DOCSLIB.ORG

Table of Contents

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Table of Contents Contents Part I. Introduction Prologue ................................................... 1 1. Mathematical Principles of Modern Natural Philosophy . 11 1.1 BasicPrinciples ................................. ....... 12 1.2 The Infinitesimal Strategy and Differential Equations . ..... 14 1.3 The Optimality Principle.......................... ...... 14 1.4 The Basic Notion of Action in Physics and the Idea of Quantization....................................... .... 15 1.5 The Method of the Green’s Function................... ... 17 1.6 Harmonic Analysis and the Fourier Method . ... 21 1.7 The Method of Averaging and the Theory of Distributions . .. 26 1.8 The SymbolicMethod ............................... ... 28 1.9 Gauge Theory – Local Symmetry and the Description of Interactions by Gauge Fields.......................... ... 34 1.10 The Challenge of Dark Matter ....................... .... 46 2. The Basic Strategy of Extracting Finite Information from Infinities – Ariadne’s Thread in Renormalization Theory . 47 2.1 Renormalization Theory in a Nutshell................ ..... 48 2.1.1 Effective Frequency and Running Coupling Constant of an Anharmonic Oscillator ....................... 48 2.1.2 The Zeta Function and Riemann’s Idea of Analytic Continuation .................................... 54 2.1.3 Meromorphic Functions and Mittag-Leffler’s Idea ofSubtractions................................... 56 2.1.4 The Square of the Dirac Delta Function . 58 2.2 Regularization of Divergent Integrals in Baby Renormalization Theory.............................. ... 60 2.2.1 Momentum Cut-off and the Method of Power-Counting 60 2.2.2 The Choice of the Normalization Momentum . 63 2.2.3 The Method of Differentiating Parameter Integrals . 63 2.2.4 The Method of Taylor Subtraction.................. 64 2.2.5 Overlapping Divergences ........................ .. 65 XXVI Contents 2.2.6 The Role of Counterterms ......................... 67 2.2.7 Euler’s Gamma Function .......................... 67 2.2.8 Integration Tricks ............................. ... 69 2.2.9 Dimensional Regularization via Analytic Continuation 73 2.2.10 Pauli–Villars Regularization .................. ..... 76 2.2.11 Analytic Regularization ....................... .... 77 2.2.12 Application to Algebraic Feynman Integrals in MinkowskiSpace................................. 80 2.2.13 Distribution-Valued Meromorphic Functions . ... 81 2.2.14 Application to Newton’s Equation of Motion . 87 2.2.15 Hints for Further Reading....................... ... 92 2.3 Further Regularization Methods in Mathematics. ..... 93 2.3.1 Euler’s Philosophy............................. ... 93 2.3.2 Adiabatic Regularization of Divergent Series . ... 94 2.3.3 Adiabatic Regularization of Oscillating Integrals. ..... 95 2.3.4 Regularization by Averaging ..................... .. 96 2.3.5 Borel Regularization........................... ... 98 2.3.6 Hadamard’s Finite Part of Divergent Integrals . 100 2.3.7 Infinite-Dimensional Gaussian Integrals and the Zeta Function Regularization........................... 101 2.4 Trouble in Mathematics ............................ ..... 102 2.4.1 Interchanging Limits........................... ... 102 2.4.2 The Ambiguity of Regularization Methods . 104 2.4.3 Pseudo-Convergence ............................ .. 104 2.4.4 Ill-PosedProblems ............................. .. 105 2.5 Mathemagics ..................................... ..... 109 3. The Power of Combinatorics .............................. 115 3.1 Algebras ........................................ ...... 115 3.2 The Algebra of Multilinear Functionals . ...... 117 3.3 Fusion, Splitting, and Hopf Algebras . ...... 122 3.3.1 The Bialgebra of Linear Differential Operators .. .. 