DOCSLIB.ORG

Relativistic Quantum Mechanics and Field Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Relativistic Quantum Mechanics and Field Theory Relativistic Quantum Mechanics and Field Theory FRANZ GROSS College of William and Mary Williamsburg, Virginia and Continuous Electron Beam Accelerator Facility Newport News, Virginia Wiley-VCH Verlag GmbH & Co. KGaA All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: Applied for British Library Cataloging-in-Publication Data: A catalogue record for this book is available from the British Library Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliogratie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. 0 1993 by John Wiley & Sons, Inc. Wiley Professional Paperback Edition Published I999 0 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form - nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printed in the Federal Republic ofGermany Printed on acid-free paper Printing Strauss GmbH, Morlenbach Bookbinding Litges & Dopf Buchbinderei GmbH, Heppenheim ISBN-13: 978-0-47 1-35386-7 ISBN-10: 0-471 -35386-8 To my parents, Genevieve and Llewellyn and to the next generation, Glen, Sue, Kathy, Caitlin, and Christina CONTENTS Preface xiii Part I QUANTUM THEORY OF RADIATION 1 1. Quantization of the Nonrelativistic String 3 1.1 The one-dimensional classical string 3 1.2 Normal modes of the string 7 1.3 Quantization of the string 10 1.4 Canonical commutation relations 12 1.5 The number operator and phonon states 13 1.6 The quanta as particles 15 1.7 The classical limit: Field-particle duality 18 1.8 Time translation 22 Problems 25 2. Quantization of the Electromagnetic Field 28 2.1 Lorentz transformations 28 2.2 Relativistic form of Maxwell’s theory 31 2.3 Interactions between particles and fields 39 2.4 Plane wave expansions 42 2.5 Massive vector fields 44 2.6 Field quantization 46 2.7 Spin of the photon 50 Problems 56 3. Interaction of Radiation with Matter 57 3.1 Time evolution and the S-matrix 57 3.2 Decay rates and cross sections 65 vii viii CONTENTS 3.3 Atomic decay 69 3.4 The Lamb shift 73 3.5 Deuteron photodisintegration 81 Problems 85 Part II RELATlVlSTlC EQUATIONS 89 4. The Klein-Gordon Equation 91 4.1 The equation 91 4.2 Conserved norm 94 4.3 Solutions for free particles 95 4.4 Pair creation from a high Coulomb barrier 97 4.5 Two-component form 102 4.6 Nonrelativistic limit 106 4.7 Coulomb scattering 108 4.8 Negative energy states 110 Problems 116 5. The Dirac Equation 119 5.1 The equation 119 5.2 Conserved norm 122 5.3 Solutions for free particles 123 5.4 Charge conjugation 126 5.5 Coulomb scattering 129 5.6 Negative energy states 131 5.7 Nonrelativistic limit 134 5.8 The Lorentz group 140 5.9 Covariance of the Dirac equation 146 5.10 Bilinear covariants 152 5.11 Chirality and massless fermions 157 Problems 169 6. Application of the Dirac Equation 163 6.1 Spherically symmetric potentials 163 6.2 Hadronic structure 171 6.3 Hydrogen-like atoms 177 Problems 183 CONTENTS iX Part 111 ELEMENTS OF QUANTUM FIELD THEORY 185 7. Second Quantization 187 7.1 Schrodinger theory 188 7.2 Identical particles 191 7.3 Charged Klein-Gordon theory 194 7.4 Dirac theory 198 7.5 Interactions: An introduction 20 1 Problems 204 8. Symmetries I 206 8.1 Noether’s theorem 206 8.2 Translations 212 8.3 Transformations of states and operators 2 14 8.4 Parity 216 8.5 Charge conjugation 219 8.6 Time reversal 223 8.7 The PCT theorem 23 1 Problems 236 9. Interacting Field Theories 238 9.1 43 theory: An example 238 9.2 Relativistic decays 240 9.3 Relativistic scattering 243 9.4 Introduction to the Feynman rules 247 9.5 Calculation of the cross section 252 9.6 Effective nonrelativistic potential 257 9.7 Identical particles 25 8 9.8 Pion-nucleon interactions and isospin 264 9.9 One-pion exchange 268 9.10 Electroweak decays 273 Problems 279 10. Quantum Electrodynamics 282 10.1 The Hamiltonian 282 10.2 Photon propagator: ep scattering 286 10.3 Antiparticles: e+e- -+ p+p- 296 x CONTENTS 10.4 e+e- annihilation 304 10.5 Fermion propagator: Compton scattering 307 Problems 316 11. Loops and Introduction to Renormalization 319 11.1 Wick’s theorem 320 11.2 QED to second order 326 11.3 Electron self-energy 330 11.4 Vacuum bubbles 336 11.5 Vacuum polarization 338 11.6 Loop integrals and dimensional regularization 342 11.7 Dispersion relations 349 1 1.8 Vertex corrections 353 11.9 Charge renormalization 358 11.10 