Geometry of Jets and Fields”
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Abstract Book
14th International Geometry Symposium 25-28 May 2016 ABSTRACT BOOK Pamukkale University Denizli - TURKEY 1 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 14th International Geometry Symposium ABSTRACT BOOK 1 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 Proceedings of the 14th International Geometry Symposium Edited By: Dr. Şevket CİVELEK Dr. Cansel YORMAZ E-Published By: Pamukkale University Department of Mathematics Denizli, TURKEY All rights reserved. No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means or whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder. Authors of papers in these proceedings are authorized to use their own material freely. Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to: Assoc. Prof. Dr. Şevket CİVELEK Pamukkale University Department of Mathematics Denizli, TURKEY Email: [email protected] 2 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 Proceedings of the 14th International Geometry Symposium May 25-28, 2016 Denizli, Turkey. Jointly Organized by Pamukkale University Department of Mathematics Denizli, Turkey 3 14th International Geometry Symposium Pamukkale University Denizli/TURKEY 25-28 May 2016 PREFACE This volume comprises the abstracts of contributed papers presented at the 14th International Geometry Symposium, 14IGS 2016 held on May 25-28, 2016, in Denizli, Turkey. 14IGS 2016 is jointly organized by Department of Mathematics, Pamukkale University, Denizli, Turkey. The sysposium is aimed to provide a platform for Geometry and its applications. -
Covariant Hamiltonian Field Theory 3
December 16, 2020 2:58 WSPC/INSTRUCTION FILE kfte COVARIANT HAMILTONIAN FIELD THEORY JURGEN¨ STRUCKMEIER and ANDREAS REDELBACH GSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH Planckstr. 1, 64291 Darmstadt, Germany and Johann Wolfgang Goethe-Universit¨at Frankfurt am Main Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany [email protected] Received 18 July 2007 Revised 14 December 2020 A consistent, local coordinate formulation of covariant Hamiltonian field theory is pre- sented. Whereas the covariant canonical field equations are equivalent to the Euler- Lagrange field equations, the covariant canonical transformation theory offers more gen- eral means for defining mappings that preserve the form of the field equations than the usual Lagrangian description. It is proved that Poisson brackets, Lagrange brackets, and canonical 2-forms exist that are invariant under canonical transformations of the fields. The technique to derive transformation rules for the fields from generating functions is demonstrated by means of various examples. In particular, it is shown that the infinites- imal canonical transformation furnishes the most general form of Noether’s theorem. We furthermore specify the generating function of an infinitesimal space-time step that conforms to the field equations. Keywords: Field theory; Hamiltonian density; covariant. PACS numbers: 11.10.Ef, 11.15Kc arXiv:0811.0508v6 [math-ph] 15 Dec 2020 1. Introduction Relativistic field theories and gauge theories are commonly formulated on the basis of a Lagrangian density L1,2,3,4. The space-time evolution of the fields is obtained by integrating the Euler-Lagrange field equations that follow from the four-dimensional representation of Hamilton’s action principle. -
Description of Bound States and Scattering Processes Using Hamiltonian Renormalization Group Procedure for Effective Particles in Quantum field Theory
Description of bound states and scattering processes using Hamiltonian renormalization group procedure for effective particles in quantum field theory Marek Wi˛eckowski June 2005 Doctoral thesis Advisor: Professor Stanisław D. Głazek Warsaw University Department of Physics Institute of Theoretical Physics Warsaw 2005 i Acknowledgments I am most grateful to my advisor, Professor Stanisław Głazek, for his invaluable advice and assistance during my work on this thesis. I would also like to thank Tomasz Masłowski for allowing me to use parts of our paper here. My heartfelt thanks go to Dr. Nick Ukiah, who supported and encouraged me during the writing of this thesis and who helped me in editing the final manuscript in English. Finally, I would like to thank my family, especially my grandmother, who has been a great support to me during the last few years. MAREK WI ˛ECKOWSKI,DESCRIPTION OF BOUND STATES AND SCATTERING... ii MAREK WI ˛ECKOWSKI,DESCRIPTION OF BOUND STATES AND SCATTERING... iii Abstract This thesis presents examples of a perturbative construction of Hamiltonians Hλ for effective particles in quantum field theory (QFT) on the light front. These Hamiltonians (1) have a well-defined (ultraviolet-finite) eigenvalue problem for bound states of relativistic constituent fermions, and (2) lead (in a scalar theory with asymptotic freedom in perturbation theory in third and partly fourth order) to an ultraviolet-finite and covariant scattering matrix, as the Feynman diagrams do. λ is a parameter of the renormalization group for Hamiltonians and qualitatively means the inverse of the size of the effective particles. The same procedure of calculating the operator Hλ applies in description of bound states and scattering. -