123 3.3.2 The Definition of Hopf Algebras.................... 128 3.4 Power Series Expansion and Hopf Algebras ............. ... 131 3.4.1 The Importance of Cancellations .................. 131 3.4.2 The Kepler Equation and the Lagrange InversionFormula ................................ 132 3.4.3 The Composition Formula for Power Series . 134 3.4.4 The Fa`adi Bruno Hopf Algebra for the Formal Diffeomorphism Group of the Complex Plane . 136 3.4.5 The Generalized Zimmermann Forest Formula . 138 3.4.6 The Logarithmic Function and Schur Polynomials . 140 3.4.7 Correlation Functions in Quantum Field Theory . 141 3.4.8 Random Variables, Moments, and Cumulants . 143 3.5 Symmetry and Hopf Algebras......................... ... 146 Contents XXVII 3.5.1 The Strategy of Coordinatization in Mathematics and Physics ......................................... 146 3.5.2 The Coordinate Hopf Algebra of a Finite Group . 148 3.5.3 The Coordinate Hopf Algebra of an Operator Group.. 150 3.5.4 The Tannaka–Krein Duality for Compact Lie Groups . 152 3.6 Regularization and Rota–Baxter Algebras ............ ..... 154 3.6.1 Regularization of the Laurent Series .............. .. 157 3.6.2 Projection Operators ........................... .. 158 3.6.3 The q-Integral ................................... 158 3.6.4 The Volterra–Spitzer Exponential Formula . 160 3.6.5 The Importance of the Exponential Function in Mathematics and Physics.......................... 161 3.7 Partially Ordered Sets and Combinatorics ............ ..... 162 3.7.1 Incidence Algebras and the Zeta Function . 162 3.7.2 The M¨obius Function as an Inverse Function . 163 3.7.3 The Inclusion–Exclusion Principle in Combinatorics .. 164 3.7.4 Applications to Number Theory.................... 166 3.8 Hints for Further Reading .......................... ..... 167 4. The Strategy of Equivalence Classes in Mathematics ...... 175 4.1 Equivalence Classes in Algebra ..................... ...... 178 4.1.1 The Gaussian Quotient Ring and the Quadratic Reciprocity Law in Number Theory................. 178 4.1.2 Application of the Fermat–Euler Theorem in Coding Theory.......................................... 182 4.1.3 Quotient Rings, Quotient Groups, and Quotient Fields 184 4.1.4 Linear Quotient Spaces .......................... 188 4.1.5 Ideals and Quotient Algebras ..................... 190 4.2 Superfunctions and the Heaviside Calculus in Electrical Engineering........................................ .... 191 4.3 Equivalence Classes in Geometry .................... ..... 194 4.3.1 The Basic Idea of Geometry Epitomized by Klein’s ErlangenProgram................................ 194 4.3.2 Symmetry Spaces, Orbit Spaces, and Homogeneous Spaces .......................................... 194 4.3.3 The Space of Quantum States...................... 199 4.3.4 Real ProjectiveSpaces .......................... .. 200 4.3.5 Complex Projective Spaces ....................... 203 4.3.6 The Shape of the Universe......................... 204 4.4 Equivalence Classes in Topology.................... ...... 205 4.4.1 Topological Quotient Spaces ..................... .. 205 4.4.2 Physical Fields, Observers, Bundles, and Cocycles . .. 208 4.4.3 Generalized Physical Fields and Sheaves . .. 216 4.4.4 Deformations, Mapping Classes, and Topological Charges......................................... 219 XXVIII Contents 4.4.5 Poincar´e’s Fundamental Group . .. 223 4.4.6 Loop Spaces and Higher Homotopy Groups .......... 225 4.4.7 Homology, Cohomology, and Electrodynamics . 227 4.4.8 Bott’s Periodicity Theorem ...................... .. 227 4.4.9 K-Theory ....................................... 228 4.4.10 Application to Fredholm Operators ............... .. 233 4.4.11 Hints for Further Reading ....................... .. 235 4.5 The Strategy of Partial Ordering .................... ..... 237 4.5.1 Feynman Diagrams............................... 