Volume 47, Issue 3, March 2006
Index of content:
 QUANTUM MECHANICS (GENERAL AND NONRELATIVISTIC)


On Shannon entropies in deformed SegalBargmann analysis
View Description Hide DescriptionWe consider a deformation of the SegalBargmann transform, which is a unitary map from a deformed quantum configuration space onto a deformed quantum phase space (the deformed SegalBargmann space). Both of these Hilbert spaces have canonical orthonormal bases. We obtain explicit formulas for the Shannon entropy of some of the elements of these bases. We also consider two reverse logSobolev inequalities in the deformed SegalBargmann space, which have been proved in a previous work, and show that a certain known coefficient in them is the best possible.

The quantum fidelity for the timeperiodic singular harmonic oscillator
View Description Hide DescriptionIn this paper we perform an exact study of “quantum fidelity” (also called Loschmidt echo) for the timeperiodic quantum harmonic oscillator of the following Hamiltonian: , when compared with the quantum evolution induced by , in the case where is a periodic function and a real constant. The reference (initial) state is taken to be an arbitrary “generalized coherent state” in the sense of Perelomov. We show that, starting with a quadratic decrease in time in the neighborhood of , this quantum fidelity may recur to its initial value 1 at an infinite sequence of times . We discuss the result when the classical motion induced by Hamiltonian is assumed to be stable versus unstable.

Effective Hamiltonians for atoms in very strong magnetic fields
View Description Hide DescriptionWe propose three effective Hamiltonians which approximate atoms in very strong homogeneous magnetic fields modelled by the Pauli Hamiltonian, with fixed total angular momentum with respect to magnetic field axis. All three Hamiltonians describe electrons and a fixed nucleus where the Coulomb interaction has been replaced by dependent onedimensional effective (vector valued) potentials but without magnetic field. Two of them are solvable in at least the one electron case. We briefly sketch how these Hamiltonians can be used to analyze the bottom of the spectrum of such atoms.

Reflection symmetries for multiqubit density operators
View Description Hide DescriptionFor multiqubit density operators in a suitable tensorial basis, we show that a number of nonunitary operations used in the detection and synthesis of entanglement are classifiable as reflection symmetries, i.e., orientation changing rotations. While onequbit reflections correspond to antiunitary symmetries, as is known, for example, from the partial transposition criterion, reflections on the joint density of two or more qubits are not accounted for by the Wigner theorem and are wellposed only for sufficiently mixed states. One example of such nonlocal reflections is the unconditional NOT operation on a multiparty density, i.e., an operation yeilding another density and such that the sum of the two is the identity operator. This nonphysical operation is admissible only for sufficiently mixed states.

Finite quantum kinematics of the harmonic oscillator^{a)}
View Description Hide DescriptionArbitrarily small changes in the commutation relations suffice to transform the usual singular quantum theories into regular quantum theories. This process is an extension of canonical quantization that we call general quantization. Here we apply general quantization to the timeindependent linear harmonic oscillator. The unstable Heisenberg group becomes the stable group SO(3). This freezes out the zeropoint energy of very soft or very hard oscillators, like those responsible for the infrared or ultraviolet divergencies of usual field theories, without much changing the medium oscillators. It produces pronounced violations of equipartition and of the usual uncertainty relations for soft or hard oscillators, and interactions between the previously uncoupled excitation quanta of the oscillator, weakly attractive for medium quanta, strongly repulsive for soft or hard quanta.

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


Exact supersymmetry in the relativistic hydrogen atom in general dimensions—supercharge and the generalized JohnsonLippmann operator
View Description Hide DescriptionA Dirac particle in general dimensions moving in a potential is shown to have an exact supersymmetry, for which the two supercharge operators are obtained in terms of (a dimensional generalization of) the JohnsonLippmann operator, an extension of the RungeLenzPauli vector that relativistically incorporates spin degrees of freedom. So the extra symmetry (S(2)) in the quantum Kepler problem, which determines the degeneracy of the levels, is so robust as to accommodate the relativistic case in arbitrary dimensions.

