Schwinger's Quantum Action Principle: From Dirac’s Formulation Through Feynman’s Path Integrals, the Schwinger-Keldysh Method, Quantum Field Theory, to Source Theory

122 Pages·2015·1.83 MB·English

by  Kimball A. Milton (auth.)

SPRINGER BRIEFS IN PHYSICS
Kimball A. Milton
Schwinger’s Quantum
Action Principle
From Dirac’s Formulation
Through Feynman’s Path
Integrals, the Schwinger-
Keldysh Method,
Quantum Field Theory, to
Source Theory
123
SpringerBriefs in Physics
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Eric Le Ru, Victoria University of Wellington, Wellington, New Zealand
Hans-Joachim Lewerenz, California Institute of Technology, Pasadena, USA
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Vesselin Petkov, Concordia University, Concordia, Canada
Charles H.-T. Wang, University of Aberdeen, Aberdeen, UK
Andrew Whitaker, Queen’s University Belfast, Belfast, UK
More information about this series at http://www.springer.com/series/8902
Kimball A. Milton
’
Schwinger s Quantum
Action Principle
’
From Dirac s Formulation Through
’
Feynman s Path Integrals,
the Schwinger-Keldysh Method,
Quantum Field Theory, to Source Theory
123
Kimball A.Milton
HomerL.DodgeDepartmentofPhysicsand
Astronomy
TheUniversity of Oklahoma
Norman
USA
ISSN 2191-5423 ISSN 2191-5431 (electronic)
SpringerBriefs inPhysics
ISBN978-3-319-20127-6 ISBN978-3-319-20128-3 (eBook)
DOI 10.1007/978-3-319-20128-3
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Abstract
Startingfromtheearliernotionsofstationaryactionprinciples,weshowhowJulian
Schwinger’sQuantumActionPrincipledescendedfromDirac’sformulation,which
independentlyledFeynmantohispath-integralformulationofquantummechanics.
Theconnectionbetweenthetwoisbroughtout,andapplicationsarediscussed.The
Keldysh-Schwinger time-cycle method of extracting matrix elements in nonequi-
libriumsituationsisdescribed.Thevariationalformulationofquantumfieldtheory
andthedevelopmentofsource theoryconstitutethelatter partofthiswork.Inthis
document, derived from Schwinger’s lectures over four decades, the continuity of
concepts, such as that of Green’s functions, becomes apparent.
v
Contents
1 Historical Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Review of Classical Action Principles. . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Lagrangian Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Hamiltonian Viewpoint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 A Third, Schwingerian, Viewpoint. . . . . . . . . . . . . . . . . . . . . . 7
2.4 Invariance and Conservation Laws. . . . . . . . . . . . . . . . . . . . . . 9
2.5 Nonconservation Laws: The Virial Theorem . . . . . . . . . . . . . . . 13
3 Classical Field Theory—Electrodynamics. . . . . . . . . . . . . . . . . . . . 15
3.1 Action of Particle in Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Electrodynamic Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Momentum and Angular Momentum Conservation. . . . . . . . . . . 21
3.5 Gauge Invariance and the Conservation of Charge. . . . . . . . . . . 24
3.6 Gauge Invariance and Local Conservation Laws . . . . . . . . . . . . 26
4 Quantum Action Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1 Harmonic Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Forced Harmonic Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3 Feynman Path Integral Formulation . . . . . . . . . . . . . . . . . . . . . 44
4.3.1 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4 Toward Source Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5 Time-Cycle or Schwinger-Keldysh Formulation. . . . . . . . . . . . . . . 51
6 Relativistic Theory of Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.1 Inference of Particle Properties . . . . . . . . . . . . . . . . . . . . . . . . 70
6.2 The Connection Between Spin and Statistics. . . . . . . . . . . . . . . 72
6.3 Fermi-Dirac Fields of Spin 1=2 . . . . . . . . . . . . . . . . . . . . . . . . 74
6.4 Spin Zero and One. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
vii
viii Contents
6.5 Spin Zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.6 Spin One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.7 Electromagnetic Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.8 Introduction of Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.9 Quantum Electrodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7 Nonrelativistic Source Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.1 Interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
7.2 Bound States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Chapter 1
Historical Introduction
Variationalprinciplesfordynamicalsystemshavealonghistory.Althoughprecursors
go back at least to Leibnitz (see for example (Euler 1752)) and Euler (1744) the
“principleofleastaction”wasgivenmodernformbydeMaupertuis(1744,1746).