238 4.5.2 The Abstract Entropy Principle in Thermodynamics . 239 4.5.3 Convergence of Generalized Sequences ............. 240 4.5.4 Inductive and Projective Topologies .............. .. 241 4.5.5 Inductive and Projective Limits . .. 243 4.5.6 Classes, Sets, and Non-Sets ...................... .. 245 4.5.7 The Fixed-Point Theorem of Bourbaki–Kneser ....... 247 4.5.8 Zorn’sLemma ................................... 248 4.6 Leibniz’s Infinitesimals and Non-Standard Analysis . ...... 248 4.6.1 Filters and Ultrafilters ......................... ... 250 4.6.2 The Full-Rigged Real Line........................ 251 Part II. Basic Ideas in Classical Mechanics 5. Geometrical Optics ....................................... 263 5.1 Ariadne’s Thread in Geometrical Optics .............. ..... 264 5.2 Fermat’s Principle of Least Time .................... ..... 268 5.3 Huygens’ Principle on Wave Fronts ................... .... 270 5.4 Carath´eodory’s Royal Road to Geometrical Optics . ..... 271 5.5 The Duality between Light Rays and Wave Fronts . 274 5.5.1 From Wave Fronts to Light Rays ................... 275 5.5.2 From Light Rays to Wave Fronts ................... 276 5.6 The Jacobi Approach to Focal Points .................. ... 276 5.7 Lie’s ContactGeometry ............................ ..... 279 5.7.1 BasicIdeas .................................... .. 279 5.7.2 Contact Manifolds and Contact Transformations . 283 5.7.3 Applications to Geometrical Optics . .. 284 5.7.4 Equilibrium Thermodynamics and Legendre Submanifolds .................................... 285 5.8 Light Rays and Non-Euclidean Geometry............... ... 289 5.8.1 Linear Symplectic Spaces........................ .. 291 5.8.2 The K¨ahler Form of a Complex Hilbert Space........ 295 5.8.3 The Refraction Index and Geodesics ................ 297 5.8.4 The Trick of Gauge Fixing......................... 299 5.8.5 GeodesicFlow .................................. 299 5.8.6 Hamilton’s Duality Trick and
Load more

Recommended publications
  • Something Is Rotten in the State of QED
    February 2020 Something is rotten in the state of QED Oliver Consa Independent Researcher, Barcelona, Spain Email: [email protected] Quantum electrodynamics (QED) is considered the most accurate theory in the his- tory of science. However, this precision is based on a single experimental value: the anomalous magnetic moment of the electron (g-factor). An examination of QED history reveals that this value was obtained using illegitimate mathematical traps, manipula- tions and tricks. These traps included the fraud of Kroll & Karplus, who acknowledged that they lied in their presentation of the most relevant calculation in QED history. As we will demonstrate in this paper, the Kroll & Karplus scandal was not a unique event. Instead, the scandal represented the fraudulent manner in which physics has been conducted from the creation of QED through today. 1 Introduction truth the hypotheses of the former members of the Manhat- tan Project were rewarded with positions of responsibility in After the end of World War II, American physicists organized research centers, while those who criticized their work were a series of three transcendent conferences for the separated and ostracized. The devil’s seed had been planted development of modern physics: Shelter Island (1947), in the scientific community, and its inevitable consequences Pocono (1948) and Oldstone (1949). These conferences were would soon grow and flourish. intended to be a continuation of the mythical Solvay confer- ences. But, after World War II, the world had changed. 2 Shelter Island (1947) The launch of the atomic bombs in Hiroshima and 2.1 The problem of infinities Nagasaki (1945), followed by the immediate surrender of Japan, made the Manhattan Project scientists true war heroes.