Turbulence, Entropy and Dynamics
TURBULENCE, ENTROPY AND DYNAMICS Lecture Notes, UPC 2014 Jose M. Redondo Contents 1 Turbulence 1 1.1 Features ................................................ 2 1.2 Examples of turbulence ........................................ 3 1.3 Heat and momentum transfer ..................................... 4 1.4 Kolmogorov’s theory of 1941 ..................................... 4 1.5 See also ................................................ 6 1.6 References and notes ......................................... 6 1.7 Further reading ............................................ 7 1.7.1 General ............................................ 7 1.7.2 Original scientific research papers and classic monographs .................. 7 1.8 External links ............................................. 7 2 Turbulence modeling 8 2.1 Closure problem ............................................ 8 2.2 Eddy viscosity ............................................. 8 2.3 Prandtl’s mixing-length concept .................................... 8 2.4 Smagorinsky model for the sub-grid scale eddy viscosity ....................... 8 2.5 Spalart–Allmaras, k–ε and k–ω models ................................ 9 2.6 Common models ........................................... 9 2.7 References ............................................... 9 2.7.1 Notes ............................................. 9 2.7.2 Other ............................................. 9 3 Reynolds stress equation model 10 3.1 Production term ............................................ 10 3.2 Pressure-strain interactions -
Ivanenko. Biography
The People of Physics Faculty Selected papers of the Journal “Soviet Physicist” 1998-2006 Dmitri Ivanenko. Scientific Biography 226 Dmitri Ivanenko (29.07.1904 - 30.12.1994), professor of Moscow State University (since 1943) , was one of the great theoreticians of XX century. He made the fundamental contribution to many areas of nuclear physics, field theory and gravitation theory. His outstanding achievements include: • The Fock - Ivanenko coefficients of parallel displacement of spinors in a curved space-time (1929) 1 . Nobel laureate Abdus Salam called it the first gauge theory. • The Ambartsumian - Ivanenko hypothesis of creation of massive particles which is a corner stone of contemporary quantum field theory (1930) 2 . • The proton-neutron model of atomic nuclei (1932) 3 . • The first shell model of nuclei (in collaboration with E. Gapon) (1932) 4 . • The first model of exchange nuclear forces by means of massive particles (in collaboration with I. Tamm) (1934) 5 . Based on this model, Nobel laureate H. Yukawa developed his meson theory. • The prediction of synchrotron radiation (in collaboration with I. Pomeranchuk) (1944) 6 and its classical theory (in collaboration with A. Sokolov). • Theory of hypernucleus (1956) 7 . • The hypothesis of quark stars (in collaboration with D. Kurdgelaidze) (1965) 8 . • The gauge gravitation theory (in collaboration with G. Sardanashvily), where gravity is treated as a Higgs field responsible for spontaneous breaking of space- 9 time symmetries (1983) . References 1. Fock V., Iwanenko D., Géometrie quantique linéaire et déplacement paralléle, Compt. Rend. Acad Sci. Paris 188 (1929) 1470. 2. Ambarzumian V., Iwanenko D., Les électrons inobservables et les rayons, Compt. -
Advances in Quantum Field Theory
ADVANCES IN QUANTUM FIELD THEORY Edited by Sergey Ketov Advances in Quantum Field Theory Edited by Sergey Ketov Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2012 InTech All chapters are Open Access distributed under the Creative Commons Attribution 3.0 license, which allows users to download, copy and build upon published articles even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. As for readers, this license allows users to download, copy and build upon published chapters even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. Notice Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published chapters. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Romana Vukelic Technical Editor Teodora Smiljanic Cover Designer InTech Design Team First published February, 2012 Printed in Croatia A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from [email protected] Advances in Quantum Field Theory, Edited by Sergey Ketov p. -
Lagrangian Dynamics of Submanifolds. Relativistic Mechanics
