PRINCIPLES OF PHYSICS From Quantum Field Theory to Classical Mechanics www.indd 1 6/11/13 8:44 AM Tsinghua Report and Review in Physics Series Editor: Bangfen Zhu (Tsinghua University, China) Vol. 1 Möbius Inversion in Physics by Nanxian Chen Vol. 2 Principles of Physics — From Quantum Field Theory to Classical Mechanics by Ni Jun www.com TSINGHUA Report and Review in Physics Vol. 2 PRINCIPLES OF PHYSICS From Quantum Field Theory to Classical Mechanics Ni Jun Tsinghua University, China World Scientific N E W J E R S E Y • L O N D O N • S I N G A P O R E • B E I J I N G • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I www.indd 2 6/11/13 8:44 AM Published by World Scientific Publishing Co.
5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Tsinghua Report and Review in Physics — Vol. 2 PRINCIPLES OF PHYSICS From Quantum Field Theory to Classical Mechanics Copyright © 2014 by World Scientific Publishing Co. All rights reserved.
This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 978-981-4579-39-1 Printed in Singapore www.com October 17, 2013 16:1 BC: 9056 - Principle of Physics ws-book9x6junni To my daughter Ruyan www.com October 17, 2013 16:1 BC: 9056 - Principle of Physics ws-book9x6junni Preface During the 20th century, physics experienced a rapid expansion.
A gen- eral theoretical physics curriculum now consists of a collection of separate courses labeled as classical mechanics, electrodynamics, quantum mechan- ics, statistical mechanics, quantum field theory, general relativity, etc., with each course taught in a different book. I consider there is a need to write a book which is compact and merge these courses into one single unified course. This book is an attempt to realize this aim. In writing this book, I focus on two purposes.
(1) Historically, physics is established from classical mechanics to quantum mechanics, from quantum mechanics to quantum field theory, from thermodynamics to statistical mechanics, and from New- tonian gravity to general relativity. However, a more logical subsequent presentation is from quantum field theory to classical mechanics, and from the physics principles on the microscopic scale to physics on the macro- scopic scale. In this book, I try to achieve this by elucidating the physics from quantum field theory to classical mechanics from a set of common ba- sic principles in a unified way. (2) Physics is considered as an experimental science.
This view, however, is being changed. In the history of physics, there are two epic heroes: Newton and Einstein. They represent two epochs in physics. In the Newtonian epoch, physical laws are deduced from exper- imental observations.
People are amazed that the observed physical laws can be described accurately by mathematical equations. At the same time, it is reasonable to ask why nature should obey the physical laws described by the mathematical equations. After wondering how accurately nature obeys the gravitational law that the gravitation force is proportional to the inverse square of the distance, one would ask why it is not found in other ways. Einstein creates a new epoch by deducing physical laws not merely from experiments but also from principles such as simplicity, symmetry vii www.com October 17, 2013 16:1 BC: 9056 - Principle of Physics ws-book9x6junni viii Principles of Physics and other understandable credos.
From the view of Einstein, physical laws should be natural and simple. It is my belief that all physics laws should be understandable. In this book, I endeavor to establish the physical for- malisms based on basic principles that are as simple and understandable as possible. The book covers all the disciplines of fundamental physics, including quantum field theory, quantum mechanics, statistical mechanics, thermo- dynamics, general relativity, electromagnetic field, and classical mechanics.
Instead of the traditional pedagogic way, the subjects and formalisms are arranged in a logical-sequential way, i. all the formulas are derived from the formulas before them. The formalisms are also kept self-contained, i. the derivations of all the physical formulas which appear in this book can be found in the same book.
Most of the required mathematical tools are also given in the appendices. Although this book covers all the disciplines of fundamental physics, the book is compact and has only about 400 pages because the contents are concise and can be treated as an integrated entity. In this book, the main emphasis is the basic formalisms of physics. The topics on applications and approximation methods are kept to a minimum and are selected based on their generality and importance.
Still it was not easy to do it when some important topics had to be omitted. Since it is impossible to provide an exhaustive bibliography, I list only the related textbooks and monographs that I am familiar with. I apologize to the authors whose books have not been included unintentionally. This book may be used as an advanced textbook by graduate students.
It is also suitable for physicists who wish to have an overview of fundamental physics. I am grateful to all my colleagues and students for their inspiration and help. I would also like to express my gratitude to World Scientific for the assistance rendered in publishing this book. Jun Ni August 8, 2013 Tsinghua, Beijing www.com October 17, 2013 16:1 BC: 9056 - Principle of Physics ws-book9x6junni Contents Preface vii 1.1 Identical particles principle .3 Creation and annihilation operators .4 Symmetrized and anti-symmetrized states .5 Commutators between creation and annihilation operators .2 The equations of motion .2 The generator of time translation .5 Path integral formulas .6 Lagrangian and action .2 Klein-Gordon equation .3 Solutions of the Klein-Gordon equation .4 The commutators for creation and annihilation operators in p-space .com October 17, 2013 16:1 BC: 9056 - Principle of Physics ws-book9x6junni x Principles of Physics 2.5 The homogeneity of spacetime .4 The complex scalar field .1 Lagrangian of the complex boson field .2 Symmetry and conservation law .2 The generator of time translation .5 Dirac-Pauli representation .6 Lorentz transformation for spinors .7 Covariance of the spinor fermion Lagrangian .9 Energy-momentum tensor and Hamiltonian operator .11 Symmetric energy-momentum tensor .13 Solutions of the free Dirac equation .14 Hamiltonian operator in p-space .19 Spin statistics relation .20 Charge of spinor particles and antiparticles .21 Representation in terms of the Weyl spinors .1 Massive vector bosons .2 Massless vector bosons .1 Lagrangian with the gauge invariance .2 Nonabelian gauge symmetry.