    [Show full text]
  • Lecture Notes on Quantum Field Theory
    Draft for Internal Circulations: v1: Spring Semester, 2012, v2: Spring Semester, 2013 v3: Spring Semester, 2014, v4: Spring Semester, 2015 Lecture Notes on Quantum Field Theory Yong Zhang1 School of Physics and Technology, Wuhan University, China (Spring 2015) Abstract These lectures notes are written for both third-year undergraduate students and first-year graduate students in the School of Physics and Technology, University Wuhan, China. They are mainly based on lecture notes of Sidney Coleman from Harvard and lecture notes of Michael Luke from Toronto and lecture notes of David Tong from Cambridge and the standard textbook of Peskin & Schroeder, so I do not claim any originality. These notes certainly have all kinds of typos or errors, so they will be updated from time to time. I do take the full responsibility for all kinds of typos or errors (besides errors in English writing), and please let me know of them. The third version of these notes are typeset by a team including: Kun Zhang2, Graduate student (Id: 2013202020002), Participant in the Spring semester, 2012 and 2014 Yu-Jiang Bi3, Graduate student (Id: 2013202020004), Participant in the Spring semester, 2012 and 2014 The fourth version of these notes are typeset by Kun Zhang, Graduate student (Id: 2013202020002), Participant in the Spring semester, 2012 and 2014-2015 1yong [email protected] 2kun [email protected] [email protected] 1 Draft for Internal Circulations: v1: Spring Semester, 2012, v2: Spring Semester, 2013 v3: Spring Semester, 2014, v4: Spring Semester, 2015 Acknowledgements I thank all participants in class including advanced undergraduate students, first-year graduate students and French students for their patience and persis- tence and for their various enlightening questions.
    [Show full text]
  • Quantum Electrodynamics - Wikipedia, the Free Encyclopedia Page 1 of 17
    Quantum electrodynamics - Wikipedia, the free encyclopedia Page 1 of 17 Quantum electrodynamics From Wikipedia, the free encyclopedia Quantum electrodynamics (QED ) is the relativistic quantum Quantum field theory field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved. QED mathematically describes all phenomena involving electrically charged particles interacting by means of exchange of photons and represents the quantum counterpart of classical electrodynamics giving a complete account of matter and light interaction. One of the founding fathers of QED, Richard Feynman, has called it "the jewel of physics" for its extremely accurate predictions of quantities like the anomalous magnetic (Feynman diagram) moment of the electron, and the Lamb shift of the energy levels History of... of hydrogen.[1] Background In technical terms, QED can be described as a perturbation theory Gauge theory of the electromagnetic quantum vacuum. Field theory Poincaré symmetry Quantum mechanics Contents Spontaneous symmetry breaking Symmetries ■ 1 History ■ 2 Feynman's view of quantum electrodynamics Crossing ■ 2.1 Introduction Charge conjugation ■ 2.2 Basic constructions Parity ■ 2.3 Probability amplitudes Time reversal ■ 2.4 Propagators ■ 2.5 Mass renormalization Tools ■ 2.6 Conclusions Anomaly ■ 3 Mathematics Effective field theory ■ 3.1 Equations of motion Expectation value ■ 3.2 Interaction picture Faddeev–Popov ghosts
    [Show full text]
  • On Perturbation Theory, Dyson Series, and Feynman Diagrams
    On Perturbation Theory, Dyson Series, and Feynman Diagrams The Interaction Picture of QM and the Dyson Series Many quantum systems have Hamiltonians of the form Hˆ = Hˆ0 + Vˆ where Hˆ0 is a free Hamiltonian with known spectrum — which is used to classify the states of the full theory — while Vˆ is a perturbation which causes transitions between the eigenstates of the Hˆ0. For example, in scattering theory ˆ 2 Preduced Hˆ0 = , Vˆ = potential V (xˆrelative). (1) 2Mreduced Similarly, for the quantum scalar field Φˆ with a (λ/24)Φˆ 4 self-interaction we have 2 m2 d3p 1 ˆ 3x 1 ˆ 2 x 1 ˆ x ˆ 2 † H0 = d 2 Π ( ) + 2 ∇Φ( ) + Φ = 3 Epaˆpaˆp + const 2 (2π) 2Ep Z Z λ while Vˆ = d3x Φˆ 4(x). 24 Z (2) To study the transitions (scattering, making new particles, decays, etc.) caused by Vˆ we want to use a fixed basis of Hˆ0 eigenstates, but we want to keep the transitions separate from the ordinary wave-packet motion and spreading — which are due to the Schr¨odinger phases e−iEt of the Hˆ0 itself. The picture of QM which separates these effects is the interaction picture. In the Schr¨odinger picture, the operators are time-independent while the quantum states −iHtˆ evolve with time as |ψiS(t)= e |ψi(0). In the Heisenberg picture it’s the other way around — the quantum states are time independent while the operators evolve with time — and the two pictures are related by the time-dependent unitary operator eiHtˆ , +iHtˆ ˆ +iHtˆ ˆ −iHtˆ |ΨiH (t) = e |ΨiS(t) ≡ |ΨiS(0) ∀t, AH (t) = e ASe .
