Volume 43, Issue 11, November 2002
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


Remarks on the connection between the additive separation of the Hamilton–Jacobi equation and the multiplicative separation of the Schrödinger equation. I. The completeness and Robertson conditions
View Description Hide DescriptionThe fundamental elements of the variable separation theory are revisited, including the Eisenhart and Robertson theorems, Kalnins–Miller theory, and the intrinsic characterization of the separation of the Hamilton–Jacobi equation, in a unitary and geometrical perspective. The general notion of complete integrability of firstorder normal systems of PDEsleads in a natural way to completeness conditions for separated solutions of the Schrödinger equation and to the Robertson condition. Two general types of multiplicative separation for the Schrödinger equation are defined and analyzed: they are called “free” and “reduced” separation, respectively. In the free separation the coordinates are necessarily orthogonal, while the reduced separation may occur in nonorthogonal coordinates, but only in the presence of symmetries (Killing vectors).

Remarks on the connection between the additive separation of the Hamilton–Jacobi equation and the multiplicative separation of the Schrödinger equation. II. First integrals and symmetry operators
View Description Hide DescriptionThe commutation relations of the firstorder and secondorder operators associated with the first integrals in involution of a Hamiltonian separable system are examined. It is shown that these operators commute if and only if a “preRobertson condition” is satisfied. This condition involves the Ricci tensor of the configuration manifold and it is implied by the Robertson condition, which is necessary and sufficient for the separability of the Schrödinger equation.

Efficient simulation of quantum state reduction
View Description Hide DescriptionThe energybased stochastic extension of the Schrödinger equation is a rather special nonlinear stochastic differential equation on Hilbert space, involving a single free parameter, that has been shown to be very useful for modeling the phenomenon of quantum state reduction. Here we construct a general closed form solution to this equation, for any given initial condition, in terms of a random variable representing the terminal value of the energy and an independent Brownian motion. The solution is essentially algebraic in character, involving no integration, and is thus suitable as a basis for efficient simulation studies of state reduction in complex systems.

The Schwinger SU(3) construction. I. Multiplicity problem and relation to induced representations
View Description Hide DescriptionThe Schwinger oscillator operator representation of SU(3) is analyzed with particular reference to the problem of multiplicity of irreducible representations. It is shown that with the use of an unitary representation commuting with the SU(3) representation, the infinity of occurrences of each SU(3) irreducible representation can be handled in complete detail. A natural “generating representation” for SU(3), containing each irreducible representation exactly once, is identified within a subspace of the Schwinger construction, and this is shown to be equivalent to an induced representation of SU(3).

The Schwinger SU(3) construction. II. Relations between Heisenberg–Weyl and SU(3) coherent states
View Description Hide DescriptionThe Schwinger oscillator operator representation of SU(3), studied in a previous paper from the representation theory point of view, is analyzed to discuss the intimate relationships between standard oscillatorcoherent state systems and systems of SU(3) coherent states. Both SU(3) standard coherent states, based on choice of highest weight vector as fiducial vector, and certain other specific systems of generalized coherent states, are found to be relevant. A complete analysis is presented, covering all the oscillatorcoherent states without exception, and amounting to SU(3) harmonic analysis of these states.

New classes of quasisolvable potentials, their exactly solvable limit, and related orthogonal polynomials
View Description Hide DescriptionWe have generated, using an sl(2,R) Liealgebraic formalism, several new classes of quasisolvable elliptic potentials, which in the appropriate limit go over to the exactly solvable forms. We have obtained exact solutions of the corresponding spectral problem for some real values of the potential parameters. We have also given explicit expressions of the families of associated orthogonal polynomials in the energy variable.