Bremsstrahlung and radiative corrections 363 Problems 37 1 12. Bound States and Unitarity 373 12.1 The ladder diagrams 374 12.2 The role of crossed ladders 382 12.3 Relativistic two-body equations 388 12.4. Normalization of bound states 39 1 12.5 The Bethe-Salpeter equation 393 12.6 The spectator equation 395 12.7 Equivalence of two-body equations 399 12.8 Unitarity 400 12.9 The Blankenbecler-Sugar equation 404 12.10 Dispersion relations and anomalous thresholds 405 Problems 41 1 Part IV SYMMETRIES AND GAUGE THEORIES 413 13. Symmetries II 415 13.1 Abelian gauge invariance 416 13.2 Non-Abelian gauge invariance 419 13.3 Yang-Mills theories 424 13.4 Chiral symmetry 421 13.5 The linear sigma model 430 CONTENTS xi 13.6 Spontaneous symmetry breaking 434 13.7 The non-linear sigma model 437 13.8 Chiral symmetry breaking and PCAC 441 Problems 446 14. Path Integrals 448 14.1 The wave function and the propagator 449 14.2 The S-Matrix 454 14.3 Time-ordered products 459 14.4 Path integrals for scalar field theories 463 14.5 Loop diagrams in 43 theory 475 14.6 Fermions 482 Problems 490 15. Quantum Chromodynamics and the Standard .-.Jdel 492 15.1 Quantization of gauge theories 492 15.2 Ghosts and the Feynman rules for QCD 50 1 15.3 Ghosts and unitarity 504 15.4 The standard electroweak model 5 10 15.5 Unitarity in the Standard Model 523 Problems 526 16. Renormalization 527 16.1 Power counting and regularization 527 16.2 d3 theory: An example 533 16.3 Proving renormalizability 547 16.4 The renormalization of QED 554 16.5 Fourth order vacuum polarization 557 16.6 The renormalization of QCD 569 Problems 57 1 17. The Renormalization Group and Asymptotic Freedom 573 17.1 The renormalization group equations 573 17.2 Scattering at large momenta 575 17.3 Behavior of the running coupling constant 578 17.4 Demonstration that QCD is asymptotically free 582 17.5 QCD corrections to the ratio R 589 Problem 592 xii CONTENTS Appendix A Relativistic Notation 593 A.1 Vectors and tensors 593 A.2 Dirac matrices 594 A.3 Dirac spinors 596 Appendix B Feynman Rules 598 B.1 Decay rates and cross sections 598 B.2 General rules 600 B.3 Special rules 602 Appendix C Evaluation of Loop Diagrams 606 Appendix D Quarks, Leptons, and All That 609 D.l Fundamental particles and forces 609 D.2 Computation of color factors 61 1 References 615 Index 619 PREFACE Relativistic Quantum Mechanics and Field Theory are among the most challenging and beautiful subjects in Physics. From their study we explain how states decay, can predict the existence of antimatter, learn about the origin of forces, and make the connection between spin and statistics. All of these are great developments which all physicists should know but it is a real challenge to learn them for the first time. This book grew out of my struggle to understand these topics and to teach them to second year graduate students. It began with notes I prepared for my personal use and later shared with my students. About two years ago I decided to have these notes typed in w, little realizing that by so doing I had committed myself to eventually producing this book. My objectives in preparing this text 'Alect the original reasons I prepared my own notes: to write a book which (i) can be understood by students learning the subject for the first time, (ii) carries the development far enough so that a student is prepared to begin research, and (iii) gives meaning to the study through examples drawn from the fields of atomic, nuclear, and particle physics. In short, the goal was to produce a book which begins at the beginning, goes to the end, and is easy to read along the way. The first two parts of this book (Part I: Quantum Theory of Radiation, and Part 11: Relativistic Equations) assume no previous experience with advanced quantum mechanics. The subjects included here are quantization of the electromagnetic field, relativistic one-body wave equations, and the theoretical explanation for atomic decay, all fundamental subjects which can be regarded as necessary to a well rounded education in physics (even for classical physicists). The presentation is modeled after the first third of a year-long course which I have taught at various times over the past 15 years and these topics are given in the beginning so that those students who must leave the course at the end of the first semester will have some knowledge of these important areas.
