The Schwinger action principle and effective action

507 Pages·2007·1.82 MB·English

by  Toms D.J.

THE SCHWINGER ACTION PRINCIPLE
AND EFFECTIVE ACTION
This book is an introduction to the Schwinger action principle in quan-
tummechanicsandquantumfieldtheory,withapplicationstoavarietyof
different models, not only those of interest to particle physics. The book
begins with a brief review of the action principle in classical mechan-
ics and classical field theory. It then moves on to quantum field theory,
focusing on the effective action method. This is introduced as simply as
possible by using the zero-point energy of the simple harmonic oscilla-
tor as the starting point. This allows the utility of the method, and the
process of regularization and renormalization of quantum field theory,
to be demonstrated with a minimum of formal development. The book
concludes with a more complete definition of the effective action, and
demonstrates how the provisional definition used earlier is the first term
in the systematic loop expansion.
Several applications of the Schwinger action principle are given,
including Bose–Einstein condensation, the Casimir effect, and trapped
Fermi gases. The renormalization of interacting scalar field theory is
presented to two-loop order. This book will interest graduate students
and researchers in theoretical physics who are familiar with quantum
mechanics.
David Toms is Reader in Mathematical Physics in the School of Math-
ematics and Statistics at Newcastle University. Prior to joining New-
castle University, Dr. Toms was a NATO Science Fellow at Imperial
CollegeLondon,andapostdoctoralFellowattheUniversityofWisconsin-
Milwaukee. His research interests include the formalism of quantum field
theory and its applications, and his most recent interests are centred
aroundKaluza–Kleintheory,basedontheideathatthereareextraspatial
dimensions beyond the three obvious ones.
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D.J.TomsThe Schwinger Action Principle and Effective Action
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†Issuedasapaperback

