Volume 46, Issue 12, December 2005
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Constraints on the mixing of bipartite states
View Description Hide DescriptionA mixed state may be represented in many different ways as a mixture of pure states . The mixing problem in quantum mechanics asks the characterization of the probability distribution and the mixed states such that for any given mixed state . Some constraints based on eigenvalues of the mixed states are established in uniparty case [see Nielsen, Phys. Rev. A.63, 052308 (2000), 63, 022144 (2000), Nielsern and Vidal Quantum Inf. Comput.176 (2001)]. We develop some new invariant sets for bipartite mixed states under local unitary operations, which are independent of eigenvalues, and prove some strong constraints based on these invariant sets for the mixing problem in bipartite case. This exhibits a remarkable difference from the uniparty case.

Ladder operators and coherent states for the JaynesCummings model in the rotatingwave approximation
View Description Hide DescriptionUsing algebraic techniques, we realize a systematic search of different types of ladder operators for the JaynesCummings model in the rotatingwave approximation. The link between our results and previous studies on the diagonalization of the associated Hamiltonian is established. Using some of the ladder operators obtained before, examples are given on the possibility of constructing a variety of interesting coherent states for this Hamiltonian.

Quantitative estimates on the enhanced binding for the PauliFierz operator
View Description Hide DescriptionFor a quantum particle interacting with a shortrange potential, we estimate from below the shift of its binding threshold, which is due to the particle interaction with a quantized radiation field.

Analytic Coulomb matrix elements in a threedimensional geometry
View Description Hide DescriptionUsing a complete basis set we have obtained an analytic expression for the matrix elements of the Coulomb interaction. These matrix elements are written in a closed form. We have used the basis set of the threedimensional isotropic quantum harmonic oscillator in order to develop our calculations, which can be useful when treating interactions in localized systems.

A combinatorial approach for studying local operations and classical communication transformations of multipartite states
View Description Hide DescriptionWe develop graph theoretic methods for analyzing maximally entangled pure states distributed between a number of different parties. We introduce a technique called bicolored merging, based on the monotonicity feature of entanglement measures, for determining combinatorial conditions that must be satisfied for any two distinct multiparticle states to be comparable under local operations and classical communication. We present several results based on the possibility or impossibility of comparability of pure multipartite states. We show that there are exponentially many such entangled multipartite states among agents. Further, we discuss a new graph theoretic metric on a class of multipartite states, and its implications.

Generalized coherent states and the control of quantum systems
View Description Hide DescriptionThe control problem for linear and nonlinear Schrödinger equations is considered. The controls are given by applying a spatially homogeneous field or varying the frequency of a quadratic trapping potential. It is demonstrated that the existence of (exact or approximate) coherentstatetype solutions may severely limit the degree to which the system can be controlled.

Decomposition of timecovariant operations on quantum systems with continuous and∕or discrete energy spectrum
View Description Hide DescriptionEvery completely positive map that commutes with the Hamiltonian time evolution is an integral or sum over (densely defined) CPmaps where is the energy that is transferred to or taken from the environment. If the spectrum is nondegenerate each is a dephasing channel followed by an energy shift. The dephasing is given by the Hadamard product of the density operator with a (formally defined) positive operator. The Kraus operator of the energy shift is a partial isometry which defines a translation on with respect to a nontranslationinvariant measure. As an example, this decomposition is explicitly calculated for the rotation invariant Gaussian channel on a single mode. The question of under what conditions a covariant channel destroys superpositions between mutually orthogonal states on the same orbit is addressed. For channels which allow mutually orthogonal output states on the same orbit, a lower bound on the quantum capacity is derived using the Fourier transform of the CPmapvalued measure.

A de Finetti representation for finite symmetric quantum states
View Description Hide DescriptionConsider a symmetricquantum state on an fold product space, that is, the state is invariant under permutations of the subsystems. We show that, conditioned on the outcomes of an informationally complete measurement applied to a number of subsystems, the state in the remaining subsystems is close to having product form. This immediately generalizes the socalled de Finetti representation to the case of finite symmetric quantum states.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


NonAbelian ChernSimons action is topological invariant on 3 simple knot
View Description Hide DescriptionUnder SU(2) gauge transformation, the nonAbelian ChernSimons action is invariant on a class of three dimensional manifold—3 simple knot.

On ChernSimons theory with an inhomogeneous gauge group and BF theory knot invariants
View Description Hide DescriptionWe study the ChernSimons topological quantum field theory with an inhomogeneous gauge group, a nonsemisimple group obtained from a semisimple one by taking its semidirect product with its Lie algebra. We find that the standard knot observable (i.e., trace of the holonomy along the knot) essentially vanishes, and yet, the nonsemisimplicity of the gauge group allows us to consider a class of unorthodox observables which breaks gauge invariance at one point and leads to a nontrivial theory on long knots in . We have two main morals. (1) In the nonsemisimple case there is more to observe in ChernSimons theory. There might be other interesting nonsemisimple gauge groups to study in this context beyond our example. (2) In the case of an inhomogeneous gauge group, we find that ChernSimons theory with the unorthodox observable is actually the same as threedimensional BF theory with the CattaneoCottaRamusinoMartellini knot observable. This leads to a simplification of their results and enables us to generalize and solve a problem they posed regarding the relation between BF theory and the AlexanderConway polynomial. We prove that the most general knot invariant coming from pure BF topological quantum field theory is in the algebra generated by the coefficients of the AlexanderConway polynomial.
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 GENERAL RELATIVITY AND GRAVITATION


