Convolution and correlation Theorem for Linear Canonical Transform and Properties

In this paper we introduce the convolution theorem for the linear canonical transform (LCT). Based on the properties of the convolution theorem for the Fourier transform we explicitly show some important properties of the relationship between the LCT and convolution. We provide an alternative form of the correlation theorem for the LC

The Lagrangian and Hamiltonian Aspects of the Electrodynamic Vacuum-Field Theory Models

We review the modern classical electrodynamics problems and present the related main fundamental principles characterizing the electrodynamical vacuum-field structure. We analyze the models of the vacuum field medium and charged point particle dynamics using the developed field theory concepts. There is also described a new approach to the classical Maxwell theory based on the derived and newly interpreted basic equations making use of the vacuum field theory approach. In particular, there are obtained the main classical special relativity theory relations and their new explanations. The well known Feynman approach to Maxwell electromagnetic equations and the Lorentz type force derivation is also discussed in detail. A related charged point particle dynamics and a hadronic string model analysis is also presented. We also revisited and reanalyzed the classical Lorentz force expression in arbitrary non-inertial reference frames and present some new interpretations of the relations between special relativity theory and its quantum mechanical aspects. Some results related with the charge particle radiation problem and the magnetic potential topological aspects are discussed. The electromagnetic Dirac-Fock-Podolsky problem of the Maxwell and Yang-Mills type dynamical systems is analyzed within the classical Dirac-Marsden-Weinstein symplectic reduction theory. The problem of constructing Fock type representations and retrieving their creation-annihilation operator structure is analyzed. An application of the suitable current algebra representation to describing the non-relativistic Aharonov-Bohm paradox is presented. The current algebra coherent functional representations are constructed and their importance subject to the linearization problem of nonlinear dynamical systems in Hilbert spaces is demonstrated.Comment: 70 p, revie

Physics of the Riemann Hypothesis

Physicists become acquainted with special functions early in their studies. Consider our perennial model, the harmonic oscillator, for which we need Hermite functions, or the Laguerre functions in quantum mechanics. Here we choose a particular number theoretical function, the Riemann zeta function and examine its influence in the realm of physics and also how physics may be suggestive for the resolution of one of mathematics' most famous unconfirmed conjectures, the Riemann Hypothesis. Does physics hold an essential key to the solution for this more than hundred-year-old problem? In this work we examine numerous models from different branches of physics, from classical mechanics to statistical physics, where this function plays an integral role. We also see how this function is related to quantum chaos and how its pole-structure encodes when particles can undergo Bose-Einstein condensation at low temperature. Throughout these examinations we highlight how physics can perhaps shed light on the Riemann Hypothesis. Naturally, our aim could not be to be comprehensive, rather we focus on the major models and aim to give an informed starting point for the interested Reader.Comment: 27 pages, 9 figure

The angular momentum controversy: What's it all about and does it matter?

The general question, crucial to an understanding of the internal structure of the nucleon, of how to split the total angular momentum of a photon or gluon into spin and orbital contributions is one of the most important and interesting challenges faced by gauge theories like Quantum Electrodynamics and Quantum Chromodynamics. This is particularly challenging since all QED textbooks state that such an splitting cannot be done for a photon (and a fortiori for a gluon) in a gauge-invariant way, yet experimentalists around the world are engaged in measuring what they believe is the gluon spin! This question has been a subject of intense debate and controversy, ever since, in 2008, it was claimed that such a gauge-invariant split was, in fact, possible. We explain in what sense this claim is true and how it turns out that one of the main problems is that such a decomposition is not unique and therefore raises the question of what is the most natural or physical choice. The essential requirement of measurability does not solve the ambiguities and leads us to the conclusion that the choice of a particular decomposition is essentially a matter of taste and convenience. In this review, we provide a pedagogical introduction to the question of angular momentum decomposition in a gauge theory, present the main relevant decompositions and discuss in detail several aspects of the controversies regarding the question of gauge invariance, frame dependence, uniqueness and measurability. We stress the physical implications of the recent developments and collect into a separate section all the sum rules and relations which we think experimentally relevant. We hope that such a review will make the matter amenable to a broader community and will help to clarify the present situation.Comment: 96 pages, 11 figures, 5 tables, review prepared for Physics Report

