Quantum Field Theory
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Geometric Approaches to Quantum Field Theory
GEOMETRIC APPROACHES TO QUANTUM FIELD THEORY A thesis submitted to The University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science and Engineering 2020 Kieran T. O. Finn School of Physics and Astronomy Supervised by Professor Apostolos Pilaftsis BLANK PAGE 2 Contents Abstract 7 Declaration 9 Copyright 11 Acknowledgements 13 Publications by the Author 15 1 Introduction 19 1.1 Unit Independence . 20 1.2 Reparametrisation Invariance in Quantum Field Theories . 24 1.3 Example: Complex Scalar Field . 25 1.4 Outline . 31 1.5 Conventions . 34 2 Field Space Covariance 35 2.1 Riemannian Geometry . 35 2.1.1 Manifolds . 35 2.1.2 Tensors . 36 2.1.3 Connections and the Covariant Derivative . 37 2.1.4 Distances on the Manifold . 38 2.1.5 Curvature of a Manifold . 39 2.1.6 Local Normal Coordinates and the Vielbein Formalism 41 2.1.7 Submanifolds and Induced Metrics . 42 2.1.8 The Geodesic Equation . 42 2.1.9 Isometries . 43 2.2 The Field Space . 44 2.2.1 Interpretation of the Field Space . 48 3 2.3 The Configuration Space . 50 2.4 Parametrisation Dependence of Standard Approaches to Quan- tum Field Theory . 52 2.4.1 Feynman Diagrams . 53 2.4.2 The Effective Action . 56 2.5 Covariant Approaches to Quantum Field Theory . 59 2.5.1 Covariant Feynman Diagrams . 59 2.5.2 The Vilkovisky–DeWitt Effective Action . 62 2.6 Example: Complex Scalar Field . 66 3 Frame Covariance in Quantum Gravity 69 3.1 The Cosmological Frame Problem . -
Arxiv:2011.03527V1 [Hep-Th] 12 Oct 2020 Uaino Eszmn W)Mdl1,Aconformal a of Model[1], Structure (WZ) Wess-Zumino of Mulation Does There and Partner
Supersymmetry in Hadron Spectroscopy and Solitons in 2 Dimensional Fermionic Media Sadataka Furui (Formerly) Graduate School of Science and Engineering, Teikyo University; [email protected] (Dated: November 9, 2020) An overview of hadron spectroscopy obtained from an extension of superconformal field theory of de Alfaro, Fubini and Furlan, which originates from the supergauge transformation of Wess and Zumino and influenced on the string theory are preseted. The standard model of hadron dynamics is based on gauge theory. In the 4 dimensional (4D) QCD Lagrangian, the Faddeev-Popov ghost that appears to compensate unphysical degrees of freedom in internal fermion propagators can be regarded as the partner of a fermion. Donaldson formulated a 5D gauge theory. Fubini and Rabinovici formulated superconformal quantum mechanics in de Sitter space, and Brodsky and de T´eramond formulated relativistic light front holographic QCD (LFHQCD) in Kaluza-Klein type 5D anti de Sitter (AdS) space, using Dirac’s light cone gauge, and succeeded in obtaining spectroscopy of baryons (fermions) and corresponding mesons (bosons), except the pion which is an axial scalar meson that remains massless. In the LFHQCD, the time parameter τ = t z/c, (z C) appears by the choice of the light-cone gauge, or the front form of Dirac’s formulation.± The theory∈ is an extension of AdS/CFT correspo- dence of Maldacena in which incorporation of Einstein’s gravity theory in Maxwell’s electromagnetic theory was tried in the framework of the string theory. Taking mathematical frameworks of treating the supersymmetry by Brodsky, Witten, Connes and their collaborators as central thema, we summarize other supersymmetric approaches like string theory in particle physics and astrophysics. -
Drawing Theories Apart: the Dispersion of Feynman Diagrams In
Drawing Theories Apart Drawing Theories Apart The Dispersion of Feynman Diagrams in Postwar Physics david kaiser The University of Chicago Press chicago and london David Kaiser is associate professor in the Program in Science, Technology, and Society and lecturer in the Department of Physics at the Massachusetts Institute of Technology. The University of Chicago Press, Chicago 60637 The University of Chicago Press, Ltd., London C 2005 by The University of Chicago All rights reserved. Published 2005 Printed in the United States of America 14 13 12 11 10 09 08 07 06 05 1 2 3 4 5 isbn: 0-226-42266-6 (cloth) isbn: 0-226-42267-4 (paper) Library of Congress Cataloging-in-Publication Data Kaiser, David. Drawing theories apart : the dispersion of Feynman diagrams in postwar physics / David Kaiser. p. cm. Includes bibliographical references and index. isbn 0-226-42266-6 (alk. paper)—isbn 0-226-42267-4 (pbk. : alk. paper) 1. Feynman diagrams. 2. Physics—United States—History— 20th century. 3. Physics—History—20th century. I. Title. QC794.6.F4K35 2005 530.0973 0904—dc22 2004023335 ∞ The paper used in this publication meets the minimum requirements of the American National Standard for Information Sciences—Permanence of Paper for Printed Library Materials, ansi z39.48-1992. He teaches his theory, not as a body of fact, but as a set of tools, to be used, and which he has actually used in his work. John Clarke Slater describing Percy Bridgman after Bridgman won the Nobel Prize in Physics, 11 January 1947 Contents Preface and Acknowledgments xi Abbreviations xvii Chapter 1 . -