JOURNAL OF GEOMETRIC MECHANICS doi:10.3934/jgm.2012.4.99 c American Institute of Mathematical Sciences Volume 4, Number 1, March 2012 pp. 99{110 LAGRANGIAN DYNAMICS OF SUBMANIFOLDS. RELATIVISTIC MECHANICS Gennadi Sardanashvily Department of Theoretical Physics Moscow State University Moscow, Russia (Communicated by Manuel de Le´on) Abstract. Geometric formulation of Lagrangian relativistic mechanics in the terms of jets of one-dimensional submanifolds is generalized to Lagrangian theory of submanifolds of arbitrary dimension. 1. Introduction. Classical non-relativistic mechanics is adequately formulated as Lagrangian and Hamiltonian theory on a fibre bundle Q ! R over the time axis R, where R is provided with the Cartesian coordinate t possessing transition functions t0 = t+const. [1,5,7,8, 11]. A velocity space of non-relativistic mechanics is a first order jet manifold J 1Q of sections of Q ! R. Lagrangians of non-relativistic mechanics are defined as densities on J 1Q. This formulation is extended to time- reparametrized non-relativistic mechanics subject to time-dependent transforma- tions which are bundle automorphisms of Q ! R [5,8]. Thus, one can think of non-relativistic mechanics as being particular classical field theory on fibre bundles over X = R. However, an essential difference between non-relativistic mechanics and field theory on fibre bundles Y ! X, dim X > 1, lies in the fact that connections on Q ! R always are flat. Therefore, they fail to be dynamic variables, but characterize non-relativistic reference frames. In comparison with non-relativistic mechanics, relativistic mechanics admits trans- formations of the time depending on other variables, e.g., Lorentz transformations in Special Relativity on a Minkowski space Q = R4. -
Classical Hamiltonian Field Theory
Classical Hamiltonian Field Theory Alex Nelson∗ Email: [email protected] August 26, 2009 Abstract This is a set of notes introducing Hamiltonian field theory, with focus on the scalar field. Contents 1 Lagrangian Field Theory1 2 Hamiltonian Field Theory2 1 Lagrangian Field Theory As with Hamiltonian mechanics, wherein one begins by taking the Legendre transform of the Lagrangian, in Hamiltonian field theory we \transform" the Lagrangian field treatment. So lets review the calculations in Lagrangian field theory. Consider the classical fields φa(t; x¯). We use the index a to indicate which field we are talking about. We should think of x¯ as another index, except it is continuous. We will use the confusing short hand notation φ for the column vector φ1; : : : ; φn. Consider the Lagrangian Z 3 L(φ) = L(φ, @µφ)d x¯ (1.1) all space where L is the Lagrangian density. Hamilton's principle of stationary action is still used to determine the equations of motion from the action Z S[φ] = L(φ, @µφ)dt (1.2) where we find the Euler-Lagrange equations of motion for the field d @L @L µ a = a (1.3) dx @(@µφ ) @φ where we note these are evil second order partial differential equations. We also note that we are using Einstein summation convention, so there is an implicit sum over µ but not over a. So that means there are n independent second order partial differential equations we need to solve. But how do we really know these are the correct equations? How do we really know these are the Euler-Lagrange equations for classical fields? We can obtain it directly from the action S by functional differentiation with respect to the field. -
Higher Order Geometric Theory of Information and Heat Based On
Article Higher Order Geometric Theory of Information and Heat Based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures: The Bedrock for Lie Group Machine Learning Frédéric Barbaresco Department of Advanced Radar Concepts, Thales Land Air Systems, Voie Pierre-Gilles de Gennes, 91470 Limours, France; [email protected] Received: 9 August 2018; Accepted: 9 October; Published: 2 November 2018 Abstract: We introduce poly-symplectic extension of Souriau Lie groups thermodynamics based on higher-order model of statistical physics introduced by Ingarden. This extended model could be used for small data analytics and machine learning on Lie groups. Souriau geometric theory of heat is well adapted to describe density of probability (maximum entropy Gibbs density) of data living on groups or on homogeneous manifolds. For small data analytics (rarified gases, sparse statistical surveys, …), the density of maximum entropy should consider higher order moments constraints (Gibbs density is not only defined by first moment but fluctuations request 2nd order and higher moments) as introduced by Ingarden. We use a poly-sympletic model introduced by Christian Günther, replacing the symplectic form by a vector-valued form. The poly-symplectic approach generalizes the Noether theorem, the existence of moment mappings, the Lie algebra structure of the space of currents, the (non-)equivariant cohomology and the classification of G-homogeneous systems. The formalism is covariant, i.e., no special coordinates or coordinate systems on the parameter space are used to construct the Hamiltonian equations. We underline the contextures of these models, and the process to build these generic structures. We also introduce a more synthetic Koszul definition of Fisher Metric, based on the Souriau model, that we name Souriau-Fisher metric. -
Arxiv:1808.08689V3
