Index of content:
Volume 45, Issue 1, January 2004
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


The space of state vectors: A hyperfinite approach
View Description Hide DescriptionWe present a version of the formalism of Dirac based on nonstandard analysis, allowing us to deal with state vectors and operators using the resources of finitedimensional linear algebra. The space of state vectors is a nonstandard Hilbert space with hyperfinite dimension, which includes all squareintegrable functions, together with vectors representing states of definite position or momentum. Every vector is normalizable, even when its norm is infinite. Observables are represented by Hermitian operators, which are always (hyper)bounded and defined on the whole space. The connection with the standard theory is established by postulating the existence of “hyperobservables” and nonstandard states. Each observable in the usual sense appears as a kind of standardscale approximation of some hyperobservable. We show that the probabilistic predictions are consistent with those of the standard theory. Consistency extends to time evolution, in the sense that if an initial nonstandard state is “nearstandard,” then the state after a finite time shall be infinitely near the standard state obtained through the Schrödinger equation.

Dissipative Schrödinger–Poisson systems
View Description Hide DescriptionWe deal with a stationary, dissipative Schrödinger–Poisson system which allows for a current flow through an open, spatially onedimensional quantum system determined by a dissipative Schrödinger operator. This dissipative Schrödinger operator can be regarded as a pseudoHamiltonian of the corresponding open quantum system. The (selfadjoint) dilation of the dissipative operator serves as a quasiHamiltonian of the system which is used to define physical quantities such as density and current for the open quantum system. The thus defined charge density in its dependence on the electrostatic potential is the nonlinear term in Poisson’s equation. We prove that the dissipative Schrödinger–Poisson system always admits a solution and all solutions are included in a ball the radius of which depends only on the data of the problem.

A polymer expansion for the quantum Heisenberg ferromagnet wave function
View Description Hide DescriptionA polymer expansion is given for the quantum Heisenberg ferromagnet wave function. Working on a finite lattice, one is dealing entirely with algebraic identities; there is no question of convergence. The conjecture to be pursued in further work is that effects of large polymers are small. This is relevant to the question of the utility of the expansion and its possible extension to the infinite volume. In themselves the constructions of the present paper are neat and elegant and have surprising simplicity.

On the Pauli operator for the Aharonov–Bohm effect with two solenoids
View Description Hide DescriptionWe consider a spin1/2 charged particle in the plane under the influence of two idealized Aharonov–Bohm fluxes. We show that the Pauli operator as a differential operator is defined by appropriate boundary conditions at the two vortices. Further we explicitly construct a basis in the deficiency subspaces of the symmetric operator obtained by restricting the domain to functions with supports separated from the vortices. This construction makes it possible to apply the Krein’s formula to the Pauli operator.

Nonadiabatic holonomy operators in classical and quantum completely integrable systems
View Description Hide DescriptionGiven a completely integrable system, we associate to any connection on a fiber bundle in invariant tori over a parameter manifold the classical and quantum holonomy operator (generalized Berry’s phase factor), without any adiabatic approximation.

Weak coherent state path integrals
View Description Hide DescriptionWeak coherent states share many properties of the usual coherent states, but do not admit a resolution of unity expressed in terms of a local integral. They arise, e.g., in the case that a group acts on an inadmissible fiducial vector. Motivated by the recent Affine Quantum Gravity Program, the present article studies the path integral representation of the affine weak coherent state matrix elements of the unitary timeevolution operator. Since weak coherent states do not admit a resolution of unity, it is clear that the standard way of constructing a path integral, by time slicing, is predestined to fail. Instead a welldefined path integral with Wiener measure, based on a continuoustime regularization, is used to approach this problem. The dynamics is rigorously established for linear Hamiltonians, and the difficulties presented by more general Hamiltonians are addressed.

Expansions about freefermion models
View Description Hide DescriptionA simple technique for expanding the free energy of general sixvertex models about freefermion points is introduced. This technique is used to verify a Coulomb gas prediction about the behavior of the leading singularity in the free energy of the staggered Fmodel at zero staggered field.

Quantum study of the spin inversion
View Description Hide DescriptionSpin motion is studied by means of the direct use of the Schrödinger equation. The solution is found in terms of Lommel’s polynomials. An expression of the tunneling splitting is obtained, in good agreement with the results coming from other calculations.

