThe work proposes a finite temperaturetheory of the quantum tunneling in a bistable quartic potential. In semiclassical approximation, the imaginary time path integral method identifies the classical background which interpolates between the potential minima and, at any , consistently fulfills antiperiodic boundary conditions. Solving the boundary problem I find that the change between the low quantum regime and the high activated regime exhibits the signatures of a first order phase transition. This is confirmed by the discontinuity in the first temperature derivative of the instantonic action. The quantum fluctuation contribution around the (anti)instantons is evaluated by the functional determinant method. The computation of the tunneling energy shows (i) a remarkable reduction at low with respect to the predictions of the infinite size canonical instantonic approach, and (ii) a steplike increase at the transition point.
 Top

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


Hidden geometric character of relativistic quantum mechanics
View Description Hide DescriptionGeometry can be an unsuspected source of equations with physical relevance, as everybody is aware since Einstein formulated the general theory of relativity. However, efforts to extend a similar type of reasoning to other areas of physics, namely, electrodynamics, quantum mechanics, and particle physics, usually had very limited success; particularly in quantum mechanics the standard formalism is such that any possible relation to geometry is impossible to detect; other authors have previously trod the geometric path to quantum mechanics, some of that work being referred to in the text. In this presentation we will follow an alternate route to show that quantum mechanics has indeed a strong geometric character. The paper makes use of geometricalgebra, also known as Clifford algebra, in fivedimensional spacetime. The choice of this space is given the character of first principle, justified solely by the consequences that can be derived from such choice and their consistency with experimental results. Given a metric space of any dimension, one can define monogenic functions, the natural extension of analytic functions to higher dimensions; such functions have null vector derivative and have previously been shown by other authors to play a decisive role in lower dimensional spaces. All monogenic functions have null Laplacian by consequence; in a hyperbolic space this fact leads inevitably to a wave equation with planelike solutions. This is also true for fivedimensional spacetime and we will explore those solutions, establishing a parallel with the solutions of the free particle Dirac equation. For this purpose we will invoke the isomorphism between the complex algebra of matrices, also known as Dirac’s matrices. There is one problem with this isomorphism, because the solutions to Dirac’s equation are usually known as spinors (column matrices) that do not belong to the matrix algebra and as such are excluded from the isomorphism. We will show that a solution in terms of Dirac spinors is equivalent to a plane wave solution. Just as one finds in the standard formulation, monogenic functions can be naturally split into positive∕negative energy together with left∕right ones. This split is provided by geometric projectors and we will show that there is a second set of projectors providing an alternate fourfold split. The possible implications of this alternate split are not yet fully understood and are presently the subject of profound research.

Large behavior of two dimensional supersymmetric YangMills quantum mechanics
View Description Hide DescriptionWe analyze the limit of supersymmetric YangMillsquantum mechanics (SYMQM) in two space time dimensions. To do so we introduce a particular class of invariant polynomials and give the solutions of twodimensional SYMQM in terms of them. We conclude that in this limit the system is not fully described by the single trace operators since there are other, bilinear operators that play a crucial role when the Hamiltonian is free.

Lorentzian version of the noncommutative geometry of the standard model of particle physics
View Description Hide DescriptionA formulation of the noncommutative geometry for the standard model of particle physics with a Lorentzian signature metric is presented. The elimination of the fermion doubling in the Lorentzian case is achieved by a modification of Connes’ internal space geometry [“Gravity coupled with matter and the foundation of noncommutative geometry,” Commun. Math. Phys.182, 155–176 (1996)] so that it has signature 6 (mod 8) rather than 0. The fermionic part of the ConnesChamseddine spectral action can be formulated, and it is shown that it allows an extension with righthanded neutrinos and the correct mass terms for the seesaw mechanism of neutrino mass generation.