Hamiltonian Nature of Monopole Dynamics J. M. Heningera, P. J. Morrisona aDepartment of Physics and Institute for Fusion Studies, The University of Texas at Austin, Austin, TX, 78712, USA Abstract Classical electromagnetism with magnetic monopoles is not a Hamiltonian field theory because the Jacobi identity for the Poisson bracket fails. The Jacobi identity is recovered only if all of the species have the same ratio of electric to magnetic charge or if an electron and a monopole can never collide. Without the Jacobi identity, there are no local canonical coordinates or Lagrangian action principle. To build a quantum field of magnetic monopoles, we either must explain why the positions of electrons and monopoles can never coincide or we must resort to new quantization techniques. 1. Introduction This letter considers the classical gauge-free theory of electromagnetic fields interacting with electrically and magnetically charged matter as a Hamiltonian field theory. We begin with a brief history of magnetic monopoles and then describe why monopole theories are not Hamiltonian field theories. The modern theory of magnetic monopoles was developed by Dirac [1, 2]. He showed that an electron in the magnetic field of a monopole is equivalent to an electron whose wave function is zero along a semi-infinite ‘string’ extending from the location of the monopole. Along this string, the electromagnetic vector potential is undefined. The phase of the electron is no longer single valued along a loop encircling the Dirac string. In order for observables to be single valued, the phase shift must be an integer multiple of 2π, so the electric and magnetic charge must be quantized. -
Background Geometry in Gauge Gravitation Theory
BACKGROUND GEOMETRY IN GAUGE GRAVITATION THEORY Gennadi Sardanashvily Department of Theoretical Physics, Moscow State University, 117234 Moscow, Russia E-mail: [email protected] Abstract Dirac fermion fields are responsible for spontaneous symmetry breaking in gauge grav- itation theory because the spin structure associated with a tetrad field is not preserved under general covariant transformations. Two solutions of this problem can be suggested. (i) There exists the universal spin structure S → X such that any spin structure Sh → X associated with a tetrad field h is a subbundle of the bundle S → X. In this model, gravitational fields correspond to different tetrad (or metric) fields. (ii) A background tetrad field h and the associated spin structure Sh are fixed, while gravitational fields are λ λ λ µ identified with additional tensor fields q µ describing deviations ha = q µha of h. One can think of h as being effective tetrad fields. We show that theree exist gauge trans- formations whiche keep the background tetrad field h and act on the effective fields by the general covariant transformation law. We come to Logunov’s Relativistic Theory of Gravity generalized to dynamic connections and fermion fields. arXiv:gr-qc/9709054v1 21 Sep 1997 1 Introduction Existence of Dirac fermion fields implies that, if a world manifold X is non-compact in order to satisfy causility conditions, it is parallelizable, that is, the tangent bundle TX is trivial and the principal bundle LX of oriented frames in TX admits a global section [1]. Dirac spinors are defined as follows [2, 3]. Let M be the Minkowski space with the metric η = diag(1, −1, −1, −1), a written with respect to a basis {e }. -
The Hamiltonian Field Theory of the Von Mises Wave Equation: Analytical and Computational Issues
Commun. Comput. Phys. Vol. 19, No. 3, pp. 758-769 doi: 10.4208/cicp.101114.140715a March 2016 The Hamiltonian Field Theory of the Von Mises Wave Equation: Analytical and Computational Issues 1,2, 1,2 Christian Cherubini ∗ and Simonetta Filippi 1 Unit of Nonlinear Physics and Mathematical Modeling, Campus Bio-Medico, University of Rome, I-00128, Rome, Italy. 2 International Center for Relativistic Astrophysics – I.C.R.A., Campus Bio-Medico, University of Rome, I-00128, Rome, Italy. Communicated by Lianjie Huang Received 10 November 2014; Accepted (in revised version) 14 July 2015 Abstract. The Von Mises quasi-linear second order wave equation, which completely describes an irrotational, compressible and barotropic classical perfect fluid, can be de- rived from a nontrivial least action principle for the velocity scalar potential only, in contrast to existing analog formulations which are expressed in terms of coupled den- sity and velocity fields. In this article, the classical Hamiltonian field theory specifically associated to such an equation is developed in the polytropic case and numerically verified in a simplified situation. The existence of such a mathematical structure sug- gests new theoretical schemes possibly useful for performing numerical integrations of fluid dynamical equations. Moreover it justifies possible new functional forms for La- grangian densities and associated Hamiltonian functions in other theoretical classical physics contexts. AMS subject classifications: 76A02, 65M60, 37K05 PACS: 47.40.-x, 02.30.Jr, 11.10.Ef, 04.90.+e Key words: Classical field theories, fluid dynamics, partial differential equations, numerical methods. 1 Introduction The dynamics of non viscous barotropic fluids is characterized by well defined Lagrangian [1] as well as Hamiltonian classical field theory formulations [1–6] developed in terms of coupled density and velocity fields.