Quantum Fields in the Riemann Spacetime 97 3.1 Lagrangian in the Riemann spacetime .2 Homogeneity of spacetime .com October 17, 2013 16:1 BC: 9056 - Principle of Physics ws-book9x6junni Contents xi 3.4 The generator of time translation .5 The relations of terms in the total action .1 Lagrangian with the scale invariance .2 Conserved current for the scale invariance .3 Scale invariance for the total Lagrangian .2 Ground state energy .1 Spontaneous symmetry breaking .4 The Higgs mechanism .5 Mass and interactions of particles. Perturbative Field Theory 123 5.1 Invariant commutation relations .2 n-point Green’s function .1 Definition of n-point Green’s function .6 Free scalar fields .3 Interacting scalar field .3 Two-point function .4 Four-point function .4 Divergency in n-point functions .1 Divergency in integrations .com October 17, 2013 16:1 BC: 9056 - Principle of Physics ws-book9x6junni xii Principles of Physics 5.1 Two-point function .2 Four-point function. From Quantum Field Theory to Quantum Mechanics 169 6.1 Non-relativistic limit of the Klein-Gordon equation .2 Non-relativistic limit of the Dirac equation .3 Spin-orbital coupling .4 The operator of time translation in quantum mechanics .5 Transformation of basis .6 One-body operators .7 Schrödinger equation .5 Radiation of electromagnetic waves .7 Electrostatic energy of charges .8 Many-body operators .9 Potentials of charge particles in the classical limit .1 Equations of motion for operators in quantum mechanics 199 8.2 Constants of motion .3 Conservation of angular momentum .2 Elementary aspects of the Schrödinger equation .5 Path integral formalism for quantum mechanics .1 Feymann’s path integral for one-particle systems 208 8.2 Lagrangian function in quantum mechanics .com October 17, 2013 16:1 BC: 9056 - Principle of Physics ws-book9x6junni Contents xiii 8.4 Path integral formalism for multi-particle systems .1 Schrödinger representation .8 de Broglie waves .9 Statistical interpretation of wave functions .10 Heisenberg uncertainty principle. Applications of Quantum Mechanics 231 9.2 Hamiltonian operator in terms of ↠and â .3 Eigenvalues and eigenstates .2 Schrödinger equation for a central potential .1 Schrödinger equation in the spherical coordinates 236 9.2 Separation of variables .3 Angular momentum operators .4 Eigenvalues of Ĵ2 and Jˆz .5 Matrix elements of angular momentum operators 241 9.1 Equi-probability principle and statistical distributions .2 Average of an observable  .2 Average using canonical distribution .3 Average using grand canonical distribution .3 Functional integral representation of partition function .4 First law of thermodynamics .5 Second law of thermodynamics .1 Entropy increase principle .com October 17, 2013 16:1 BC: 9056 - Principle of Physics ws-book9x6junni xiv Principles of Physics 10.2 Extensiveness of ln Z .3 Thermodynamic quantities in terms of partition function .4 Kelvin formulation of the second law of thermodynamics .9 Gibbs-Duhem relation .11 Derivatives of thermodynamic quantities .6 Third law of thermodynamics .7 Thermodynamic quantities expressed in terms of grand partition function .8 Relation between grand partition function and partition function .9 Systems with particle number changeable .1 Thermodynamic relations for open systems .2 Equilibrium conditions of two systems .3 Phase equilibrium conditions .10 Equilibrium distributions of nearly independent particle systems .1 Derivations of the distribution functions of single particle from the macro-canonical distribution .2 Partition function of independent particle systems .3 About summations in calculations of independent particle system .1 Absolute and relative fluctuations .2 Fluctuations in systems of canonical ensemble .3 Fluctuations in systems of grand canonical ensemble .12 Classic statistical mechanics and quantum corrections .1 Classic limit of statistical distribution functions .com October 17, 2013 16:1 BC: 9056 - Principle of Physics ws-book9x6junni Contents xv 11.
Applications of Statistical Mechanics 301 11.1 Partition function for mass center motion .2 Ideal gas of single-atom molecules .3 Internal degrees of freedom .2 Weakly degenerate quantum gas .1 Bose-Einstein condensation .2 Thermodynamic properties of BEC .1 Classical energy-momentum tensor .2 Equation of motion in the Riemann spacetime .3 Weak field limit .1 Static weak field limit-Newtonian gravitation .2 Equation of motion in Newtonian approximation 337 12.4 Weak field approximation in the harmonic gauge 339 12.4 Spherical solutions for stars .1 Spherically symmetric spacetime .2 Einstein equations for static fluid .3 The metric outside a star .4 Interior structure of a star .5 Structure of a Newtonian star .6 Simple model for the interior structure of stars .7 Pressure of relativistic Fermi gas .2 Solutions with void .2 Higher rank tensors .com October 17, 2013 16:1 BC: 9056 - Principle of Physics ws-book9x6junni xvi Principles of Physics A.1 Infinitesimal Lorentz transformation .2 Finite Lorentz transformation .9 Riemann curvature tensor. 383 Appendix B Functional Formula 385 Appendix C Gaussian Integrals 387 C.3 Gaussian integrations with source. 389 Appendix D Grassmann Algebra 391 Appendix E Euclidean Representation 397 Appendix F Some Useful Formulas 399 Appendix G Jacobian 403 Appendix H Geodesic Equation 405 Bibliography 409 Index 413 www.