Dirac: a Scientific Biography in Geneva, Bringing His Wife and Two Children with Him
Dirac A SCIENTIFIC BIOGRAPHY HELGE KRAGH CAMBRIDGE UNIVERSITY PRESS Published by the Press Syndic:a1e or the Uaiversity ol Cambridge The Pin Building, Tru~ton Streec, Cambridge CB2 1RP 40 West200l Slrl:el, New York. NY 10011-4211. USA 10 Stamford Road, Oakleigh, Victoria 3166, Australia 0 Cambridge Uaiversity Press 1990 First published 1990 Reprinted 1991, 1992 Printed in the Uaited Stales of America Ubrary of Congress Clllaloging-in-Publiwtion Data Kragh, Helge, 1944- Dirac : a scientific biography I He1ge Kragh. P. em. Bibliography: p. Includes indexes. ISBN M21-38089-8 I. Dirac, P. A.M. (Paul AdrieD Maurice), 1902- 2. Physicists- Great Britaia - Biography. I. Tille. DCI6.051K73 1990 530'.092- dc20 [B) 89-17257 CIP ISBN 0.521-38089-8 hardback TO BODIL CONTENTS Preface page lX I. Early years 2. Discovery of quantum mechanics 14 3. Relativity and spinning electrons 48 4. Travels and thinking 67 5. The dream of philosophers 87 6. Quanta and fields 118 7. Fifty years of a physicist's life 151 8. ''The so-called quantum electrodynamics" 165 9. Electrons and ether 189 10. Just a disappointment 205 II. Adventures in cosmology 223 12. The purest soul 247 13. Philosophy in physics 260 14. The principle of mathematical beauty 275 Appendix I. Dirac bibliometrics 293 Appendix II. Bibliography of P. A. M. Dirac 304 Notes and references 315 General bibliography 364 Index of names 383 Index ofsubjects 387 vii PREFACE NE of the greatest physicists who ever lived, P. A. M. Dirac 0 ( 1902-84) made contributions that may well be compared with those of other, better known giants of science such as Newton, Maxwell, Einstein, and Bohr. -
A Closer Look at Gluons 3
A Closer Look at Gluons 1 Chapter 6 ACLOSER LOOK AT GLUONS Sadataka Furui* (formerly) Teikyo University, Graduate School of Science and Engineering Utsunomiya, Tochigi, Japan Abstract: Gluons are strong interaction gauge fields which interact between quarks, i.e. constituents of baryons and mesons. Interaction of matters is phenomenologically described by gauge theory of strong, electromagnetic, weak and gravitational interactions. In electro- weak theory, left handed leptons lL and neutrino νL, right handed leptons lR and left handed quarks u , d and right handed quarks u , d follow SU(2) U(1) symmetry. Charge of L L R R × leptons and quarks define hypercharge Y , and via Higgs mechanism SU(2) U(1) L × Y symmetry forms U(1)em symmetry. Presence of J P = 3/2+ baryons, or N ∗++ uuu suggests a new degree of free- ∼ dom “color” for quarks, which follows SU(3) symmetry group. Hence the gluon fields are expressed as a where a = 1, 2, , 8 specify the color SU(3) bases, and µ are 4- Aµ · · · dimensional space-time coordinates. The quantum electrodynamics was extended to quan- tum chromo dynamics (QCD). Since ∂µ a is not a free field, the gauge theory requires Aµ ghosts that compensates unphysical degrees of freedom of gluons. Gluons, ghosts, leptons and quarks are related by Becchi-Rouet-Stora -Tyuitin (BRST) transformation of electro- weak and strong interaction of the U(1) SU(2) SU(3) symmetric Faddeev-Popov × × Lagrangian. In order to describe Hadron dynamics properly, embedding of 4-dimensional space to 5-dimensional space was tried in lattice simulations, and in light front holographic QCD (LFHQCD) approach in which conformally symmetric light-front dynamics without ghost are embedded in AdS5, and a parameter that fixes a mass scale was chosen from the Prin- ciple of Maximum Conformality. -
Werner Heisenberg's Ideas on Particle Physics in the Light of Recent Achievements Konrad Bleuler Institut Für Theoretische Kernphysik, Universität Bonn
Werner Heisenberg's Ideas on Particle Physics in the Light of Recent Achievements Konrad Bleuler Institut für Theoretische Kernphysik, Universität Bonn Z. Naturforsch. 