Wigner distributions and quantum mechanics on Lie groups: The case of the regular representation
View Description Hide DescriptionWe consider the problem of setting up the Wigner distribution for states of a quantum system whose configuration space is a Lie group. The basic properties of Wigner distributions in the familiar Cartesian case are systematically generalized to accommodate new features which arise when the configuration space changes from dimensional Euclidean space to a Lie group The notion of canonical momentum is carefully analyzed, and the meanings of marginal probability distributions and their recovery from the Wigner distribution are clarified. For the case of compact an explicit definition of the Wigner distribution is proposed, possessing all the required properties. Geodesic curves in which help introduce a notion of the midpoint of two group elements play a central role in the construction.

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


Lattice approximations and continuum limits of quantum fields
View Description Hide DescriptionLattice quantum field models with different lattice cutoffs in the free and interacting parts are constructed and their continuum limits are studied. A comparision with previously constructed continuum limits is given, in the spirit of a discussion on how limit models depend on chosen regularizations.

Effective Lagrangians for scalar fields and finite size effects in field theory
View Description Hide DescriptionWe first discuss the approach of effective field theory in a dimensional Euclidean space. We consider a model with two interacting scalar fields with masses and Assuming we show that there is a decoupling in the effective theory describing the dynamic of the light mass field. Furthermore, we consider the presence of two parallel hyperplanes which break translational symmetry, with a natural cutoff satisfying Then imposing Dirichlet and also Neumann boundary conditions, we study the perturbative renormalization of the effective theory in a region bounded by the two parallel hyperplanes in the oneloop approximation.

Free field dynamics in the generalized AdS (super)space
View Description Hide DescriptionPure gauge representation for general vacuum background fields (Cartan forms) in the generalized AdS superspace identified with is found. This allows us to formulate dynamics of free massless fields in the generalized AdS space–time and to find their (generalized) conformal and higher spin field transformation laws. Generic solution of the field equations is also constructed explicitly. The results are obtained with the aid of the star product realization of ortosymplectic superalgebras.

String theory extensions of Einstein–Maxwell fields: The stationary case
View Description Hide DescriptionWe present a new approach for generating solutions in heterotic string theory compactified down to three dimensions on a torus with where and stand for the number of compactified space–time dimensions and Abelian gauge fields, respectively. It is shown that in the case when and is arbitrary, one can apply a solutiongenerating procedure which consists of mapping seed solutions of the stationary Einstein theory with Maxwell fields to the heterotic string realm by using pure field redefinitions. A novel feature of this method is that it is precisely the electromagnetic sector of the stationary electrovacuum that mainly gives rise to a nontrivial multidimensional metric. This approach leads to classes of solutions which are invariant with respect to the total group of threedimensional charging symmetries of the heterotic string theory, i.e., to all finite transformations which generate charged solutions from neutral ones and preserve the asymptotics of the starting field configurations. As an application of the presented approach we generate a particular extension of the stationary Einstein–multiMaxwell theory obtained on the basis of the Kerr–multiNewman–NUT special class of solutions and establish the conditions under which the resulting multidimensional metric of the heterotic string theory is asymptotically flat.

Holography and SL(2,R) symmetry in 2D Rindler space–time
View Description Hide DescriptionIt is shown that it is possible to define quantum field theory of a massless scalar free field on the Killing horizon of a 2D Rindler space–time. Free quantum field theory on the horizon enjoys diffeomorphism invariance and turns out to be unitarily and algebraically equivalent to the analogous theory of a scalar field propagating inside Rindler space–time, no matter the value of the mass of the field in the bulk. More precisely, there exists a unitary transformation that realizes the bulkboundary correspondence upon an appropriate choice for Fock representation spaces. Second, the found correspondence is a subcase of an analogous algebraic correspondence described by injective * homomorphisms of the abstract algebras of observables generated by abstract quantum freefield operators. These field operators are smeared with suitable test functions in the bulk and exact oneforms on the horizon. In this sense the correspondence is independent from the chosen vacua. It is proven that, under that correspondence, the “hidden” SL(2,R) quantum symmetry found in a previous work gets a clear geometric meaning, it being associated with a group of diffeomorphisms of the horizon itself.