Density and current of a dissipative Schrödinger operator
View Description Hide DescriptionWe regard a current flow through an open onedimensional quantum system which is determined by a dissipative Schrödinger operator. The imaginary part of the corresponding form originates from Robin boundary conditions with certain complex valued coefficients imposed on Schrödinger’s equation. This dissipative Schrödinger operator can be regarded as a pseudoHamiltonian of the corresponding open quantum system. The dilation of the dissipative operator provides a (selfadjoint) quasiHamiltonian of the system, more precisely, the Hamiltonian of the minimal closed system which contains the open one is used to define physical quantities such as density and current for the open quantum system. The carrier density turns out to be an expression in the generalized eigenstates of the dilation while the current density is related to the characteristic function of the dissipative operator. Finally a rigorous setup of a dissipative Schrödinger–Poisson system is outlined.

coherent states
View Description Hide DescriptionWe generalize Schwinger boson representation of SU(2) algebra to and define coherent states of using bosonic harmonic oscillator creation and annihilation operators. We give an explicit construction of all Casimirs of in terms of these creation and annihilation operators. The coherent states belonging to any irreducible representations of are labeled by the eigenvalues of the Casimir operators and are characterized by complex orthonormal vectors describing the manifold. The coherent states provide a resolution of identity, satisfy the continuity property, and possess a variety of group theoretic properties.

On bound states for systems of weakly coupled Schrödinger equations in one space dimension
View Description Hide DescriptionWe establish the Birman–Schwinger relation for a class of Schrödinger operators on where H is an auxiliary Hilbert space and is an operatorvalued potential. As an application we give an asymptotic formula for the bound states which may arise for a weakly coupled Schrödinger operator with a matrix potential (having one or more thresholds). In addition, for a twochannel system with eigenvalues embedded in the continuous spectrum we show that, under a small perturbation, such eigenvalues turn into resonances.

Scattering into cones and flux across surfaces in quantum mechanics: A pathwise probabilistic approach
View Description Hide DescriptionWe show how the scatteringintocones and fluxacrosssurfaces theorems in quantum mechanics have very intuitive pathwise probabilistic versions based on some results by Carlen about large time behavior of paths of Nelson’s diffusions. The quantum mechanical results can then be recovered by taking expectations in our pathwise statements.

Coherent state triplets and their inner products
View Description Hide DescriptionIt is shown that if H is a Hilbert space for a representation of a group then there are triplets of spaces in which is a space of coherent state or vector coherent statewave functions and is its dual relative to a conveniently defined measure. It is shown also that there is a sequence of maps which facilitates the construction of the corresponding inner products. After completion if necessary, the spaces become isomorphic Hilbert spaces. It is shown that the inner product for H is often easier to evaluate in than in Thus, we obtain integral expressions for the inner products of coherent state and vector coherent state representations. These expressions are equivalent to the algebraic expressions of Kmatrix theory, but they are frequently more efficient to apply. The construction is illustrated by many examples.

Weak pseudoHermiticity and antilinear commutant
View Description Hide DescriptionWe inquire into some properties of diagonalizable pseudoHermitian operators, showing that their definition can be relaxed and that the pseudoHermiticity property is strictly connected with the existence of an antilinear symmetry. This result is then illustrated by considering the particular case of the complex Morse potential.

Quantum optical versus quantum Brownian motion master equation in terms of covariance and equilibrium properties
View Description Hide DescriptionStructures of quantum Fokker–Planck equations are characterized with respect to the properties of complete positivity, covariance under symmetry transformations and satisfaction of equipartition, referring to recent mathematical work on structures of unbounded generators of covariant quantum dynamical semigroups. In particular the quantum optical master equation and the quantum Brownian motion master equation are shown to be associated to U(1) and R symmetry, respectively. Considering the motion of a Brownian particle, where the expression of the quantum Fokker–Planck equation is not completely fixed by the aforementioned requirements, a recently introduced microphysical kinetic model is briefly recalled, where a quantum generalization of the linear Boltzmann equation in the small energy and momentum transfer limit straightforwardly leads to quantum Brownian motion.

