Volume 41, Issue 4, April 2000
Index of content:
 QUANTUM PHYSICS; PARTICLES AND FIELDS


Darboux transformation for the Schrödinger equation with steplike potentials
View Description Hide DescriptionThe onedimensional Schrödinger equation is considered when the potential is asymptotic to a positive constant on the right half line. The corresponding Darboux transformation is established by showing how the scattering solutions, the scattering coefficients, and the potential change when bound states are added or removed. The scattering coefficients are represented as certain integrals, from which their properties can be directly extracted.

Field theory in
View Description Hide DescriptionWe study field theory on the quantum group which is considered as a physical space. We construct deformed equations and Lagrangians for scalar, Dirac, and gauge fields and study their properties. It seems important for us that nontrivial structure of noncommutative differential calculus leads to an appearance of an additional spin 0 gauge field.

Quantum field theory on the noncommutative plane with symmetry
View Description Hide DescriptionWe study properties of a scalar quantum field theory on the twodimensional noncommutative plane with quantum symmetry. We start from the consideration of a firstly quantized quantum particle on the noncommutative plane. Then we define quantum fields depending on noncommutative coordinates and construct a field theoretical action using the invariant measure on the noncommutative plane. With the help of the partial wave decomposition we show that this quantum field theory can be considered as a second quantization of the particle theory on the noncommutative plane and that this field theory has (contrary to the common belief) even more severe ultraviolet divergences than its counterpart on the usual commutative plane. Finally we introduce the symmetry transformations of physical states on noncommutative spaces and discuss them in detail for the case of the quantum group.

Neutral particles and super Schwinger terms
View Description Hide Descriptiongraded Schwinger terms for neutral particles in 1 and 3 space dimensions are considered.

Simple spin networks as Feynman graphs
View Description Hide DescriptionWe show how spin networks can be described and evaluated as Feynman integrals over an internal space. This description can, in particular, be applied to the socalled simple SO(D) spin networks that are of importance for higherdimensional generalizations of loop quantum gravity. As an illustration of the power of the new formalism, we use it to obtain the asymptotics of an amplitude for the D simplex and show that its oscillatory part is given by the Regge action.

Chiral limit of the twodimensional fermionic determinant in a general magnetic field
View Description Hide DescriptionWe consider the effective action for massive twodimensional QED in flat Euclidean space–time in the background of a general squareintegrable magnetic field with finite range. It is shown that its small mass limit is controlled by the chiral anomaly. New results for the lowenergy scattering of electrons in dimensions in static, inhomogeneous magnetic fields are also presented.

Generic Bell correlation between arbitrary local algebras in quantum field theory
View Description Hide DescriptionWe prove that for any two commuting von Neumann algebras of infinite type, the open set of Bell correlated states for the two algebras is norm dense. We then apply this result to algebraic quantum field theory—where all local algebras are of infinite type—in order to show that for any two spacelike separated regions, there is an open dense set of field states that dictate Bell correlations between the regions. We also show that any vector state cyclic for one of a pair of commuting nonAbelian von Neumann algebras is entangled (i.e., nonseparable) across the algebras—from which it follows that every field state with bounded energy is entangled across any two spacelike separated regions.

Exactly solvable models of sphere interactions in relativistic quantum mechanics
View Description Hide DescriptionWe introduce and develop a systematic theory of sphere interactions formally given by the Hamiltonian with boundary conditions of the second type, as a logical continuation of the work performed [J. Math. Phys. 40, 4255 (1999)]. First, we give the mathematical definition of the model, selfadjointness of the Hamiltonian, the indicial equation, and the useful scattering elements. Next, we extend the model by adding a Coulomb potential and provide useful mathematical definitions and corresponding stationary scattering elements.

Relativistic scattering theory for finitely many sphere interactions supported by concentric spheres
View Description Hide DescriptionFor the model formally expressed as we give a precise mathematical definition, provide basic properties and main scattering elements.

The Reeh–Schlieder property for thermal field theories
View Description Hide DescriptionWe show that the Reeh–Schlieder property w.r.t. KMS states is a direct consequence of locality, additivity, and the relativistic KMS condition. The latter characterizes the thermal equilibrium states of a relativistic quantum field theory. The statement remains valid even if the given equilibrium state breaks spatial translation invariance.

Relating Green’s functions in axial and Lorentz gauges using finite fielddependent BRS transformations
View Description Hide DescriptionWe use finite fielddependent BRS transformations (FFBRS) to connect the Green functions in a set of two otherwise unrelated gauge choices. We choose the Lorentz and the axial gauges as examples. We show how the Green functions in axial gauge can be written as a series in terms of those in Lorentz gauges. Our method also applies to operator Green’s functions. We show that this process involves another set of related FFBRS transformations that is derivable from infinitesimal FFBRS. We suggest possible applications.

NonKolmogorov probability models and modified Bell’s inequality
View Description Hide DescriptionWe analyze the proof of Bell’s inequality and demonstrate that this inequality is related to one particular model of probability theory, namely Kolmogorov measuretheoretical axiomatics from 1933. We found a (numerical) statistical correction to Bell’s inequality. Such an additional term on the righthand side of Bell’s inequality can be considered as a probability invariant of a quantum state φ. This is a measure of nonreproducibility of hidden variables in different runs of experiments. Experiments to verify Bell’s inequality can be considered as just experiments to estimate the constant It seems that Bell’s inequality could not be used as a crucial reason to deny local realism. We consider deterministic as well as stochastic hidden variables models.

The onedimensional spinless relativistic Coulomb problem
View Description Hide DescriptionMotivated by a recent analysis that presents explicitly the general solution, we consider the eigenvalue problem of the spinless Salpeter equation with a (“hardcore amended”) Coulomb interaction potential in one dimension. We prove the existence of a critical coupling constant (which contradicts the assertions of the previous analysis) and give analytic upper bounds on the energy eigenvalues. These upper bounds seem to disprove the previous explicit solution.