Virasoro algebra with central charge on the horizon of a twodimensionalRindler space–time
View Description Hide DescriptionUsing the holographic machinery built up in a previous work, we show that the hidden symmetry of a scalar quantum field propagating in a Rindler space–time admits an enlargement in terms of a unitary positiveenergy representation of Virasoro algebra defined in the Fock representation. That representation has central charge The Virasoro algebra of operators gets a manifest geometrical meaning if referring to the holographically associated quantum field theory on the horizon: It is nothing but a representation of the algebra of vector fields defined on the horizon equipped with a point at infinity. All that happens provided the Virasoro ground energy vanishes and, in that case, the Rindler Hamiltonian is associated with a certain Virasoro generator. If a suitable regularization procedure is employed, for the ground state of that generator seems to correspond to a thermal state when examined in the Rindler wedge, taking the expectation value with respect to Rindler time. Finally, under Wick rotation in Rindler time, the pair of quantum field theories which are built up on the future and past horizon defines a proper twodimensional conformal quantum field theory on a cylinder.

Searching for a connection between matroid theory and string theory
View Description Hide DescriptionWe make a number of observations about matterghost string phase, which may eventually lead to a formal connection between matroid theory and string theory. In particular, in order to take advantage of the already established connection between matroid theory and Chern–Simons theory, we propose a generalization of string theory in terms of some kind of Kahler metric. We show that this generalization is closely related to the Kahler–Chern–Simons action due to Nair and Schiff. In addition, we discuss matroid/string connection via matroid bundles and a Schild type action, and we add new information about the relationship between matroid theory, supergravity and Chern–Simons formalism.

Monopole–antimonopole solutions of the Skyrmed SU(2) Yang–Mills–Higgs model
View Description Hide DescriptionAxially symmetric monopole–antimonopole dipole solutions to the secondorder equations of a simple SU(2) Yang–Mills–Higgs model featuring a quartic Skyrmelike term are constructed numerically. The effect of varying the Skyrme coupling constant on these solutions is studied in some detail.

 GENERAL RELATIVITY AND GRAVITATION


The Weyl–Lanczos relations and the fourdimensional Lanczos tensor wave equation and some symmetry generators
View Description Hide DescriptionWe examine symmetry generators for exterior differential systems and for systems of partial differential equations and apply the Cartan theory of exterior differential systems to the Weyl–Lanczos equations and to the Lanczos wave equation in four dimensions. We look at a number of examples of symmetries for the Weyl–Lanczos equations in four dimensions and give examples of isovectors when the solutionmanifold is the Schwarzschild, Kasner or Gödel space–time. Solutions of the Weyl–Lanczos system are automatically solutions of the Lanczos wave equation. We give examples of symmetry generators for the Lanczos wave equation and find that they are not automatically symmetry generators for the Weyl–Lanczos equations.

WKB analysis of the Regge–Wheeler equation down in the frequency plane
View Description Hide DescriptionThe Regge–Wheeler equation for blackhole gravitational waves is analyzed for large negative imaginary frequencies, leading to a calculation of the cut strength for waves outgoing to infinity. In the—limited—region of overlap, the results agree well with numerical findings [Leung et al., Class. Quantum Grav. 20, L217 (2003)]. Requiring these waves to be outgoing into the horizon as well subsequently yields an analytic formula for the highly damped Schwarzschild quasinormal modes, including the leading correction. Just as in the WKB quantization of, e.g., the harmonic oscillator, solutions in different regions of space have to be joined through a connection formula, valid near the boundary between them where WKB breaks down. For the oscillator, this boundary is given by the classical turning points; fascinatingly, the connection here involves an expansion around the blackhole singularity

 DYNAMICAL SYSTEMS


Continuous symmetries of Lagrangians and exact solutions of discrete equations
View Description Hide DescriptionOne of the difficulties encountered when studying physical theories in discrete space–time is that of describing the underlying continuous symmetries (like Lorentz, or Galilei invariance). One of the ways of addressing this difficulty is to consider point transformations acting simultaneously on difference equations and lattices. In a previous article we have classified ordinary difference schemes invariant under Lie groups of point transformations. The present article is devoted to an invariant Lagrangian formalism for scalar singlevariable difference schemes. The formalism is used to obtain first integrals and explicit exact solutions of the schemes. Equations invariant under two and threedimensional groups of Lagrangian symmetries are considered
