Noncommutative Dynamics of Random Operators

We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra ${\cal A}$ on a transformation groupoid $\Gamma = E \times G$ where $E$ is the total space of a principal fibre bundle over spacetime, and $G$ a suitable group acting on $\Gamma$. We show that every $a \in {\cal A}$ defines a random operator, and we study the dynamics of such operators. In the noncommutative regime, there is no usual time but, on the strength of the Tomita-Takesaki theorem, there exists a one-parameter group of automorphisms of the algebra ${\cal A}$ which can be used to define a state dependent dynamics; i.e., the pair $({\cal A}, \phi)$, where $\phi$ is a state on ${\cal A}$, is a ``dynamic object''. Only if certain additional conditions are satisfied, the Connes-Nikodym-Radon theorem can be applied and the dependence on $\phi$ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair $({\cal A}, \phi)$ defines the so-called free probability calculus, as developed by Voiculescu and others, with the state $\phi$ playing the role of the noncommutative probability measure. This shows that in the noncommutative regime dynamics and probability are unified. This also explains probabilistic properties of the usual quantum mechanics.Comment: 13 pages, LaTe

Quantum Flux and Reverse Engineering of Quantum Wavefunctions

An interpretation of the probability flux is given, based on a derivation of its eigenstates and relating them to coherent state projections on a quantum wavefunction. An extended definition of the flux operator is obtained using coherent states. We present a "processed Husimi" representation, which makes decisions using many Husimi projections at each location. The processed Husimi representation reverse engineers or deconstructs the wavefunction, yielding the underlying classical ray structure. Our approach makes possible interpreting the dynamics of systems where the probability flux is uniformly zero or strongly misleading. The new technique is demonstrated by the calculation of particle flow maps of the classical dynamics underlying a quantum wavefunction.Comment: Accepted to EP

Scarring by homoclinic and heteroclinic orbits

In addition to the well known scarring effect of periodic orbits, we show here that homoclinic and heteroclinic orbits, which are cornerstones in the theory of classical chaos, also scar eigenfunctions of classically chaotic systems when associated closed circuits in phase space are properly quantized, thus introducing strong quantum correlations. The corresponding quantization rules are also established. This opens the door for developing computationally tractable methods to calculate eigenstates of chaotic systems.Comment: 5 pages, 4 figure

Application of serious games to sport, health and exercise

Use of interactive entertainment has been exponentially expanded since the last decade. Throughout this 10+ year evolution there has been a concern about turning entertainment properties into serious applications, a.k.a "Serious Games". In this article we present two set of Serious Game applications, an Environment Visualising game which focuses solely on applying serious games to elite Olympic sport and another set of serious games that incorporate an in house developed proprietary input system that can detect most of the human movements which focuses on applying serious games to health and exercise

Improving the Quark Number Susceptibilities for Staggered Fermions

Quark number susceptibilities approach their ideal gas limit at sufficiently high temperatures. As in the case of other thermodynamic quantities, this limit itself is altered substantially on lattices with small temporal extent, N_t = 4-8, making it thus difficult to check the validity of perturbation theory. Unlike other observables, improving susceptibilities or number densities is subject to constraints of current conservation and absence of chemical potential dependent divergences. We construct such an improved number density and susceptibility for staggered fermions and show that they approximate the continuum ideal gas limit better on small temporal lattices.Comment: Lattice2002(nonzerot), 3 pages, 3 figure

Orbit bifurcations and the scarring of wavefunctions

We extend the semiclassical theory of scarring of quantum eigenfunctions psi_{n}(q) by classical periodic orbits to include situations where these orbits undergo generic bifurcations. It is shown that |psi_{n}(q)|^{2}, averaged locally with respect to position q and the energy spectrum E_{n}, has structure around bifurcating periodic orbits with an amplitude and length-scale whose hbar-dependence is determined by the bifurcation in question. Specifically, the amplitude scales as hbar^{alpha} and the length-scale as hbar^{w}, and values of the scar exponents, alpha and w, are computed for a variety of generic bifurcations. In each case, the scars are semiclassically wider than those associated with isolated and unstable periodic orbits; moreover, their amplitude is at least as large, and in most cases larger. In this sense, bifurcations may be said to give rise to superscars. The competition between the contributions from different bifurcations to determine the moments of the averaged eigenfunction amplitude is analysed. We argue that there is a resulting universal hbar-scaling in the semiclassical asymptotics of these moments for irregular states in systems with a mixed phase-space dynamics. Finally, a number of these predictions are illustrated by numerical computations for a family of perturbed cat maps.Comment: 24 pages, 6 Postscript figures, corrected some typo