We will not attempt to trace the history here; a brief useful account is given in
Sommerfeld’slectures(Sommerfeld1964).Themostimportantnamesinthehistory
ofthedevelopmentofdynamicalsystems,oratleastthosethatwillbearmostdirectly
onthefollowingdiscussion,arethoseofLagrange(1788)andHamilton(1834,1835).
Here we are concentrating on the work of Julian Schwinger (1918–1994), who
had profound and pervasive influence on 20th century physics, and whose many
studentshavebecomeleadersindiversefields.1Forbiographicalinformationabout
his life and work see (Mehra 2000; Milton 2007). Therefore, we will take up the
storyinthemodernera.ShortlyafterDirac’sworkwithFockandPodolsky(Dirac
1932),inwhichthedemonstrationoftheequivalencebetweenhistheoryofquantum
electrodynamics, and that of Heisenberg and Pauli (Heisenberg 1929), P. A. M.
Dirac wrote a paper on “The Lagrangian in Quantum Mechanics” (Dirac 1933).
ThispaperhadaprofoundinfluenceonRichardFeynman’sdoctoraldissertationat
Princeton on “The Principles of Least Action in Quantum Mechanics” (Feynman
1942), and on his later work on the formulations of the“Space-Time Approach to
QuantumElectrodynamics”(Feynman1949).Dirac’spaperfurtherformedthebasis
forSchwinger’sdevelopmentofthequantumactionprinciple,whichfirstappeared
in his final operator field formulation of quantum field theory (Schwinger 1951b),
whichwewilldescribeinChap.6.
The response of Feynman and Schwinger to Dirac’s inspiring paper was
completely different. Feynman was to give a global “solution” to the problem of
1Forcomplexreasons,Schwinger’sinfluenceonmodernphysicsisnotwidelyappreciated.His
contributionstoourcurrentunderstandingofnatureareunderrepresentedintextbooks,withsome
notableexceptions(Toms2007).
©TheAuthor(s)2015 1
K.A.Milton,Schwinger’sQuantumActionPrinciple,
SpringerBriefsinPhysics,DOI10.1007/978-3-319-20128-3_1
2 1 HistoricalIntroduction
determining the transformation function, the probability amplitude connecting the
stateofthesystematonetimetothatatalatertime,intermsofasumoverclassical
trajectories, the famous path integral. Schwinger, instead, derived (initially postu-
lated)adifferentialequationforthattransformationfunctionintermsofaquantum
actionfunctional.ThisdifferentialequationpossessedFeynman’spathintegralasa
formalsolution,whichremainedpoorlydefined;butSchwingerbelievedthroughout
hislifethathisapproachwas“moregeneral,moreelegant,moreuseful,andmore
tiedtothehistoricallineofdevelopmentasthequantumtranscriptionofHamilton’s
actionprinciple”(Schwinger1973).
Later,inatributetoFeynman,Schwingercommentedfurther.Dirac,ofcourse,
wasthefather of transformationtheory (Dirac1927). Thetransformation function
fromadescriptionofthesystemattimet toadescriptionattimet is“theproductof
2 1
allthetransformationsfunctionsassociatedwiththesuccessiveinfinitesimalincre-
mentsintime.”Diracsaidthelatter,thatis,thetransformationfunctionfromtimet
totimet +dt correspondstoexp[(i/(cid:2))dt L],where L istheLagrangianexpressed
intermsofthecoordinatesatthetwotimes.Forthetra(cid:2)nsformationfunctionbetween
t andt “theintegrandisexp[(i/(cid:2))W].whereW = t1dt L.”“Nowweknow,and
2 1 t2
Dirac surely knew, that to within a constant factor the ‘correspondence,’ for infin-
itesimal dt, is an equality when we deal with a system of nonrelativistic particles
possessingacoordinate-dependentpotentialenergyV ….Whythen,didDiracnot
makeamoreprecise,iflessgeneralstatement?Becausehewasinterestedinagen-
eralquestion:What,inquantummechanics,correspondstotheclassicalprincipleof
stationaryaction?”
“Why, in the decade that followed, didn’t someone pick up the computational
possibilities offered by this integral approach to the time transformation function?
To answer this question bluntly, perhaps no one needed it—until Feynman came
along.”(Schwinger1989a).
ButSchwinger followedthedifferentialroute,andstartinginearly1950 began
a new, his third, formulation of quantum electrodynamics, based on a variational
approach.Thiswasfirstpublishedin1951(Schwinger1951b).Abitlaterhestarted
developinganewformulationofquantumkinematics,whichhecalledMeasurement
Algebra, which got its first public presentation at École de Physique des Houches
in the summer of 1955. There were several short notes in the Proceedings of the
USNationalAcademypublishedin1960,explainingboththequantumkinematical
approach and the dynamical action principle (Schwinger 1960a–d), but although
he often promised to write a book on the subject (as he also promised a book on
quantum field theory) nothing came of it. Les Houches lectures, based on notes
takenbyRobertKohler,eventuallyappearedin1970(Schwinger1970a).Lectures
based on a UCLA course on quantum mechanics by Schwinger were eventually
publishedunderEnglert’seditorship(Schwinger2001).Theincompletenessofthe
writtenrecordmaybepartlyalleviatedbythepresentessay.
Westartonaclassicalfooting.