State Vector Reduction as a Shadow of a Noncommutative Dynamics

A model, based on a noncommutative geometry, unifying general relativity with quantum mechanics, is further develped. It is shown that the dynamics in this model can be described in terms of one-parameter groups of random operators. It is striking that the noncommutative counterparts of the concept of state and that of probability measure coincide. We also demonstrate that the equation describing noncommutative dynamics in the quantum gravitational approximation gives the standard unitary evolution of observables, and in the "space-time limit" it leads to the state vector reduction. The cases of the spin and position operators are discussed in details.Comment: 20 pages, LaTex, no figure

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

SU(3) Lattice Gauge Theory in the Fundamental--Adjoint Plane and Scaling Along the Wilson Axis

We present further evidence for the bulk nature of the phase transition line in the fundamental--adjoint action plane of SU(3) lattice gauge theory. Computing the string tension and some glueball masses along the thermal phase transition line of finite temperature systems with $N_t=4$, which was found to join onto the bulk transition line at its endpoint, we find that the ratio $\sqrt{\sigma} / T_c$ remains approximately constant. However, the mass of the $0^{++}$ glueball decreases as the endpoint of the bulk transition line is approached, while the other glueball masses appear unchanged. This is consistent with the notion that the bulk transition line ends in a critical endpoint, with the continuum limit there being a $\phi^4$ theory with a diverging correlation length only in the $0^{++}$ channel. We comment on the implications for the scaling behavior along the fundamental or Wilson axis.Comment: 10 pages. Plain LaTeX file. 4 postscript figures added in a separate fil

Group invariant inferred distributions via noncommutative probability

One may consider three types of statistical inference: Bayesian, frequentist, and group invariance-based. The focus here is on the last method. We consider the Poisson and binomial distributions in detail to illustrate a group invariance method for constructing inferred distributions on parameter spaces given observed results. These inferred distributions are obtained without using Bayes' method and in particular without using a joint distribution of random variable and parameter. In the Poisson and binomial cases, the final formulas for inferred distributions coincide with the formulas for Bayes posteriors with uniform priors.Comment: Published at http://dx.doi.org/10.1214/074921706000000563 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

SVIM: Structural Variant Identification using Mapped Long Reads

Motivation: Structural variants are defined as genomic variants larger than 50bp. They have been shown to affect more bases in any given genome than SNPs or small indels. Additionally, they have great impact on human phenotype and diversity and have been linked to numerous diseases. Due to their size and association with repeats, they are difficult to detect by shotgun sequencing, especially when based on short reads. Long read, single molecule sequencing technologies like those offered by Pacific Biosciences or Oxford Nanopore Technologies produce reads with a length of several thousand base pairs. Despite the higher error rate and sequencing cost, long read sequencing offers many advantages for the detection of structural variants. Yet, available software tools still do not fully exploit the possibilities. Results: We present SVIM, a tool for the sensitive detection and precise characterization of structural variants from long read data. SVIM consists of three components for the collection, clustering and combination of structural variant signatures from read alignments. It discriminates five different variant classes including similar types, such as tandem and interspersed duplications and novel element insertions. SVIM is unique in its capability of extracting both the genomic origin and destination of duplications. It compares favorably with existing tools in evaluations on simulated data and real datasets from PacBio and Nanopore sequencing machines. Availability and implementation: The source code and executables of SVIM are available on Github: github.com/eldariont/svim. SVIM has been implemented in Python 3 and published on bioconda and the Python Package Index. Supplementary information: Supplementary data are available at Bioinformatics online