The Schwinger Action Principle
and Effective Action
DAVID J. TOMS
School of Mathematics and Statistics
Newcastle University
cambridge university press
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,S˜aoPaulo
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
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(cid:2)c D.J.Toms2007
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithout
thewrittenpermissionofCambridgeUniversityPress.
Firstpublished2007
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
A catalogue record for this publication is available from the British Library
Library of Congress Cataloguing in Publication data
Toms,DavidJ.,1953–
TheSchwingeractionprincipleandeffectiveaction/DavidJ.Toms.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-521-87676-6
1.Schwingeractionprinciple. 2.Quantumtheory. 3.Mathematicalphysics. I.Title.
QC174.17.S32T662007
530.12—dc22 2007013097
ISBN978-0-521-87676-6hardback
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Contents
Preface page x
1 Action principle in classical mechanics 1
1.1 Euler–Lagrange equations 1
1.2 Hamilton’s principle 5
1.3 Hamilton’s equations 8
1.4 Canonical transformations 12
1.5 Conservation laws and symmetries 22
Notes 38
2 Action principle in classical field theory 40
2.1 Continuous systems 40
2.2 Lagrangian and Hamiltonian formulation for
continuous systems 43
2.3 Some examples 48
2.4 Functional differentiation and Poisson brackets for
field theory 54
2.5 Noether’s theorem 60
2.6 The stress–energy–momentum tensor 66
2.7 Gauge invariance 75
2.8 Fields of general spin 83
2.9 The Dirac equation 88
Notes 99
3 Action principle in quantum theory 100
3.1 States and observables 100
3.2 Schwinger action principle 111
3.3 Equations of motion and canonical commutation relations 113
3.4 Position and momentum eigenstates 120
3.5 Simple harmonic oscillator 124
3.6 Real scalar field 132
vii
viii Contents
3.7 Complex scalar field 140
3.8 Schro¨dinger field 144
3.9 Dirac field 151
3.10 Electromagnetic field 157
Notes 168
4 The effective action 169
4.1 Introduction 169
4.2 Free scalar field in Minkowski spacetime 174
4.3 Casimir effect 178
4.4 Constant gauge field background 181
4.5 Constant magnetic field 186
4.6 Self-interacting scalar field 196
4.7 Local Casimir effect 203
Notes 207
5 Quantum statistical mechanics 208
5.1 Introduction 208
5.2 Simple harmonic oscillator 213
5.3 Real scalar field 217
5.4 Charged scalar field 221
5.5 Non-relativistic field 228
5.6 Dirac field 234
5.7 Electromagnetic field 235
Notes 237
6 Effective action at finite temperature 238
6.1 Condensate contribution 238
6.2 Free homogeneous non-relativistic Bose gas 241
6.3 Internal energy and specific heat 245
6.4 Bose gas in a harmonic oscillator confining potential 247
6.5 Density of states method 258
6.6 Charged non-relativistic Bose gas in a constant
magnetic field 267
6.7 The interacting Bose gas 278
6.8 The relativistic non-interacting charged scalar field 289
6.9 The interacting relativistic field 293
6.10 Fermi gases at finite temperature in a magnetic field 298
6.11 Trapped Fermi gases 309
Notes 322
7 Further applications of the Schwinger action principle 323
7.1 Integration of the action principle 323
7.2 Application of the action principle to the
free particle 325
Contents ix
7.3 Application to the simple harmonic oscillator 329
7.4 Application to the forced harmonic oscillator 332
7.5 Propagators and energy levels 337
7.6 General variation of the Lagrangian 344
7.7 The vacuum-to-vacuum transition amplitude 348
7.8 More general systems 352
Notes 367
8 General definition of the effective action 368
8.1 Generating functionals for free field theory 368
8.2 Interacting fields and perturbation theory 374
8.3 Feynman diagrams 383
8.4 One-loop effective potential for a real scalar field 388
8.5 Dimensional regularization and the derivative expansion 394
8.6 Renormalization of λφ4 theory 403
8.7 Finite temperature 415
8.8 Generalized CJT effective action 425
8.9 CJT approach to Bose–Einstein condensation 433
Notes 446
Appendix1 Mathematical appendices 447
Appendix2 Review of special relativity 462
Appendix3 Interaction picture 469
Bibliography 479
Index 486
Preface
Then Jurgen mounted this horse and rode away from the plowed field wherein
nothing grew as yet. As they left the furrows they came to a signboard with
writing on it, in a peculiar red and yellow lettering.
Jurgen paused to decipher this.
‘Read me!’ was written on the signboard: ‘read me, and judge if you under-
stand! So you stopped in your journey because I called, scenting something
unusual,somethingdroll.Thus,althoughIamnothing,andevenless,thereisno
onethatseesmebutlingershere.Stranger,Iamalawoftheuniverse.Stranger,
render the law what is due the law!’
Jurgenfeltcheated.‘Averyfoolishsignboard,indeed!forhowcanitbe‘alaw
of the universe’, when there is no meaning to it!’ says Jurgen. ‘Why, for any law
to be meaningless would not be fair.’
(James Branch Cabell)
The quantum theory of fields is now a mature and well-developed subject
with a wide range of applications to physical systems. The predominant
approach that is taught to students today is the Feynman path inte-
gral. There is another approach that receives much less attention due to
Schwinger, and his method is the main emphasis of the present book.
The intention of this book is to present the material in a manner that
is accessible to final-year undergraduates or beginning postgraduate stu-
dents.Theprospectivereaderisexpectedtoalreadyhavehadanexposure
toclassicalmechanics,quantummechanics,andstatisticalmechanics;the
present book is not to be viewed as an introduction to any of these sub-
jects. The assumed background is that found in a typical physics under-
graduate in the UK at the end of the third year. Some material that is
includedherewillalreadybefamiliartoasuitablypreparedstudentandis
intended as a refresher. Other material goes beyond that typically taught
attheundergraduatelevel,butispresentedinawaythatleadsondirectly
fromthemorefamiliarmaterial.Thenecessarymathematicalbackground
is also typical of that found in the third-year physics undergraduate.
I have included some of the necessary background material in appendices
x