45a, 1051-1058 (1990); received December 26, 1989 Dedicated to Hans-Peter Dürr on the occasion of his 60th birthday W. Heisenberg's unconventional - at his time even revolutionary - ideas on "elementary" particles are reviewed - or, better, rediscovered - within the framework of present day gauge- and string- theories. I. Looking back at a period long past, i.e. more than atmosphere and the outer space) by Pacini, Piccioni four decades ago, unforgettable hours of private dis- and Conversi (difficult to pronounce - the third name cussions with Werner Heisenberg come back to my was usually replaced by "Conversly" - and even more mind: I now realize that his then perfectly new physi- difficult to understand) had clearly shown that the cal ideas which, at that time, were not really appreci- then well-known cosmic ray particles (later on called ated, acquire by now, i.e. if re-interpreted in the light muons), whose mass and creation cross sections were of recent and breathtaking developments, a perfectly apparently in accordance with Yukawa's famous pre- new and unexpected physical meaning. In other diction, did, however, not exhibit the expected rela- words: Heisenberg's original principles [1] - if re- tively large absorption cross section as foreseen formulated within the framework of present day new through a most elementary quantum-mechanical and far reaching mathematical concepts and experi- argument. mental achievements - represent, in fact, an intuitive After a few months of questioning and discussions, insight into a future theory long before time was ready Occhialini's wonderful combined photographic plates for its implementation. -
Werner Heisenberg's Ideas on Particle Physics in the Light of Recent Achievements Konrad Bleuler Institut Für Theoretische Kernphysik, Universität Bonn
Werner Heisenberg's Ideas on Particle Physics in the Light of Recent Achievements Konrad Bleuler Institut für Theoretische Kernphysik, Universität Bonn Z. Naturforsch. 45a, 1051-1058 (1990); received December 26, 1989 Dedicated to Hans-Peter Dürr on the occasion of his 60th birthday W. Heisenberg's unconventional - at his time even revolutionary - ideas on "elementary" particles are reviewed - or, better, rediscovered - within the framework of present day gauge- and string- theories. I. Looking back at a period long past, i.e. more than atmosphere and the outer space) by Pacini, Piccioni four decades ago, unforgettable hours of private dis- and Conversi (difficult to pronounce - the third name cussions with Werner Heisenberg come back to my was usually replaced by "Conversly" - and even more mind: I now realize that his then perfectly new physi- difficult to understand) had clearly shown that the cal ideas which, at that time, were not really appreci- then well-known cosmic ray particles (later on called ated, acquire by now, i.e. if re-interpreted in the light muons), whose mass and creation cross sections were of recent and breathtaking developments, a perfectly apparently in accordance with Yukawa's famous pre- new and unexpected physical meaning. In other diction, did, however, not exhibit the expected rela- words: Heisenberg's original principles [1] - if re- tively large absorption cross section as foreseen formulated within the framework of present day new through a most elementary quantum-mechanical and far reaching mathematical concepts and experi- argument. mental achievements - represent, in fact, an intuitive After a few months of questioning and discussions, insight into a future theory long before time was ready Occhialini's wonderful combined photographic plates for its implementation. -
Chapter 8 Maxwell Field
Chapter 8 Maxwell Field All electrodynamic processes are described by the Maxwell equations. Surprisingly they repre- sent the equations of motion of afirst-quantized theory, although the Planck constant ¯h does not appear explicitly. This apparent contradiction is resolved by the following consideration. If the quanta of the Maxwellfield, i.e. the photons, had afinite rest massM, then it would appear due to dimensional reasons together with spatio-temporal derivatives as a mass term in the equations of motion in form of the inverse Compton wave length (7.21). Thus, performing the limit of a vanishing rest mass, i.e.M 0, also the Planck constant ¯h vanishes automatically → from the respective equations of motion. In this chapter wefirst review the relativistic covariant formulation of thisfirst-quantized Maxwell theory. Afterwards, we invoke the canonicalfield quantization formalism and work out systematically the second quantization of the Maxwell theory. In particular, we have to deal with the intricate consequences of the underlying local gauge symmetry, which is due to the vanishing rest mass of the quanta of the Maxwellfield. In this way we determine step by step the respective properties of a single photon as, for instance, its energy, its momentum, and its spin. Finally, we discuss the photon propagator, which represents an important building block in the Feynman diagrams of quantum electrodynamics describing the interaction between light and matter. 8.1 Maxwell Equations Forces of an electromagneticfield upon electric charges, which are at rest or move, are mediated by both the electricfield strengthE and the magnetic inductionB. Physically both vector fields are generated by the charge densityρ and the current densityj.