Load more

Recommended publications
  • The Second Quantized Approach to the Study of Model Hamiltonians in Quantum Hall Regime
    Washington University in St. Louis Washington University Open Scholarship Arts & Sciences Electronic Theses and Dissertations Arts & Sciences Winter 12-15-2016 The Second Quantized Approach to the Study of Model Hamiltonians in Quantum Hall Regime Li Chen Washington University in St. Louis Follow this and additional works at: https://openscholarship.wustl.edu/art_sci_etds Recommended Citation Chen, Li, "The Second Quantized Approach to the Study of Model Hamiltonians in Quantum Hall Regime" (2016). Arts & Sciences Electronic Theses and Dissertations. 985. https://openscholarship.wustl.edu/art_sci_etds/985 This Dissertation is brought to you for free and open access by the Arts & Sciences at Washington University Open Scholarship. It has been accepted for inclusion in Arts & Sciences Electronic Theses and Dissertations by an authorized administrator of Washington University Open Scholarship. For more information, please contact [email protected]. WASHINGTON UNIVERSITY IN ST. LOUIS Department of Physics Dissertation Examination Committee: Alexander Seidel, Chair Zohar Nussinov Michael Ogilvie Xiang Tang Li Yang The Second Quantized Approach to the Study of Model Hamiltonians in Quantum Hall Regime by Li Chen A dissertation presented to The Graduate School of Washington University in partial fulfillment of the requirements for the degree of Doctor of Philosophy December 2016 Saint Louis, Missouri c 2016, Li Chen ii Contents List of Figures v Acknowledgements vii Abstract viii 1 Introduction to Quantum Hall Effect 1 1.1 The classical Hall effect . .1 1.2 The integer quantum Hall effect . .3 1.3 The fractional quantum Hall effect . .7 1.4 Theoretical explanations of the fractional quantum Hall effect . .9 1.5 Laughlin's gedanken experiment .
    [Show full text]
  • Path Integrals in Quantum Mechanics
    Path Integrals in Quantum Mechanics Dennis V. Perepelitsa MIT Department of Physics 70 Amherst Ave. Cambridge, MA 02142 Abstract We present the path integral formulation of quantum mechanics and demon- strate its equivalence to the Schr¨odinger picture. We apply the method to the free particle and quantum harmonic oscillator, investigate the Euclidean path integral, and discuss other applications. 1 Introduction A fundamental question in quantum mechanics is how does the state of a particle evolve with time? That is, the determination the time-evolution ψ(t) of some initial | i state ψ(t ) . Quantum mechanics is fully predictive [3] in the sense that initial | 0 i conditions and knowledge of the potential occupied by the particle is enough to fully specify the state of the particle for all future times.1 In the early twentieth century, Erwin Schr¨odinger derived an equation specifies how the instantaneous change in the wavefunction d ψ(t) depends on the system dt | i inhabited by the state in the form of the Hamiltonian. In this formulation, the eigenstates of the Hamiltonian play an important role, since their time-evolution is easy to calculate (i.e. they are stationary). A well-established method of solution, after the entire eigenspectrum of Hˆ is known, is to decompose the initial state into this eigenbasis, apply time evolution to each and then reassemble the eigenstates. That is, 1In the analysis below, we consider only the position of a particle, and not any other quantum property such as spin. 2 D.V. Perepelitsa n=∞ ψ(t) = exp [ iE t/~] n ψ(t ) n (1) | i − n h | 0 i| i n=0 X This (Hamiltonian) formulation works in many cases.