Becchi–Rouet–Stora–Tyutin invariant formulation of spontaneously broken gauge theory in a generalized differential geometry
View Description Hide DescriptionNoncommutative geometry (NCG) on a discrete space successfully reproduces the Higgs mechanism of the spontaneously broken gauge theory, in which the Higgs boson field is regarded as a kind of gauge field on the discrete space. People also know how to construct the generalized differential geometry (a GDG) as one version of NCG on a discrete space where denotes the N points discrete space. A GDG is a direct generalization of the differential geometry on the ordinary manifold into the discrete one. In this paper, we attempt to construct the Becchi–Rouet–Stora–Tyutin (BSRT) invariant formulation of spontaneously broken gauge theory based on a GDG and obtain the BSRT invariant Lagrangian with the ’t Hooft–Feynman gauge fixing term.

Wave packet revivals and quasirevivals in onedimensional power law potentials
View Description Hide DescriptionThe harmonic oscillator and the infinite square well are two of the simplest onedimensional quantum systems which exhibit wave packet revivals (trivially so in the case of the oscillator.) These two potentials can be thought of as special cases of the general onedimensional powerlaw potential given by (with and for the oscillator and square well, respectively.) Using an autocorrelation function approach and the WKB approximation for the quantized energy levels in such potentials, we exhibit numerical evidence for wave packet revivals in the case of arbitrary which agree well with more analytic results. We derive expressions for the revival and collapse time scales in terms of the physical quantities of the system as well as the wave packet parameters. We find that both times scale with the powerlaw exponent k as In this way, we can explicitly exhibit the approach to the two familiar limiting cases. We also briefly consider the case of a “half” well where an infinite wall in added at the origin.

Reduction of quantum systems on Riemannian manifolds with symmetry and application to molecular mechanics
View Description Hide DescriptionThis paper deals with a general method for the reduction of quantum systems with symmetry. For a Riemannian manifoldM admitting a compact Lie groupG as an isometry group, the quotient space is not a smooth manifold in general but stratified into a collection of smooth manifolds of various dimensions. If the action of the compact group G is free, M is made into a principal fiber bundle with structure group G. In this case, reduced quantum systems are set up as quantum systems on the associated vector bundles over This idea of reduction fails, if the action of G on M is not free. However, the Peter–Weyl theorem works well for reducing quantum systems on M. When applied to the space of wave functions on M, the Peter–Weyl theorem provides the decomposition of the space of wave functions into spaces of equivariant functions on M, which are interpreted as Hilbert spaces for reduced quantum systems on Q. The concept of connection on a principal fiber bundle is generalized to be defined well on the stratified manifoldM. Then the reduced Laplacian is well defined as a selfadjoint operator with the boundary conditions on singular sets of lower dimensions. Application to quantum molecular mechanics is also discussed in detail. In fact, the reduction of quantum systems studied in this paper stems from molecular mechanics. If one wishes to consider the molecule which is allowed to lie in a line when it is in motion, the reduction method presented in this paper works well.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Exceptional Lagrangians for spin2 field: Standard variables
View Description Hide DescriptionExceptional Lagrangians for spin2 field in terms of standard second order tensor may be obtained using the method proposed some years ago by Lax. We found several nonlinear theories which can be used to describe the gravitational field. Among them, we discovered a particularly interesting one which has (in a strike analogy with Born model for spin one) an upper field value.

Precession of a freely rotating rigid body. Inelastic relaxation in the vicinity of poles
View Description Hide DescriptionWhen a solid body is freely rotating at an angular velocity Ω, the ellipsoid of constant angular momentum, in the space has poles corresponding to spinning about the minimalinertia and maximalinertia axes. The first pole may be considered stable if we neglect the inner dissipation, but becomes unstable if the dissipation is taken into account. This happens because the bodies dissipate energy when they rotate about any axis different from principal. In the case of an oblate symmetrical body, the angular velocity describes a circular cone about the vector of (conserved)angular momentum. In the course of relaxation, the angle of this cone decreases, so that both the angular velocity and the maximalinertia axis of the body align along the angular momentum. The generic case of an asymmetric body is far more involved. Even the symmetrical prolate body exhibits a sophisticated behavior, because an infinitesimally small deviation of the body’s shape from a rotational symmetry (i.e., a small difference between the largest and second largest moments of inertia) yields libration: the precession trajectory is not a circle but an ellipse. In this article we show that often the most effective internal dissipation takes place at twice the frequency of the body’s precession. Applications to precessing asteroids, cosmicdust alignment, and rotating satellites are discussed.

Dual Lagrangian field theories
View Description Hide DescriptionWe investigate how, under suitable regularity conditions, firstorder Lagrangian field theories can be recasted in terms of a secondorder Lagrangian, called the dual Lagrangian of the theory, depending on canonical conjugate momenta together with their derivatives. The necessary and sufficient conditions which allow such a (local) reformulation, obtained through a suitable generalization of the Legendre transformation, are analyzed. The global geometric framework is also investigated in detail. As an example, we apply the dual Lagrangian formulation to the Hilbert Lagrangian and to Euclidean selfdual gravity.

A new look at the Schouten–Nijenhuis, Frölicher–Nijenhuis, and Nijenhuis–Richardson brackets
View Description Hide DescriptionIn this paper we reexpress the Schouten–Nijenhuis, the Frölicher–Nijenhuis, and the Nijenhuis–Richardson brackets on a symplectic space using the extended Poisson brackets structure present in the pathintegral formulation of classical mechanics.
