Quantum Spacetime,Quantum Geometry and Planck scales

The Principles of Quantum Mechanics and of Classical General Relativity indicate that Spacetime in the small (Planck scale) ought to be described by a noncommutative C* Algebra, implementing spacetime uncertainty relations. A model C* algebra of Quantum Spacetime and its Quantum Geometry is described. Interacting Quantum Field Theory on such a background is discussed, with open problems and recent progress. Applications to cosmology suggest that the Planck scale ought to depend upon dynamics, and possible consequences in the large of the quantum structure in the small are outlined.     

Microscopic Quantum Hydrodynamics of Systems of Fermions: Part I

The fundamental equations of the microscopic quantum hydrodynamics of fermions in an external electromagnetic field (i.e., the particle balance equation, the momentum balance equation, the energy balance equation, and the magnetic moment balance equation) are derived using the Schrödinger equation. The form of the spin–spin interaction Hamiltonian is specified. To close the system of the balance equations for a multiparticle fermion system, the effective one-particle Schrödinger equation must be introduced.     

Jones Invariant,Cantorian Geometry and Quantum Spacetime

The relation between Jones knot polynominals and statistical mechanics is discussed in the light of Cantorian geometry. It is further shown that von Neumanns continuous geometry may be regarded as being a quantum spacetime akin to Cantorian space E ^(∞) and noncommutative geometry.     

The Golden Mean in Quantum Geometry,KnotTheory and Related Topics

The note summarizes some very recent and other not so well known results related tothe Golden Mean, φ, in theoretical physics. The subjects considered are knot theory,Noncommutative Geometry, four manifolds, Cantorian spacetime and quasi crystallography. Theinvolvement of the Golden Mean in these subjects is seen as an indication for an underlayingcoherent unity and unsuspected deep relationship which exists between various disciplines inphysics and which may appear when examined superficially to be unrelated although it is deeplyrelated.     

Nonadiabatic quantum chaos in atom optics

Coherent dynamics of atomic matter waves in a standing-wave laser field is studied. In the dressed-state picture, wave packets of ballistic two-level atoms propagate simultaneously in two optical potentials. The probability to make a transition from one potential to another one is maximal when centroids of wave packets cross the field nodes and is given by a simple formula with the single exponent, the Landau-Zener parameter κ. If κ ? 1, the motion is essentially adiabatic. If κ ? 1, it is (almost) resonant and periodic. If κ ? 1, atom makes nonadiabatic transitions with a splitting of its wave packet at each node and strong complexification of the wave function as compared to the two other cases. This effect is referred as nonadiabatic quantum chaos. Proliferation of wave packets at κ ? 1 is shown to be connected closely with chaotic center-of-mass motion in the semiclassical theory of point-like atoms with positive values of the maximal Lyapunov exponent. The quantum-classical correspondence established is justified by the fact that the Landau-Zener parameter κ specifies the regime of the semiclassical dynamical chaos in the map simulating chaotic center-of-mass motion. Manifestations of nonadiabatic quantum chaos are found in the behavior of the momentum and position probabilities.     

Hydrodynamics of artificial heart valves

The use of a pulsating-type hydrodynamic bench apparatus for studying the functioning of heart-valve prostheses at various values of the volume flow rate is described. A complex of hydrodynamic characteristics is developed and the dependence of these characteristics on the minute volume, flow regime, and the ratio of the linear dimensions of the prosthesis is examined for various types of artificial heart valves. The operation of the heart-value prostheses under different hydrodynamic conditions is analyzed, and the design defects of certain types of prostheses are noted.     

Riemann-Finsler Geometry with Applications to Information Geometry

Information geometry is a new branch in mathematics, originated from the applications of differential geometry to statistics. In this paper we briefly introduce Riemann-Finsler geometry, by which we establish Information Geometry on a much broader base, so that the potential applications of Information Geometry will be beyond statistics.     

Nonadiabatic couplings and chaos in a dynamical potential well model

The mechanism of nonadiabatic couplings between quantum states of a potential well model with finite heights and a dynamical width coordinate is investigated in detail. The system is described in a mixed quantum-classical approach in which the oscillations of the classical width coordinate induce transitions between the quantum states of a particle trapped inside the well. The dynamics of the system is considered in detail for transitions between two quantum states and resulting coupled Bloch-oscillator equations. Poincaré sections showing a mixed phase space with chaotic and regular behaviour are found by a numerical investigation. In particular, chaos results for high energies of the well width oscillations when the mixing between the adiabatic reference states is strong. The inclusion of relaxation is considered and shown that in this case the regimes of chaotic and regular dynamics are not separated as in the relaxation free case. In particular, for some initial conditions chaos can become a transient phenomena placed in a time window between regular oscillations of the system.     

Pseudoquaternion Geometry

We describe complex holomorphic transformations of a quaternion vector space taking left quaternion lines to left quaternion lines and real linear transformations of the quaternion plane simultaneously preserving the sets of left and right quaternion lines.     

Symplectic Geometry

Book Review

Symplectic GeometryA. T. Fomenko: Advanced Studies in Contemporary Mathematics, Volume 5, Gordon and Breach Science Publishers, 1988, 387 pp