Quantum Mechanics and Quantum Field Theory

240 Pages·2011·0.91 MB·English

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QuantumMechanicsandQuantumFieldTheory
Explainingtheconceptsofquantummechanicsandquantumfieldtheoryinaprecise
mathematical language, this textbook is an ideal introduction for graduate students
inmathematics,helpingtopreparethemforfurtherstudiesinquantumphysics.
The textbook covers topics that are central to quantum physics: non-relativistic
quantummechanics,quantumstatisticalmechanics,relativisticquantummechanics,
and quantum field theory. There is also background material on analysis, classical
mechanics,relativity,andprobability.Eachtopicisexploredthroughastatementof
basicprinciplesfollowedbysimpleexamples.Around100problemsthroughoutthe
textbookhelpreadersdeveloptheirunderstanding.
JonathanDimock is Professor of Mathematics, SUNY at Buffalo. He has carried out
research in various areas of mathematical physics, including constructive quantum
fieldtheory,quantumfieldtheoryonmanifolds,renormalizationgroupmethods,and
stringtheory.

Quantum Mechanics
and Quantum Field Theory
A Mathematical Primer
JONATHAN DIMOCK
SUNYatBuffalo
CAMBRIDGE UNIVERSITY PRESS
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(cid:2)c J.Dimock2011
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Firstpublished2011
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloginginPublicationdata
Dimock,Jonathan,1945–
Quantummechanicsandquantumfieldtheory:
amathematicalprimer/JonathanDimock.
p. cm.
ISBN978-1-107-00509-9(hardback)
1. Quantumtheory–Mathematics. I. Title.
QC174.17.M35D56 2011
530.12–dc22
2010041723
ISBN978-1-107-00509-9Hardback
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forBenjamin,Christina,Gregory,andAnn

Contents
Preface pagexi
Introduction 1
PartI Non-relativistic 3
1 Mathematicalprelude 5
1.1 Boundedoperators 5
1.2 Unboundedoperators 11
1.3 Self-adjointoperators 14
1.4 Compactoperators 22
2 Classicalmechanics 28
2.1 Hamiltonianmechanics 28
2.2 Examples 29
2.3 Canonicaltransformations 31
2.4 Symmetries 34
3 Quantummechanics 38
3.1 Principlesofquantummechanics 38
3.2 Canonicalquantization 41
3.3 Symmetries 43
3.4 Perspectivesandproblems 45
4 Singleparticle 47
4.1 Freeparticle 47
4.2 Particleinapotential 48
4.3 Spectrum 51
4.4 Theharmonicoscillator 53
vii
viii Contents
(cid:2)
4.5 Scattering 55
4.6 Spin 58
5 Manyparticles 63
5.1 Twoparticles 63
5.2 Identicalparticles 66
5.3 n-particles 67
5.4 Fockspace 70
6 Statisticalmechanics 78
6.1 Mixedstates 78
6.2 Equilibriumstates 79
6.3 Freebosongas 83
6.4 Freefermiongas 86
6.5 Interactingbosons 88
6.6 Furtherdevelopments 90
PartII Relativistic 93
7 Relativity 95
7.1 Principlesofrelativity 95
7.2 Minkowskispace 96
7.3 Classicalfreefields 103
7.4 Interactingclassicalfields 107
7.5 Fundamentalsolutions 111
8 Scalarparticlesandfields 114
8.1 Scalarparticles 114
8.2 Scalarfields 118
8.3 Chargedscalarfield 126
9 Electronsandphotons 130
9.1 Spinors 130
9.2 Electrons 132
9.3 Diracfields 139
9.4 Photons 144
9.5 Electromagneticfield 148
10 Fieldtheoryonamanifold 152
10.1 Lorentzianmanifolds 152