    [Show full text]
  • Canonical Quantization of Two-Dimensional Gravity S
    Journal of Experimental and Theoretical Physics, Vol. 90, No. 1, 2000, pp. 1–16. Translated from Zhurnal Éksperimental’noœ i Teoreticheskoœ Fiziki, Vol. 117, No. 1, 2000, pp. 5–21. Original Russian Text Copyright © 2000 by Vergeles. GRAVITATION, ASTROPHYSICS Canonical Quantization of Two-Dimensional Gravity S. N. Vergeles Landau Institute of Theoretical Physics, Russian Academy of Sciences, Chernogolovka, Moscow oblast, 142432 Russia; e-mail: [email protected] Received March 23, 1999 Abstract—A canonical quantization of two-dimensional gravity minimally coupled to real scalar and spinor Majorana fields is presented. The physical state space of the theory is completely described and calculations are also made of the average values of the metric tensor relative to states close to the ground state. © 2000 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION quantization) for observables in the highest orders, will The quantum theory of gravity in four-dimensional serve as a criterion for the correctness of the selected space–time encounters fundamental difficulties which quantization route. have not yet been surmounted. These difficulties can be The progress achieved in the construction of a two- arbitrarily divided into conceptual and computational. dimensional quantum theory of gravity is associated The main conceptual problem is that the Hamiltonian is with two ideas. These ideas will be formulated below a linear combination of first-class constraints. This fact after the necessary notation has been introduced. makes the role of time in gravity unclear. The main computational problem is the nonrenormalizability of We shall postulate that space–time is topologically gravity theory. These difficulties are closely inter- equivalent to a two-dimensional cylinder.
    [Show full text]
  • On the Reality of Quantum Collapse and the Emergence of Space-Time
    Article On the Reality of Quantum Collapse and the Emergence of Space-Time Andreas Schlatter Burghaldeweg 2F, 5024 Küttigen, Switzerland; [email protected]; Tel.: +41-(0)-79-870-77-75 Received: 22 February 2019; Accepted: 22 March 2019; Published: 25 March 2019 Abstract: We present a model, in which quantum-collapse is supposed to be real as a result of breaking unitary symmetry, and in which we can define a notion of “becoming”. We show how empirical space-time can emerge in this model, if duration is measured by light-clocks. The model opens a possible bridge between Quantum Physics and Relativity Theory and offers a new perspective on some long-standing open questions, both within and between the two theories. Keywords: quantum measurement; quantum collapse; thermal time; Minkowski space; Einstein equations 1. Introduction For a hundred years or more, the two main theories in physics, Relativity Theory and Quantum Mechanics, have had tremendous success. Yet there are tensions within the respective theories and in their relationship to each other, which have escaped a satisfactory explanation until today. These tensions have also prevented a unified view of the two theories by a more fundamental theory. There are, of course, candidate-theories, but none has found universal acceptance so far [1]. There is a much older debate, which concerns the question of the true nature of time. Is reality a place, where time, as we experience it, is a mere fiction and where past, present and future all coexist? Is this the reason why so many laws of nature are symmetric in time? Or is there really some kind of “becoming”, where the present is real, the past irrevocably gone and the future not yet here? Admittedly, the latter view only finds a minority of adherents among today’s physicists, whereas philosophers are more balanced.