    [Show full text]
  • Quantum Field Theory Notes
    QFT Lectures Notes Jared Kaplan Department of Physics and Astronomy, Johns Hopkins University Abstract Various QFT lecture notes to be supplemented by other course materials. Contents 1 Introduction 2 1.1 What are we studying? . .2 1.2 Useful References and Texts . .3 2 Fall Semester 4 2.1 Philosophy . .4 2.2 Review of Lagrangians and Harmonic Oscillators . .8 2.3 Balls and Springs . 11 2.4 Special Relativity and Anti-particles . 17 2.5 Canonical Quantization and Noether's Theorem . 24 2.6 Free Scalar Quantum Field Theory with Special Relativity . 27 2.7 Dimensional Analysis, or Which Interactions Are Important? . 33 2.8 Interactions in Classical Field Theory with a View Towards QFT . 36 2.9 Overview of Scattering and Perturbation Theory . 39 2.10 Relating the S-Matrix to Cross Sections and Decay Rates . 41 2.11 Old Fashioned Perturbation Theory . 46 2.12 LSZ Reduction Formula { S-Matrix from Correlation Functions . 51 2.13 Feynman Rules as a Generalization of Classical Physics . 56 2.14 The Hamiltonian Formalism for Perturbation Theory . 59 2.15 Feynman Rules from the Hamiltonian Formalism . 64 2.16 Particles with Spin 1 . 72 2.17 Covariant Derivatives and Scalar QED . 78 2.18 Scattering and Ward Identities in Scalar QED . 83 2.19 Spinors and QED . 87 2.20 Quantization and Feynman Rules for QED . 95 2.21 Overview of `Renormalization' . 100 2.22 Casimir Effect . 102 2.23 The Exact 2-pt Correlator { Kallen-Lehmann Representation . 106 2.24 Basic Loop Effects { the 2-pt Correlator . 109 2.25 General Loop Effects, Renormalization, and Interpretation .
    [Show full text]
  • Quantum Field Theory II an Introduction to Feynman Diagrams
    Quantum Field Theory II An introduction to Feynman diagrams A course for MPAGS Dr Sam T Carr University of Birmingham [email protected] February 9, 2009 Preface This course follows on from Quantum Field Theory I, and will introduce more advanced techniques such as Green’s functions, and Feynman diagrams. It will begin with outlining the formalism required for such an approach, and follow on by applying the formalism to many examples within condensed matter physics, such as Fermi-liquid theory, disordered systems, and phonon interactions. Prerequisites: A familiarity with basic complex analysis - such as the residue theorem, and eval- • uation of integrals by completing the contour in the upper or lower half planes. A familiarity with Fourier analysis - both Fourier series and Fourier transforms. • The basics of second quantization, as taught last term in the Quantum Field • Theory I module. The course will also suppose some knowledge of condensed matter physics, at an un- dergraduate level (e.g. that given in either Kittel or Ashcroft and Mermin). While not strictly speaking necessary to follow the course, the entire content will be somewhat obscured if the reader has never met condensed matter physics before. Suggested further reading Mattuck: A guide to Feynman diagrams in the many body problem - a very • pedagogical introduction to the concepts behind diagrams. However, sometimes overly wordy when it comes to learning to actually calculate things. Mahan: Many Particle systems - contains basically everything in the subject area. • Very terse to read, and too big to learn directly from, but as a reference book, is one of the best available.