The ubiquitous role of linear algebra within applied mathematics

Die vorliegende Diplomarbeit kombiniert elementare Methoden der Linearen Algebra mit der Angewandten Mathematik. Die Lineare Algebra wird dabei als unverzichtbares Werkzeug der modernen Anwendungsgebiete der Mathematik verstanden. Das erste Kapitel beschäftigt sich mit drei fundamentalen Gesetzen der Physik und Elektrotechnik von Kirchho beziehungsweise Ohm, die sich zu einer Gleichung zusammenfassen lassen, nämlich der Netzwerkgleichung. Hierfür werden die vier Teilräume, Graphen und Inzidenzmatrizen in Betracht gezogen. Das zweite Kapitel widmet sich der Eigenwerte und Eigenvektoren. Markov Ketten, die Singulärwertzerlegung und die Methode der kleinsten Quadrate (minimaler Länge) werden vorgestellt. Statistik und demographische Prozesse, Bildkompression beziehungsweise Anwendungen der kleinsten Quadrate (minimaler Länge) sind unter den mathematischen Anwendungen, die hier zutrage kommen. Das dritte Kapitel beschäftigt sich mit Grundlagen der Bild- und Signalverarbeitung. Es wird in die Fourier-Analysis eingeleitet und einige wichtige Elemente der Signalverarbeitung, wie Filter, besprochen. Das letzte Kapitel unternimmt einen Versuch diese Arbeit in die Diskussion über Angewandte Mathematik in der Schule einzubinden.This thesis combines Linear Algebra with Applied Mathematics. The author presents mathematical applications from physics and electrical engineering, statistics and demography, signal and image processing by means of linear algebra. The first chapter presents three fundamental laws of Kirchhoff and Ohm, respectively, which combine into the fundamental network equation. Therefore, the author considers the four subspaces, graphs and incidence matrices. The second chapter is all about applications to eigenvalues and eigenvectors. The author introduces Markov chains, the singular value decomposition and the method of (minimal norm) least squares. Statistics and demographic processes, image compres- sion and (minimal norm) least squares applications, respectively, are among mathematical applications introduced in this chapter. The third chapter deals with basics of signal and image processing. The author provides an introduction to Fourier analysis and discusses some important tools for signal processing, e.g., filters. The last chapter constitutes an attempt to review this thesis with respect to mathematics education, teaching and, in general, didactics

The ℏ→0\hbar\to 0 limit of open quantum systems with general Lindbladians: vanishing noise ensures classicality beyond the Ehrenfest time

Quantum and classical systems evolving under the same formal Hamiltonian $H$ may exhibit dramatically different behavior after the Ehrenfest timescale $t_E \sim \log(\hbar^{-1})$, even as $\hbar \to 0$. Coupling the system to a Markovian environment results in a Lindblad equation for the quantum evolution. Its classical counterpart is given by the Fokker-Planck equation on phase space, which describes Hamiltonian flow with friction and diffusive noise. The quantum and classical evolutions may be compared via the Wigner-Weyl representation. Due to decoherence, they are conjectured to match closely for times far beyond the Ehrenfest timescale as $\hbar \to 0$. We prove a version of this correspondence, bounding the error between the quantum and classical evolutions for any sufficiently regular Hamiltonian $H(x,p)$ and Lindblad functions $L_k(x,p)$. The error is small when the strength of the diffusion $D$ associated to the Lindblad functions satisfies $D \gg \hbar^{4/3}$, in particular allowing vanishing noise in the classical limit. We use a time-dependent semiclassical mixture of variably squeezed Gaussian states evolving by a local harmonic approximation to the Lindblad dynamics. Both the exact quantum trajectory and its classical counterpart can be expressed as perturbations of this semiclassical mixture, with the errors bounded using Duhamel's principle. We present heuristic arguments suggesting the $4/3$ exponent is optimal and defines a boundary in the sense that asymptotically weaker diffusion permits a breakdown of quantum-classical correspondence at the Ehrenfest timescale. Our presentation aims to be comprehensive and accessible to both mathematicians and physicists. In a shorter companion paper, we treat the special case of Hamiltonians of the form $H=p^2/2m + V(x)$ and linear Lindblad operators, with explicit bounds that can be applied directly to physical systems.Comment: 53 pages + appendices, 2 figures. Companion to arXiv:2306.1371

The Lagrangian and Hamiltonian Aspects of the Electrodynamic Vacuum-Field Theory Models

The  important classical  Ampère’s and Lorentz laws derivations  are revisited and their relationships with  the modern vacuum field theory approach to modern relativistic electrodynamics are demonstrated.  The relativistic models of the vacuum field medium and charged point particle dynamics as well as  related classical electrodynamics problems  jointly with the fundamental principles, characterizing the electrodynamical vacuum-field structure,  based on  the developed field theory concepts are reviewed and analyzed detail. There is also described a new approach to the classical Maxwell theory based on the derived and newly interpreted basic equations making use of the vacuum field theory approach. There are obtained the main classical special relativity theory relationships and their new explanations. The well known Feynman approach to Maxwell electromagnetic equations and the Lorentz type force derivation is also. A related charged point particle dynamics and a hadronic string model analysis is also presented. We also revisited and reanalyzed the classical Lorentz force expression in arbitrary non-inertial reference frames and present some new interpretations of the relations between special relativity theory and its quantum mechanical aspects. Some results related with the charge particle radiation problem and the magnetic potential topological aspects are discussed. The electromagnetic Dirac-Fock-Podolsky problem of the Maxwell and Yang-Mills type dynamical systems is analyzed within the classical Dirac-Marsden-Weinstein symplectic reduction theory. Based on the Gelfand-Vilenkin representation theory of infinite dimensional groups and the Goldin-Menikoff-Sharp theory of generating Bogolubov type functionals the problem of constructing Fock type representations and retrieving their creation-annihilation operator structure is analyzed. An application of the suitable current algebra representation to describing the non-relativistic Aharonov-Bohm paradox is demonstrated. The current algebra coherent functional representations are constructed and their importance subject to the linearization problem of nonlinear dynamical systems in Hilbert spaces is also presented