    [Show full text]
  • Introduction to Quantum Field Theory
    Utrecht Lecture Notes Masters Program Theoretical Physics September 2006 INTRODUCTION TO QUANTUM FIELD THEORY by B. de Wit Institute for Theoretical Physics Utrecht University Contents 1 Introduction 4 2 Path integrals and quantum mechanics 5 3 The classical limit 10 4 Continuous systems 18 5 Field theory 23 6 Correlation functions 36 6.1 Harmonic oscillator correlation functions; operators . 37 6.2 Harmonic oscillator correlation functions; path integrals . 39 6.2.1 Evaluating G0 ............................... 41 6.2.2 The integral over qn ............................ 42 6.2.3 The integrals over q1 and q2 ....................... 43 6.3 Conclusion . 44 7 Euclidean Theory 46 8 Tunneling and instantons 55 8.1 The double-well potential . 57 8.2 The periodic potential . 66 9 Perturbation theory 71 10 More on Feynman diagrams 80 11 Fermionic harmonic oscillator states 87 12 Anticommuting c-numbers 90 13 Phase space with commuting and anticommuting coordinates and quanti- zation 96 14 Path integrals for fermions 108 15 Feynman diagrams for fermions 114 2 16 Regularization and renormalization 117 17 Further reading 127 3 1 Introduction Physical systems that involve an infinite number of degrees of freedom are usually described by some sort of field theory. Almost all systems in nature involve an extremely large number of degrees of freedom. A droplet of water contains of the order of 1026 molecules and while each water molecule can in many applications be described as a point particle, each molecule has itself a complicated structure which reveals itself at molecular length scales. To deal with this large number of degrees of freedom, which for all practical purposes is infinite, we often regard a system as continuous, in spite of the fact that, at certain distance scales, it is discrete.
    [Show full text]
  • Lecture 1 – Symmetries & Conservation
    LECTURE 1 – SYMMETRIES & CONSERVATION Contents • Symmetries & Transformations • Transformations in Quantum Mechanics • Generators • Symmetry in Quantum Mechanics • Conservations Laws in Classical Mechanics • Parity Messages • Symmetries give rise to conserved quantities . Symmetries & Conservation Laws Lecture 1, page1 Symmetry & Transformations Systems contain Symmetry if they are unchanged by a Transformation . This symmetry is often due to an absence of an absolute reference and corresponds to the concept of indistinguishability . It will turn out that symmetries are often associated with conserved quantities . Transformations may be: Active: Active • Move object • More physical Passive: • Change “description” Eg. Change Coordinate Frame • More mathematical Passive Symmetries & Conservation Laws Lecture 1, page2 We will consider two classes of Transformation: Space-time : • Translations in (x,t) } Poincaré Transformations • Rotations and Lorentz Boosts } • Parity in (x,t) (Reflections) Internal : associated with quantum numbers Translations: x → 'x = x − ∆ x t → 't = t − ∆ t Rotations (e.g. about z-axis): x → 'x = x cos θz + y sin θz & y → 'y = −x sin θz + y cos θz Lorentz (e.g. along x-axis): x → x' = γ(x − βt) & t → t' = γ(t − βx) Parity: x → x' = −x t → t' = −t For physical laws to be useful, they should exhibit a certain generality, especially under symmetry transformations. In particular, we should expect invariance of the laws to change of the status of the observer – all observers should have the same laws, even if the evaluation of measurables is different. Put differently, the laws of physics applied by different observers should lead to the same observations. It is this principle which led to the formulation of Special Relativity.
    [Show full text]
  • Noether: the More Things Change, the More Stay the Same
    Noether: The More Things Change, the More Stay the Same Grzegorz G luch∗ R¨udigerUrbanke EPFL EPFL Abstract Symmetries have proven to be important ingredients in the analysis of neural networks. So far their use has mostly been implicit or seemingly coincidental. We undertake a systematic study of the role that symmetry plays. In particular, we clarify how symmetry interacts with the learning algorithm. The key ingredient in our study is played by Noether's celebrated theorem which, informally speaking, states that symmetry leads to conserved quantities (e.g., conservation of energy or conservation of momentum). In the realm of neural networks under gradient descent, model symmetries imply restrictions on the gradient path. E.g., we show that symmetry of activation functions leads to boundedness of weight matrices, for the specific case of linear activations it leads to balance equations of consecutive layers, data augmentation leads to gradient paths that have \momentum"-type restrictions, and time symmetry leads to a version of the Neural Tangent Kernel. Symmetry alone does not specify the optimization path, but the more symmetries are contained in the model the more restrictions are imposed on the path. Since symmetry also implies over-parametrization, this in effect implies that some part of this over-parametrization is cancelled out by the existence of the conserved quantities. Symmetry can therefore be thought of as one further important tool in understanding the performance of neural networks under gradient descent. 1 Introduction There is a large body of work dedicated to understanding what makes neural networks (NNs) perform so well under versions of gradient descent (GD).