Noncommutative unification of general relativity and quantum mechanics
View Description Hide DescriptionWe present a model unifying general relativity and quantum mechanics based on a noncommutative geometry. This geometry is developed in terms of a noncommutative algebra which is defined on a transformation groupoid given by the action of a noncompact group on the total space of a principal fiber bundle over spacetime. The case is important since to obtain physical effects predicted by the model we should assume that is a Lorentz group or some of its representations. We show that the generalized Einstein equation of the model has the form of the eigenvalueequation for the generalized Ricci operator, and all relevant operators in the quantum sector of the model are random operators; we study their dynamics. We also show that the model correctly reproduces general relativity and the usual quantum mechanics. It is interesting that the latter is recovered by performing the measurement of any observable. In the act of such a measurement the model “collapses” to the usual quantum mechanics.

Causal sites as quantum geometry
View Description Hide DescriptionWe propose a structure called a causal site to use as a setting for quantum geometry, replacing the underlying point set. The structure has an interesting categorical form, and a natural “tangent 2bundle,” analogous to the tangent bundle of a smooth manifold. Examples with reasonable finiteness conditions have an intrinsic geometry, which can approximate classical solutions to general relativity. We propose an approach to quantization of causal sites as well.

Universal homogeneous causal sets
View Description Hide DescriptionCausal sets are particular partially ordered sets which have been proposed as a basic model for discrete spacetime in quantum gravity. We show that the class of all countable pastfinite causal sets contains a unique causal set which is universal (i.e., any member of can be embedded into ) and homogeneous (i.e., has maximal degree of symmetry). Moreover, can be constructed both probabilistically and explicitly.
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 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Nonstandard connections in cosympletic field theory
View Description Hide DescriptionIn the jetbundle description of timedependent mechanics there are some elements, such as the Lagrangian energy and the construction of the Hamiltonian formalism, which require the prior choice of a connection. This situation is analyzed by EcheverríaEnríquez et al. [J. Phys. A28, 5553–5567 (1995)]. The aim of this paper is to extend the results in that paper to first order field theory, using the cosymplectic formalism described by de León and coworkers [J. Math. Phys.39, 876–893 (1998); 42, 2092–2104 (2001)]. If the trivial configuration bundle of a Lagrangian system is endowed with one connection, different from the trivial one given by the product structure, we study the consequences on the geometric elements of the theory, the dynamical equations and the variational principles.
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 STATISTICAL PHYSICS


On the sharpness of the zeroentropydensity conjecture
View Description Hide DescriptionThe zeroentropydensity conjecture states that the entropy density defined as vanishes for all translationinvariant pure states on the spin chain. Or equivalently, , the von Neumann entropy of such a state restricted to consecutive spins, is sublinear. In this paper it is proved that this conjecture cannot be sharpened, i.e., translationinvariant states give rise to arbitrary fast sublinear entropy growth. The proof is constructive, and is based on a class of states derived from quasifree states on a CAR algebra. The question whether the entropy growth of pure quasifree states can be arbitrary fast sublinear was first raised by Fannes et al. [J. Math. Phys.44, 6005 (2003)]. In addition to the main theorem it is also shown that the entropy asymptotics of all pure shiftinvariant nontrivial quasifree states is at least logarithmic.

Extensive ground state entropy in supersymmetric lattice models
View Description Hide DescriptionWe present the result of calculations of the Witten index for a supersymmetric lattice model on lattices of various type and size. Because the model remains supersymmetric at finite lattice size, the Witten index can be calculated using rowtorow transfer matrices and the calculations are similar to calculations of the partition function at negative activity . The Witten index provides a lower bound on the number of ground states. We find strong numerical evidence that the Witten index grows exponentially with the number of sites of the lattice, implying that the model has extensive entropy in the ground state.

Nonlinear diffusion equation, Tsallis formalism and exact solutions
View Description Hide DescriptionWe address this work to analyze a nonlinear diffusionequation in the presence of an absorption term taking external forces and spatial timedependent diffusion coefficient into account. The nonlinear terms present in this equation are due to a nonlinear generalization of the Darcy law and the presence of an absorbent (source) term. We obtain new exact solutions and investigate nonlinear effects produced on the solutions by these terms. We also connect the results found here within the Tsallis formalism.
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 METHODS OF MATHEMATICAL PHYSICS


Eigenvalues of Casimir invariants for
View Description Hide DescriptionFor each quantum superalgebra with , an infinite family of Casimir invariants is constructed. This is achieved by using an explicit form for the Lax operator. The eigenvalue of each Casimir invariant on an arbitrary irreducible highest weight module is also calculated.

Multiple nodal bound states for a quasilinear Schrödinger equation
View Description Hide DescriptionNehari techniques are used to prove the existence of multiple (indeed infinitely many) nodal type bound states for the quasilinear Schrödinger equation with prescribed number of nodes.

Invariant noncommutative connections
View Description Hide DescriptionIn this paper we classify invariant noncommutative connections in the framework of the algebra of endomorphisms of a complex vector bundle. It has been proven previously that this noncommutative algebra generalizes in a natural way the ordinary geometry of connections. We use explicitly some geometric constructions usually introduced to classify ordinary invariant connections, and we expand them using algebraic objects coming from the noncommutative setting. The main result is that the classification can be performed using a “reduced” algebra, an associated differential calculus and a module over this algebra.