Quantum group covariance and the braided structure of deformed oscillators
View Description Hide DescriptionThe connection between braided Hopf algebrastructure and quantum group covariance of the deformed oscillators is constructed explicitly. In this context we provide deformations of the Hopf algebra of functions on SU(1,1). Quantum subgroups and their representations are also discussed.
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 RELATIVISTIC QUANTUM MECHANICS, FIELD THEORY, BRANE THEORY (INCLUDING STRINGS)


The global existence in the Cauchy problem of the Maxwell–Chern–Simons–Higgs system
View Description Hide DescriptionIn this article we prove the global existence of solution of the classical Maxwell–Chern–Simons–Higgsequations in dimensional Minkowski space–time in the Lorentz gauge. We also prove that the topological solution of the Maxwell–Chern–Simons–Higgs system converges to that of Maxwell–Higgs system as the Chern–Simons constant κ goes to zero, reproducing the classical result by Moncrief [J. Math. Phys. 21, 2291 (1980)] on the global existence of the Maxwell–Klein–Gordon system in dimension.

Scale invariant Euclidean field theory in any dimension
View Description Hide DescriptionWe discuss a dimensional scalar field interacting with a metric field. The metric depends on dimensional coordinates (where We choose formal Gaussian scale invariant correlation functions for the metric field. By a projection to a lower dimensional subspace we obtain a scale invariant nonGaussian model of Euclidean quantum field theory in or dimensions.

The foliation operator in history quantum field theory
View Description Hide DescriptionAs a preliminary to discussing the quantization of the foliation in a history form of general relativity, we show how the discussion in an earlier work [J. Math. Phys. 43, 3053 (2002)] of a history version of free, scalar quantum field theory can be augmented in such a way as to include the quantization of the unitlength, timelike vector that determines a Lorentzian foliation of Minkowski space–time. We employ a Hilbert bundle construction that is motivated by (i) discussing the role of the external Lorentz group in the existing history quantum field theory [J. Math. Phys. 43, 3053 (2002)] and (ii) considering a specific representation of the extended history algebra obtained from the multisymplectic representation of scalar field theory.

Microlocal analysis of quantum fields on curved space–times: Analytic wave front sets and Reeh–Schlieder theorems
View Description Hide DescriptionWe show in this article that the Reeh–Schlieder property holds for states of quantum fields on real analytic curved space–times if they satisfy an analytic microlocal spectrum condition. This result holds in the setting of general quantum field theory, i.e., without assuming the quantum field to obey a specific equation of motion. Moreover, quasifree states of the Klein–Gordon field are further investigated in the present work and the (analytic) microlocal spectrum condition is shown to be equivalent to simpler conditions. We also prove that any quasifree ground or KMS state of the Klein–Gordon field on a stationary real analytic space–time fulfills the analytic microlocal spectrum condition.
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 GENERAL RELATIVITY AND GRAVITATION


Covariant Hamiltonian boundary conditions in General Relativity for spatially bounded space–time regions
View Description Hide DescriptionWe investigate the covariant Hamiltonian symplectic structure of General Relativity for spatially bounded regions of space–time with a fixed timeflow vector. For existence of a welldefined Hamiltonian variational principle taking into account a spatial boundary, it is necessary to modify the standard Arnowitt–Deser–Misner Hamiltonian by adding a boundary term whose form depends on the spatial boundary conditions for the gravitational field. The most general mathematically allowed boundary conditions and corresponding boundary terms are shown to be determined by solving a certain equation obtained from the symplectic current pulled back to the hypersurface boundary of the space–time region. A main result is that we obtain a covariant derivation of Dirichlet, Neumann, and mixed type boundary conditions on the gravitational field at a fixed boundary hypersurface, together with the associated Hamiltonian boundary terms. As well, we establish uniqueness of these boundary conditions under certain assumptions motivated by the form of the symplectic current. Our analysis uses a Noether charge method which extends and unifies several results developed in recent literature for General Relativity. As an illustration of the method, we apply it to the Maxwell field equations to derive allowed boundary conditions and boundary terms for the existence of a welldefined Hamiltonian variational principle for an electromagnetic field in a fixed spatially bounded region of Minkowski space–time.

The maximum dimension of the inheriting algebra in perfect fluid space–times
View Description Hide DescriptionWe determine the maximum dimension of the Lie algebra of inheriting conformal Killing vectors in perfect fluid space–times. For the case of conformally flat space–times the maximum dimension is eight and for the case of nonconformally flat space–times the maximum dimension is found to be five. We illustrate each case with examples.