    [Show full text]
  • The Homology Theory of the Closed Geodesic Problem
    J. DIFFERENTIAL GEOMETRY 11 (1976) 633-644 THE HOMOLOGY THEORY OF THE CLOSED GEODESIC PROBLEM MICHELINE VIGUE-POIRRIER & DENNIS SULLIVAN The problem—does every closed Riemannian manifold of dimension greater than one have infinitely many geometrically distinct periodic geodesies—has received much attention. An affirmative answer is easy to obtain if the funda- mental group of the manifold has infinitely many conjugacy classes, (up to powers). One merely needs to look at the closed curve representations with minimal length. In this paper we consider the opposite case of a finite fundamental group and show by using algebraic calculations and the work of Gromoll-Meyer [2] that the answer is again affirmative if the real cohomology ring of the manifold or any of its covers requires at least two generators. The calculations are based on a method of the second author sketched in [6], and he wishes to acknowledge the motivation provided by conversations with Gromoll in 1967 who pointed out the surprising fact that the available tech- niques of algebraic topology loop spaces, spectral sequences, and so forth seemed inadequate to handle the "rational problem" of calculating the Betti numbers of the space of all closed curves on a manifold. He also acknowledges the help given by John Morgan in understanding what goes on here. The Gromoll-Meyer theorem which uses nongeneric Morse theory asserts the following. Let Λ(M) denote the space of all maps of the circle S1 into M (not based). Then there are infinitely many geometrically distinct periodic geodesies in any metric on M // the Betti numbers of Λ(M) are unbounded.
    [Show full text]
  • Quantum Mechanics
    Quantum Mechanics Richard Fitzpatrick Professor of Physics The University of Texas at Austin Contents 1 Introduction 5 1.1 Intendedaudience................................ 5 1.2 MajorSources .................................. 5 1.3 AimofCourse .................................. 6 1.4 OutlineofCourse ................................ 6 2 Probability Theory 7 2.1 Introduction ................................... 7 2.2 WhatisProbability?.............................. 7 2.3 CombiningProbabilities. ... 7 2.4 Mean,Variance,andStandardDeviation . ..... 9 2.5 ContinuousProbabilityDistributions. ........ 11 3 Wave-Particle Duality 13 3.1 Introduction ................................... 13 3.2 Wavefunctions.................................. 13 3.3 PlaneWaves ................................... 14 3.4 RepresentationofWavesviaComplexFunctions . ....... 15 3.5 ClassicalLightWaves ............................. 18 3.6 PhotoelectricEffect ............................. 19 3.7 QuantumTheoryofLight. .. .. .. .. .. .. .. .. .. .. .. .. .. 21 3.8 ClassicalInterferenceofLightWaves . ...... 21 3.9 QuantumInterferenceofLight . 22 3.10 ClassicalParticles . .. .. .. .. .. .. .. .. .. .. .. .. .. .. 25 3.11 QuantumParticles............................... 25 3.12 WavePackets .................................. 26 2 QUANTUM MECHANICS 3.13 EvolutionofWavePackets . 29 3.14 Heisenberg’sUncertaintyPrinciple . ........ 32 3.15 Schr¨odinger’sEquation . 35 3.16 CollapseoftheWaveFunction . 36 4 Fundamentals of Quantum Mechanics 39 4.1 Introduction ..................................
