A Unifying Approach to Quaternion Adaptive Filtering: Addressing the Gradient and Convergence

A novel framework for a unifying treatment of quaternion valued adaptive filtering algorithms is introduced. This is achieved based on a rigorous account of quaternion differentiability, the proposed I-gradient, and the use of augmented quaternion statistics to account for real world data with noncircular probability distributions. We first provide an elegant solution for the calculation of the gradient of real functions of quaternion variables (typical cost function), an issue that has so far prevented systematic development of quaternion adaptive filters. This makes it possible to unify the class of existing and proposed quaternion least mean square (QLMS) algorithms, and to illuminate their structural similarity. Next, in order to cater for both circular and noncircular data, the class of widely linear QLMS (WL-QLMS) algorithms is introduced and the subsequent convergence analysis unifies the treatment of strictly linear and widely linear filters, for both proper and improper sources. It is also shown that the proposed class of HR gradients allows us to resolve the uncertainty owing to the noncommutativity of quaternion products, while the involution gradient (I-gradient) provides generic extensions of the corresponding real- and complex-valued adaptive algorithms, at a reduced computational cost. Simulations in both the strictly linear and widely linear setting support the approach

Exact Solution of Quantum Field Theory on Noncommutative Phase Spaces

We present the exact solution of a scalar field theory defined with noncommuting position and momentum variables. The model describes charged particles in a uniform magnetic field and with an interaction defined by the Groenewold-Moyal star-product. Explicit results are presented for all Green's functions in arbitrary even spacetime dimensionality. Various scaling limits of the field theory are analysed non-perturbatively and the renormalizability of each limit examined. A supersymmetric extension of the field theory is also constructed in which the supersymmetry transformations are parametrized by differential operators in an infinite-dimensional noncommutative algebra.Comment: 70 pages AMSTe

Noncommutative geometry and Painlevé equations

We construct the elliptic Painlevé equation and its higher dimensional analogs as the action of line bundles on 1 -dimensional sheaves on noncommutative surfaces

Stable Unitary Integrators for the Numerical Implementation of Continuous Unitary Transformations

The technique of continuous unitary transformations has recently been used to provide physical insight into a diverse array of quantum mechanical systems. However, the question of how to best numerically implement the flow equations has received little attention. The most immediately apparent approach, using standard Runge-Kutta numerical integration algorithms, suffers from both severe inefficiency due to stiffness and the loss of unitarity. After reviewing the formalism of continuous unitary transformations and Wegner's original choice for the infinitesimal generator of the flow, we present a number of approaches to resolving these issues including a choice of generator which induces what we call the "uniform tangent decay flow" and three numerical integrators specifically designed to perform continuous unitary transformations efficiently while preserving the unitarity of flow. We conclude by applying one of the flow algorithms to a simple calculation that visually demonstrates the many-body localization transition.Comment: 13 pages, 4 figures, Comments welcom

Quantum Gravity, Field Theory and Signatures of Noncommutative Spacetime

A pedagogical introduction to some of the main ideas and results of field theories on quantized spacetimes is presented, with emphasis on what such field theories may teach us about the problem of quantizing gravity. We examine to what extent noncommutative gauge theories may be regarded as gauge theories of gravity. UV/IR mixing is explained in detail and we describe its relations to renormalization, to gravitational dynamics, and to deformed dispersion relations in models of quantum spacetime of interest in string theory and in doubly special relativity. We also discuss some potential experimental probes of spacetime noncommutativity.Comment: 26 pages, 4 figures; v2: comments and references added; v3: typos corrected, clarifying comments and references added; Based on Plenary Lecture delivered at the XXIX Encontro Nacional de Fisica de Particulas e Campos, Sao Lourenco, Brasil, September 22-26, 2008; Final version to be published in General Relativity and Gravitatio

Quantum Spacetime and Algebraic Quantum Field Theory

We review the investigations on the quantum structure of spactime, to be found at the Planck scale if one takes into account the operational limitations to localization of events which result from the concurrence of Quantum Mechanics and General Relativity. We also discuss the different approaches to (perturbative) Quantum Field Theory on Quantum Spacetime, and some of the possible cosmological consequences.Comment: 49 pages, 2 figure