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

Conceptual Unification of Gravity and Quanta

We present a model unifying general relativity and quantum mechanics. The model is based on the (noncommutative) algebra \mbox{{\cal A}} on the groupoid \Gamma = E \times G where E is the total space of the frame bundle over spacetime, and G the Lorentz group. The differential geometry, based on derivations of \mbox{{\cal A}}, is constructed. The eigenvalue equation for the Einstein operator plays the role of the generalized Einstein's equation. The algebra \mbox{{\cal A}}, when suitably represented in a bundle of Hilbert spaces, is a von Neumann algebra \mathcal{M} of random operators representing the quantum sector of the model. The Tomita-Takesaki theorem allows us to define the dynamics of random operators which depends on the state \phi . The same state defines the noncommutative probability measure (in the sense of Voiculescu's free probability theory). Moreover, the state \phi satisfies the Kubo-Martin-Schwinger (KMS) condition, and can be interpreted as describing a generalized equilibrium state. By suitably averaging elements of the algebra \mbox{{\cal A}}, one recovers the standard geometry of spacetime. We show that any act of measurement, performed at a given spacetime point, makes the model to collapse to the standard quantum mechanics (on the group G). As an example we compute the noncommutative version of the closed Friedman world model. Generalized eigenvalues of the Einstein operator produce the correct components of the energy-momentum tensor. Dynamics of random operators does not ``feel'' singularities.Comment: 28 LaTex pages. Substantially enlarged version. Improved definition of generalized Einstein's field equation

Framework for non-perturbative analysis of a Z(3)-symmetric effective theory of finite temperature QCD

We study a three dimensional Z(3)-symmetric effective theory of high temperature QCD. The exact lattice-continuum relations, needed in order to perform lattice simulations with physical parameters, are computed to order O(a^0) in lattice perturbation theory. Lattice simulations are performed to determine the phase structure of a subset of the parameter space.Comment: 28 pages, 11 figures; v3: references rearranged, typos corrected, figs changed, published versio

Anatomy of Malicious Singularities

As well known, the b-boundaries of the closed Friedman world model and of Schwarzschild solution consist of a single point. We study this phenomenon in a broader context of differential and structured spaces. We show that it is an equivalence relation $\rho$, defined on the Cauchy completed total space $\bar{E}$ of the frame bundle over a given space-time, that is responsible for this pathology. A singularity is called malicious if the equivalence class $[p_0]$ related to the singularity remains in close contact with all other equivalence classes, i.e., if $p_0 \in \mathrm{cl}[p]$ for every $p \in E$. We formulate conditions for which such a situation occurs. The differential structure of any space-time with malicious singularities consists only of constant functions which means that, from the topological point of view, everything collapses to a single point. It was noncommutative geometry that was especially devised to deal with such situations. A noncommutative algebra on $\bar{E}$, which turns out to be a von Neumann algebra of random operators, allows us to study probabilistic properties (in a generalized sense) of malicious singularities. Our main result is that, in the noncommutative regime, even the strongest singularities are probabilistically irrelevant.Comment: 16 pages in LaTe

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

Lattice quark propagator with staggered quarks in Landau and Laplacian gauges

We report on the lattice quark propagator using standard and improved Staggered quark actions, with the standard, Wilson gauge action. The standard Kogut-Susskind action has errors of \oa{2} while the ``Asqtad'' action has \oa{4}, \oag{2}{2} errors. The quark propagator is interesting for studying the phenomenon of dynamical chiral symmetry breaking and as a test-bed for improvement. Gauge dependent quantities from lattice simulations may be affected by Gribov copies. We explore this by studying the quark propagator in both Landau and Laplacian gauges. Landau and Laplacian gauges are found to produce very similar results for the quark propagator.Comment: 11 pages, 15 figure

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