    [Show full text]
  • Feynman Diagrams for Beginners∗
    Feynman Diagrams for Beginners∗ Krešimir Kumerickiˇ y Department of Physics, Faculty of Science, University of Zagreb, Croatia Abstract We give a short introduction to Feynman diagrams, with many exer- cises. Text is targeted at students who had little or no prior exposure to quantum field theory. We present condensed description of single-particle Dirac equation, free quantum fields and construction of Feynman amplitude using Feynman diagrams. As an example, we give a detailed calculation of cross-section for annihilation of electron and positron into a muon pair. We also show how such calculations are done with the aid of computer. Contents 1 Natural units 2 2 Single-particle Dirac equation 4 2.1 The Dirac equation . 4 2.2 The adjoint Dirac equation and the Dirac current . 6 2.3 Free-particle solutions of the Dirac equation . 6 3 Free quantum fields 9 3.1 Spin 0: scalar field . 10 3.2 Spin 1/2: the Dirac field . 10 3.3 Spin 1: vector field . 10 arXiv:1602.04182v1 [physics.ed-ph] 8 Feb 2016 4 Golden rules for decays and scatterings 11 5 Feynman diagrams 13 ∗Notes for the exercises at the Adriatic School on Particle Physics and Physics Informatics, 11 – 21 Sep 2001, Split, Croatia [email protected] 1 2 1 Natural units 6 Example: e+e− ! µ+µ− in QED 16 6.1 Summing over polarizations . 17 6.2 Casimir trick . 18 6.3 Traces and contraction identities of γ matrices . 18 6.4 Kinematics in the center-of-mass frame . 20 6.5 Integration over two-particle phase space .
    [Show full text]
  • Lightcone Effective Hamiltonians and RG Flows
    CERN-TH-2018-030 Lightcone Effective Hamiltonians and RG Flows A. Liam Fitzpatrick1, Jared Kaplan2, Emanuel Katz1, Lorenzo G. Vitale1, Matthew T. Walters1;3 1 Department of Physics, Boston University, Boston, MA 02215 2 Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218 Center for Quantum Mathematics and Physics (QMAP), UC Davis, CA 95616 Stanford Institute for Theoretical Physics, Stanford University, Palo Alto, CA 94305 3 CERN, Theoretical Physics Department, 1211 Geneva 23, Switzerland We present a prescription for an effective lightcone (LC) Hamiltonian that includes the effects of zero modes, focusing on the case of Conformal Field Theories (CFTs) deformed by relevant operators. We show how the prescription resolves a number of issues with LC quantization, including i) the apparent non-renormalization of the vacuum, ii) discrepancies in critical values of bare parameters in equal-time vs LC quantization, and iii) an inconsistency at large N in CFTs with simple AdS duals. We describe how LC quantization can drastically arXiv:1803.10793v2 [hep-th] 9 May 2018 simplify Hamiltonian truncation methods applied to some large N CFTs, and discuss how the prescription identifies theories where these simplifications occur. We demonstrate and check our prescription in a number of examples. May 2018 Contents 1. Introduction and Summary 2 1.1. Advantages of Lightcone Quantization . 4 1.2. The Problem of Zero Modes . 7 1.3. Our Prescription . 9 2. Lightcone Quantization and its Discontents 11 2.1. CFT Definition of Zero Modes . 11 2.2. Concrete Bare Parameter Discrepancies . 14 2.3. A Problem with Holographic Models . 15 2.4.
    [Show full text]