A New Algebraic Structure of Finite Quantum Systems and the Modified Bessel Functions

In this paper we present a new algebraic structure (a super hyperbolic system in our terminology) for finite quantum systems, which is a generalization of the usual one in the two-level system. It fits into the so-called generalized Pauli matrices, so they play an important role in the theory. Some deep relation to the modified Bessel functions of integer order is pointed out. By taking a skillful limit finite quantum systems become quantum mechanics on the circle developed by Ohnuki and Kitakado.Comment: Latex ; 14 pages ; no figure ; minor changes. To appear in International Journal of Geometric Methods in Modern Physics, (Vo.4, No.7), 200

Flow Equations for Uplifting Half-Flat to Spin(7) Manifolds

In this short supplement to [1], we discuss the uplift of half-flat six-folds to Spin(7) eight-folds 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) eight-fold is a double G_2 seven-fold fibration over an interval, the G_2 seven-fold itself being the half-flat six-fold fibered over the other interval, and second, by simply considering the fibration of the half-flat six-fold over a product of two intervals. The flow equations one gets are an obvious generalization of the Hitchin's flow equations (to obtain seven-folds of G_2 holonomy from half-flat six-folds [2]). We explicitly show the uplift of the Iwasawa using both methods, thereby proposing the form of the new Spin(7) metrics. We give a plausibility argument ruling out the uplift of the Iwasawa manifold to a Spin(7) eight fold at the "edge", using the second method. For $Spin(7)$ eight-folds of the type $X_7\times S^1$, $X_7$ being a seven-fold of SU(3) structure, we motivate the possibility of including elliptic functions into the "shape deformation" functions of seven-folds of SU(3) structure of [1] via some connections between elliptic functions, the Heisenberg group, theta functions, the already known $D7$-brane metric [3] and hyper-K\"{a}hler metrics obtained in twistor spaces by deformations of Atiyah-Hitchin manifolds by a Legendre transform in [4].Comment: 12 pages, LaTeX; v3: (JMP) journal version which includes clarifying remarks related to connection between Spin(7)-folds and SU(3)structur

Gravity-induced resonances in a rotating trap

It is shown that in an anisotropic harmonic trap that rotates with the properly chosen rotation rate, the force of gravity leads to a resonant behavior. Full analysis of the dynamics in an anisotropic, rotating trap in 3D is presented and several regions of stability are identified. On resonance, the oscillation amplitude of a single particle, or of the center of mass of a many-particle system (for example, BEC), grows linearly with time and all particles are expelled from the trap. The resonances can only occur when the rotation axis is tilted away from the vertical position. The positions of the resonances (there are always two of them) do not depend on the mass but only on the characteristic frequencies of the trap and on the direction of the angular velocity of rotation.Comment: 10 pages, 12 figures, to appear in Physical Review

Rigid motions: action-angles, relative cohomology and polynomials with roots on the unit circle

Revisiting canonical integration of the classical solid near a uniform rotation, canonical action angle coordinates, hyperbolic and elliptic, are constructed in terms of various power series with coefficients which are polynomials in a variable $r^2$ depending on the inertia moments. Normal forms are derived via the analysis of a relative cohomology problem and shown to be obtainable without the use of ellitptic integrals (unlike the derivation of the action-angles). Results and conjectures also emerge about the properties of the above polynomials and the location of their roots. In particular a class of polynomials with all roots on the unit circle arises.Comment: 26 pages, 1 figur

Static, massive fields and vacuum polarization potential in Rindler space

In Rindler space, we determine in terms of special functions the expression of the static, massive scalar or vector field generated by a point source. We find also an explicit integral expression of the induced electrostatic potential resulting from the vacuum polarization due to an electric charge at rest in the Rindler coordinates. For a weak acceleration, we give then an approximate expression in the Fermi coordinates associated with the uniformly accelerated observer.Comment: 11 pages, latex, no figure

The AdS_5xS^5 superstring worldsheet S-matrix and crossing symmetry

An S-matrix satisying the Yang-Baxter equation with symmetries relevant to the AdS_5xS^5 superstring has recently been determined up to an unknown scalar factor. Such scalar factors are typically fixed using crossing relations, however due to the lack of conventional relativistic invariance, in this case its determination remained an open problem. In this paper we propose an algebraic way to implement crossing relations for the AdS_5xS^5 superstring worldsheet S-matrix. We base our construction on a Hopf-algebraic formulation of crossing in terms of the antipode and introduce generalized rapidities living on the universal cover of the parameter space which is constructed through an auxillary, coupling constant dependent, elliptic curve. We determine the crossing transformation and write functional equations for the scalar factor of the S-matrix in the generalized rapidity plane.Comment: 27 pages, no figures; v2: sign typo fixed in (24), everything else unchange

Static and Dynamic Properties of Trapped Fermionic Tonks-Girardeau Gases

We investigate some exact static and dynamic properties of one-dimensional fermionic Tonks-Girardeau gases in tight de Broglie waveguides with attractive p-wave interactions induced by a Feshbach resonance. A closed form solution for the one-body density matrix for harmonic trapping is analyzed in terms of its natural orbitals, with the surprising result that for odd, but not for even, numbers of fermions the maximally occupied natural orbital coincides with the ground harmonic oscillator orbital and has the maximally allowed fermionic occupancy of unity. The exact dynamics of the trapped gas following turnoff of the p-wave interactions are explored.Comment: 4 pages, 2 figures, submitted to PR

Integrable Hamiltonian systems with vector potentials

We investigate integrable 2-dimensional Hamiltonian systems with scalar and vector potentials, admitting second invariants which are linear or quadratic in the momenta. In the case of a linear second invariant, we provide some examples of weakly-integrable systems. In the case of a quadratic second invariant, we recover the classical strongly-integrable systems in Cartesian and polar coordinates and provide some new examples of integrable systems in parabolic and elliptical coordinates.Comment: 23 pages, Submitted to Journal of Mathematical Physic

Vacuum polarization induced by a uniformly accelerated charge

We consider a point charge fixed in the Rindler coordinates which describe a uniformly accelerated frame. We determine an integral expression of the induced charge density due to the vacuum polarization at the first order in the fine structure constant. In the case where the acceleration is weak, we give explicitly the induced electrostatic potential.Comment: 13 pages, latex, no figures, to appear in Int. J. Theor. Phys

Nonexistence of an integral of the 6th degree in momenta for the Zipoy-Voorhees metric

We prove nonexistence of a nontrivial integral that is polynomial in momenta of degree less than 7 for the Zipoy-Voorhees spacetime with the parameter $\delta=2$Comment: 7 pages, no figure