BoseEinstein condensate and spontaneous breaking of conformal symmetry on Killing horizons II
View Description Hide DescriptionIn the paper cited in the title [J. Math. Phys.46, 062303 (2005)] local scalar QFT (in Weyl algebraic approach) has been constructed on degenerate semiRiemannian manifolds corresponding to the extension of Killing horizons by adding points at infinity to the null geodesic forming the horizon. It has been proved that the theory admits a natural representation of in terms of automorphisms and this representation is unitarily implementable if referring to a certain invariant state . Among other results it has been proved that the theory admits a class of inequivalent algebraic (coherent) states , with , which break part of the symmetry, in the sense that each of them is not invariant under the full group and so there is no unitary representation of whole group which leaves fixed the cyclic GNS vector. These states, if restricted to suitable portions of are invariant and extremal KMS states with respect to a surviving oneparameter group symmetry. In this paper we clarify the nature of symmetry breakdown. We show that, in fact, spontaneous symmetry breaking occurs in the natural sense of algebraic quantum field theory: if , there is no unitary representation of whole group which implements the automorphism representation of itself in the GNS representation of (leaving fixed or not the state).

NonAbelian gauge field theory in scale relativity
View Description Hide DescriptionGauge field theory is developed in the framework of scale relativity. In this theory,spacetime is described as a nondifferentiable continuum, which implies it is fractal, i.e., explicitly dependent on internal scale variables. Owing to the principle of relativity that has been extended to scales, these scale variables can themselves become functions of the spacetime coordinates. Therefore, a coupling is expected between displacements in the fractalspacetime and the transformations of these scale variables. In previous works, an Abelian gauge theory (electromagnetism) has been derived as a consequence of this coupling for global dilations and/or contractions. We consider here more general transformations of the scale variables by taking into account separate dilations for each of them, which yield nonAbelian gauge theories. We identify these transformations with the usual gauge transformations. The gauge fields naturally appear as a new geometric contribution to the total variation of the action involving these scale variables, while the gauge charges emerge as the generators of the scale transformation group. A generalized action is identified with the scalerelativistic invariant. The gauge charges are the conservative quantities, conjugates of the scale variables through the action, which find their origin in the symmetries of the “scalespace.” We thus found in a geometric way and recover the expression for the covariant derivative of gauge theory. Adding the requirement that under the scale transformations the fermion multiplets and the boson fields transform such that the derived Lagrangian remains invariant, we obtain gauge theories as a consequence of scale symmetries issued from a geometric spacetime description.

The generalized Ricci flow for threedimensional manifolds with one Killing vector
View Description Hide DescriptionWe consider threedimensional (3D) flow equations inspired by the renormalization group (RG) equations of string theory with a threedimensional target space. By modifying the flow equations to include a U(1) gauge field, and adding carefully chosen De Turck terms, we are able to extend recent twodimensional results of Bakas to the case of a 3D Riemannian metric with one Killing vector. In particular, we show that the RG flow with De Turck terms can be reduced to two equations: the continual Toda flow solved by Bakas, plus its linearizaton. We find exact solutions which flow to homogeneous but not always isotropic geometries.

 GENERAL RELATIVITY AND GRAVITATION


Casimir effect in a twodimensional signature changing spacetime
View Description Hide DescriptionWe study the Casimir effect for free massless scalar fields propagating on a twodimensional cylinder with a metric that admits a change of signature from Lorentzian to Euclidean. We obtain a nonzero pressure, on the hypersurfaces of signature change, which destabilizes the signature changing region and so alters the energy spectrum of scalar fields. The modified region and spectrum, themselves, back react on the pressure. Moreover, the central term of diffeomorphism algebra of corresponding infinite conserved charges changes correspondingly.