Note on symmetries of the KnizhnikZamolodchikov equation
View Description Hide DescriptionWe continue the study of hidden symmetries of the fourpoint KnizhnikZamolodchikov equation initiated by Giribet [Phys. Lett. B628, 148 (2005)]. Here, we focus our attention on the fourpoint correlation function in those cases where one spectral flowed state of the sector is involved. We give a formula that shows how this observable can be expressed in terms of the fourpoint function of non spectral flowed states. This means that the formula holding for the winding violating fourstring scattering processes in has a simple expression in terms of the one for the conservative case, generalizing what is known for the case of threepoint functions, where the violating and the nonviolating structure constants turn out to be connected one to each other in a similar way. What makes this connection particularly simple is the fact that, unlike what one would naively expect, it is not necessary to explicitly solve the fivepoint function containing a single spectral flow operator to this end. Instead, nondiagonal functional relations between different solutions of the KnizhnikZamolodchikov equation turn out to be the key point for this short path to exist. Considering such functional relation is necessary but it is not sufficient; besides, the formula also follows from the relation existing between correlators in both WessZuminoNovikovWitten (WZNW) and Liouville conformal theories.
 Top

 GENERAL RELATIVITY AND GRAVITATION


Multipole structure of current vectors in curved spacetime
View Description Hide DescriptionA method is presented which allows the exact construction of conserved (i.e., divergencefree) current vectors from appropriate sets of multipole moments. Physically, such objects may be taken to represent the flux of particles or electric charge inside some classical extended body. Several applications are discussed. In particular, it is shown how to easily write down the class of all smooth and spatially bounded currents with a given total charge. This implicitly provides restrictions on the moments arising from the smoothness of physically reasonable vector fields. We also show that requiring all of the moments to be constant in an appropriate sense is often impossible. This likely limits the applicability of the EhlersRudolphDixon notion of quasirigid motion. A simple condition is also derived that allows currents to exist in two different spacetimes with identical sets of multipole moments (in a natural sense).
 Top

 DYNAMICAL SYSTEMS


Incursive discretization, system bifurcation, and energy conservation
View Description Hide DescriptionIncursive discretization of the classical harmonic oscillator leads to system bifurcation. The resulting hyperincursive representation has two alternative distinct algorithms of ordered, serial, noncommuting instructions, and admits solutions having a discretized classical total energy that is perfectly conserved and phase space trajectories that are fully stable at all time scales. Hyperincursive representations can be generated for any Hamiltonian system.

Geometric integration of the electromagnetic twobody problem
View Description Hide DescriptionThe equations of motion of the twobody problem of Dirac’s electrodynamics of point charges consist of a delay equation for the proton and a delay equation for the electron. These equations involve the third derivative of the charges’s position and have runaway solutions, which make forward numerical integration troublesome. Dirac’s equations of motion are algebraicdelay equations, involving a degenerate linear form of the past accelerations. A Fredholm alternative yields a system of secondorder delay equations of motion plus a constraint on the initial segment of trajectory. Here we use the Fredholm constraint as a geometric tool to derive covariant secondorder equations of motion in position for backward time integration. We also extend the backward integration scheme to include a generalized version of Dirac’s theory that includes two delays.

Dynamical behavior for the threedimensional generalized HasegawaMima equations
View Description Hide DescriptionThe long time behavior of solution of the threedimensional generalized HasegawaMima [Phys. Fluids21, 87 (1978)]equations with dissipation term is considered. The global attractor problem of the threedimensional generalized HasegawaMima equations with periodic boundary condition was studied. Applying the method of uniform a priori estimates, the existence of global attractor of this problem was proven, and also the dimensions of the global attractor are estimated.
 Top

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Massless particles in threedimensional Lorentzian warped products
View Description Hide DescriptionThe model of a massless relativistic particle with curvaturedependent Lagrangian is well known in dimensional Minkowski space. For other gravitational fields less rigid than those with constant (zero) curvature only a few results are known. In this paper, we give a geometric approach in order to solve the field equations associated with that Lagrangian in the setting of an interesting threedimensional background, namely, a threedimensional warped product with Lorentzian fibers. When some rigidity conditions are imposed to the fiber (constant Gauss curvature), the trajectories can be totally described. Several examples help us clarify this.
