5,222 research outputs found

### 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

### 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

### 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

### 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

### Modular symmetry and temperature flow of conductivities in quantum Hall systems with varying Zeeman energy

The behaviour of the critical point between quantum Hall plateaux, as the
Zeeman energy is varied, is analysed using modular symmetry of the Hall
conductivities following from the law of corresponding states. Flow diagrams
for the conductivities as a function of temperature, with the magnetic field
fixed, are constructed for different Zeeman energies, for samples with
particle-hole symmetry.Comment: 15 pages, 13 figure

### 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

### PT-Symmetric Sinusoidal Optical Lattices at the Symmetry-Breaking Threshold

The $PT$ symmetric potential $V_0[\cos(2\pi x/a)+i\lambda\sin(2\pi x/a)]$ has
a completely real spectrum for $\lambda\le 1$, and begins to develop complex
eigenvalues for $\lambda>1$. At the symmetry-breaking threshold $\lambda=1$
some of the eigenvectors become degenerate, giving rise to a Jordan-block
structure for each degenerate eigenvector. In general this is expected to
result in a secular growth in the amplitude of the wave. However, it has been
shown in a recent paper by Longhi, by numerical simulation and by the use of
perturbation theory, that for a broad initial wave packet this growth is
suppressed, and instead a saturation leading to a constant maximum amplitude is
observed. We revisit this problem by explicitly constructing the Bloch
wave-functions and the associated Jordan functions and using the method of
stationary states to find the dependence on the longitudinal distance $z$ for a
variety of different initial wave packets. This allows us to show in detail how
the saturation of the linear growth arises from the close connection between
the contributions of the Jordan functions and those of the neighbouring Bloch
waves.Comment: 15 pages, 7 figures Minor corrections, additional reference

### An Invertible Linearization Map for the Quartic Oscillator

The set of world lines for the non-relativistic quartic oscillator satisfying
Newton's equation of motion for all space and time in 1-1 dimensions with no
constraints other than the "spring" restoring force is shown to be equivalent
(1-1-onto) to the corresponding set for the harmonic oscillator. This is
established via an energy preserving invertible linearization map which
consists of an explicit nonlinear algebraic deformation of coordinates and a
nonlinear deformation of time coordinates involving a quadrature. In the
context stated, the map also explicitly solves Newton's equation for the
quartic oscillator for arbitrary initial data on the real line. This map is
extended to all attractive potentials given by even powers of the space
coordinate. It thus provides classes of new solutions to the initial value
problem for all these potentials

### Time evolution of the QED vacuum in a uniform electric Field: Complete analytic solution by spinorial decomposition

Exact analytical solutions are presented for the time evolution of the
density of pairs produced in the QED vacuum by a time-independent, uniform
electric field. The mathematical tool used here to describe the pair production
is the Dirac-Heisenberg-Wigner function introduced before [Phys. Rev. D 44,
1825 (1991)]. The initial value problem for this function is solved by
decomposing the solution into a product of spinors. The equations for spinors
are much simpler and are solved analytically. These calculations are
nonperturbative since pair production is due to quantum-mechanical tunneling
and the explicit solutions clearly exhibit their nonanalytic behavior.Comment: 6 pages, 1 figur

- âŠ