Nonintegrability of density perturbations in the FriedmannRobertsonWalker universe
View Description Hide DescriptionWe investigate the evolution equation of linear density perturbations in the FriedmannRobertsonWalker universe with matter, radiation, and the cosmological constant. The concept of solvability by quadratures is defined and used to prove that there are no “closed form” solutions except for the known Chernin, Heath, Meszaros and simple degenerate ones. The analysis is performed applying Kovacic’s algorithm. The possibility of the existence of other, more general solutions involving special functions is also investigated.

 CLASSICAL MECHANICS AND CLASSICAL FIELDS


Analytical integrability and physical solutions of equation
View Description Hide DescriptionA new idea of electron inertiainduced ion sound wave excitation for transonic plasma equilibrium has already been reported. In such unstable plasma equilibrium, a linear source driven Kortewegde Vries equation describes the nonlinear ion sound wave propagation behavior. By numerical techniques, two distinct classes of solution (soliton and oscillatory shocklike structures) are obtained. Present contribution deals with the systematic methodological efforts to find out its analytical solutions. As a first step, we apply the Painleve method to test whether the derived equation is analytically integrable or not. We find that the derived equation is indeed analytically integrable since it satisfies Painleve property. Hirota’s bilinearization method and the modified sineGordon method (also termed as sinecosine method) are used to derive the analytical results. Perturbative technique is also applied to find out quasistationary solutions. A few graphical plots are provided to offer a glimpse of the structural profiles obtained by different methods applied. It is conjectured that these solutions may open a new scope of acoustic spectroscopy in plasma hydrodynamics.

On Burnett coefficients in periodic media
View Description Hide DescriptionThe aim of this work is to demonstrate a curious property of general periodic structures. It is well known that the corresponding homogenized matrix is positive definite. We calculate here the next order Burnett coefficients associated with such structures. We prove that these coefficients form a tensor which is negative semidefinite. We also provide some examples showing degeneracy in multidimension.

Real irreducible sesquilinearquadratic tensor concomitants of complex bivectors
View Description Hide DescriptionIrreducible tensor concomitants of an arbitrary complex antisymmetric second rank tensor, or bivector, in a Minkowski spacetime are presented. These tensors are quadratic in the complex bivector and invariant under an overall multiplicative phase change of the bivector; in other words, they are sesquilinearquadratic tensor concomitants of the complex bivector. The tensors are real and irreducible under the full real Lorentz group. Particular consideration is given to when the complex bivector is the electromagnetic field strength tensor (complex description is appropriate for radiation), and it is found that some of these irreducible tensors are novel electromagnetic observables not previously mentioned in the literature.

From Stäckel systems to integrable hierarchies of PDE’s: Benenti class of separation relations
View Description Hide DescriptionWe propose a general scheme of constructing of soliton hierarchies from finite dimensional Stäckel systems and related separation relations. In particular, we concentrate on the simplest class of separation relations, called Benenti class, i.e., certain Stäckel systems with quadratic in momenta integrals of motion.

 FLUIDS


Extended KleinGordon action, gravity and nonrelativistic fluid
View Description Hide DescriptionWe consider a scalar field action for which the Lagrangian density is a power of the massless KleinGordon Lagrangian. The coupling of gravity to this matter action is considered. In this case, we show the existence of nontrivial scalar field configurations with vanishing energymomentum tensor on any static, spherically symmetric vacuum solutions of the Einstein equations. These configurations in spite of being coupled to gravity do not affect the curvature of spacetime. The properties of this particular matter action are also analyzed. For a particular value of the exponent, the extended KleinGordon action is shown to exhibit a conformal invariance without requiring the introduction of a nonminimal coupling. We also establish a correspondence between this action and a nonrelativistic isentropic fluid in one fewer dimension. This fluid can be identified with the (generalized) Chaplygin gas for a particular value of the power. It is also shown that the nonrelativistic fluid admits, apart from the Galileo symmetry, an additional symmetry whose action is a rescaling of the time.