    [Show full text]
  • 10 Group Theory and Standard Model
    Physics 129b Lecture 18 Caltech, 03/05/20 10 Group Theory and Standard Model Group theory played a big role in the development of the Standard model, which explains the origin of all fundamental particles we see in nature. In order to understand how that works, we need to learn about a new Lie group: SU(3). 10.1 SU(3) and more about Lie groups SU(3) is the group of special (det U = 1) unitary (UU y = I) matrices of dimension three. What are the generators of SU(3)? If we want three dimensional matrices X such that U = eiθX is unitary (eigenvalues of absolute value 1), then X need to be Hermitian (real eigenvalue). Moreover, if U has determinant 1, X has to be traceless. Therefore, the generators of SU(3) are the set of traceless Hermitian matrices of dimension 3. Let's count how many independent parameters we need to characterize this set of matrices (what is the dimension of the Lie algebra). 3 × 3 complex matrices contains 18 real parameters. If it is to be Hermitian, then the number of parameters reduces by a half to 9. If we further impose traceless-ness, then the number of parameter reduces to 8. Therefore, the generator of SU(3) forms an 8 dimensional vector space. We can choose a basis for this eight dimensional vector space as 00 1 01 00 −i 01 01 0 01 00 0 11 λ1 = @1 0 0A ; λ2 = @i 0 0A ; λ3 = @0 −1 0A ; λ4 = @0 0 0A (1) 0 0 0 0 0 0 0 0 0 1 0 0 00 0 −i1 00 0 01 00 0 0 1 01 0 0 1 1 λ5 = @0 0 0 A ; λ6 = @0 0 1A ; λ7 = @0 0 −iA ; λ8 = p @0 1 0 A (2) i 0 0 0 1 0 0 i 0 3 0 0 −2 They are called the Gell-Mann matrices.
    [Show full text]
  • Zero-Point Energy and Interstellar Travel by Josh Williams
    ;;;;;;;;;;;;;;;;;;;;;; itself comes from the conversion of electromagnetic Zero-Point Energy and radiation energy into electrical energy, or more speciÞcally, the conversion of an extremely high Interstellar Travel frequency bandwidth of the electromagnetic spectrum (beyond Gamma rays) now known as the zero-point by Josh Williams spectrum. ÒEre many generations pass, our machinery will be driven by power obtainable at any point in the universeÉ it is a mere question of time when men will succeed in attaching their machinery to the very wheel work of nature.Ó ÐNikola Tesla, 1892 Some call it the ultimate free lunch. Others call it absolutely useless. But as our world civilization is quickly coming upon a terrifying energy crisis with As you can see, the wavelengths from this part of little hope at the moment, radical new ideas of usable the spectrum are incredibly small, literally atomic and energy will be coming to the forefront and zero-point sub-atomic in size. And since the wavelength is so energy is one that is currently on its way. small, it follows that the frequency is quite high. The importance of this is that the intensity of the energy So, what the hell is it? derived for zero-point energy has been reported to be Zero-point energy is a type of energy that equal to the cube (the third power) of the frequency. exists in molecules and atoms even at near absolute So obviously, since weÕre already talking about zero temperatures (-273¡C or 0 K) in a vacuum. some pretty high frequencies with this portion of At even fractions of a Kelvin above absolute zero, the electromagnetic spectrum, weÕre talking about helium still remains a liquid, and this is said to be some really high energy intensities.
    [Show full text]
  • Conservation of Total Momentum TM1
    L3:1 Conservation of total momentum TM1 We will here prove that conservation of total momentum is a consequence Taylor: 268-269 of translational invariance. Consider N particles with positions given by . If we translate the whole system with some distance we get The potential energy must be una ected by the translation. (We move the "whole world", no relative position change.) Also, clearly all velocities are independent of the translation Let's consider to be in the x-direction, then Using Lagrange's equation, we can rewrite each derivative as I.e., total momentum is conserved! (Clearly there is nothing special about the x-direction.) L3:2 Generalized coordinates, velocities, momenta and forces Gen:1 Taylor: 241-242 With we have i.e., di erentiating w.r.t. gives us the momentum in -direction. generalized velocity In analogy we de✁ ne the generalized momentum and we say that and are conjugated variables. Note: Def. of gen. force We also de✁ ne the generalized force from Taylor 7.15 di ers from def in HUB I.9.17. such that generalized rate of change of force generalized momentum Note: (generalized coordinate) need not have dimension length (generalized velocity) velocity momentum (generalized momentum) (generalized force) force Example: For a system which rotates around the z-axis we have for V=0 s.t. The angular momentum is thus conjugated variable to length dimensionless length/time 1/time mass*length/time mass*(length)^2/time (torque) energy Cyclic coordinates = ignorable coordinates and Noether's theorem L3:3 Cyclic:1 We have de ned the generalized momentum Taylor: From EL we have 266-267 i.e.
    [Show full text]