 METHODS OF MATHEMATICAL PHYSICS


Sojourn time for rank one perturbations
View Description Hide DescriptionWe consider a selfadjoint, purely absolutely continuous operator . Let be a rank one operator such that for has a simple eigenvalue embedded in its absolutely continuous spectrum, with corresponding eigenvector. Let be a rank one perturbation of the operator , namely, . Under suitable conditions, the operator has no point spectrum in a neighborhood of , for small. Here, we study the evolution of the state under the Hamiltonian , in particular, we obtain explicit estimates for its sojourn time . By perturbation theory, we prove that is finite for , and that for small it is of order . Finally, by using an analytic deformation technique, we estimate the sojourn time for the Friedrichs model in .

Exact and quasiexact solvability of secondorder superintegrable quantum systems: I. Euclidean space preliminaries
View Description Hide DescriptionWe show that secondorder superintegrable systems in twodimensional and threedimensional Euclidean space generate both exactly solvable (ES) and quasiexactly solvable (QES) problems in quantum mechanics via separation of variables, and demonstrate the increased insight into the structure of such problems provided by superintegrability. A principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable. Most past separation of variables treatments of QES problems via partial differential equations have only incorporated separability, not multiseparability. Also, we propose another definition of ES and QES. The quantum mechanical problem is called ES if the solution of Schrödinger equation can be expressed in terms of hypergeometric functions and is QES if the Schrödinger equation admits polynomialsolutions with coefficients necessarily satisfying a threeterm or higher order of recurrence relations. In three dimensions we give an example of a system that is QES in one set of separable coordinates, but is not ES in any other separable coordinates. This example encompasses Ushveridze’s tenthorder polynomial QES problem in one set of separable coordinates and also leads to a fourthorder polynomial QES problem in another separable coordinate set.

Intrinsic spectral geometry of the KerrNewman event horizon
View Description Hide DescriptionWe uniquely and explicitly reconstruct the instantaneous intrinsic metric of the KerrNewman event horizon from the spectrum of its Laplacian. In the process we find that the angular momentum parameter, radius, area; and in the uncharged case, mass, can be written in terms of these eigenvalues. In the uncharged case this immediately leads to the unique and explicit determination of the Kerr metric in terms of the spectrum of the event horizon. Robinson’s “no hair” theorem now yields the corollary: One can “hear the shape” of noncharged stationary axially symmetric black holespacetimes by listening to the vibrational frequencies of its event horizon only.

Flow equations for uplifting halfflat to Spin(7) manifolds
View Description Hide DescriptionIn this supplement to the paper by Franzen et al. [Fortschr. Phys. (to be published)], we discuss the uplift of halfflat sixfolds to Spin(7) eightfolds by fibration of the former over a product of two intervals. We show that the same can be done in two ways—one, such that the required Spin(7) eightfold is a double sevenfold fibration over an interval, the sevenfold itself being the halfflat sixfold fibered over the other interval, and second, by simply considering the fibration of the halfflat sixfold over a product of two intervals. The flow equations one gets are an obvious generalization of the Hitchin’s flow equations [to obtain sevenfolds of holonomy from halfflat sixfolds [Hitchin (2001)]]. We explicitly show the uplift of the Iwasawa using both methods, thereby proposing the form of new Spin(7) metrics. We give a plausibility argument ruling out the uplift of the Iwasawa manifold to a Spin(7) eightfold at the “edge,” using the second method. For Spin(7) eightfolds of the type , being a sevenfold of SU(3) structure, we motivate the possibility of including elliptic functions into the “shape deformation” functions of sevenfolds of SU(3) structure of Franzen et al. via some connections between elliptic functions, the Heisenberg group, theta functions, the already known brane metric [Greene et al., Nucl. Phys. B 337, 1 (1990)], and hyperKähler metrics obtained in twistor spaces by deformations of AtiyahHitchin manifolds by a Legendre transform [Chalmers, Phys. Rev. D 58, 125011 (1998)].
