Feynman’s Operational Calculus and Beyond: Noncommutativity and Time-Ordering

384 Pages·2015·4.1 MB·English

by  Gerald W Johnson

OXFORDMATHEMATICALMONOGRAPHS
SeriesEditors
J.M.BALL W.T.GOWERS
N.J.HITCHIN L.NIRENBERG
R.PENROSE A.WILES
OXFORDMATHEMATICALMONOGRAPHS
Forafulllistoftitlespleasevisit
http://ukcatalogue.oup.com/category/academic/series/science/maths/omm.do
DonaldsonandKronheimer:TheGeometryofFour-Manifolds,paperback
Woodhouse:GeometricQuantization,SecondEdition,paperback
Hirschfeld:ProjectiveGeometriesoverFiniteFields,SecondEdition
EvansandKawahigashi:QuantumSymmetriesofOperatorAlgebras
Klingen:ArithmeticalSimilarities:PrimeDecompositionandFiniteGroupTheory
MatsuzakiandTaniguchi:HyperbolicManifoldsandKleinianGroups
Macdonald:SymmetricFunctionsandHallPolynomials,SecondEdition,paperback
Catto,LeBris,andLions:MathematicalTheoryofThermodynamicLimits:Thomas-FermiTypeModels
McDuffandSalamon:IntroductiontoSymplecticTopology,paperback
Holschneider:Wavelets:AnAnalysisTool,paperback
Goldman:ComplexHyperbolicGeometry
ColbournandRosa:TripleSystems
Kozlov,Maz’ya,andMovchan:AsymptoticAnalysisofFieldsinMulti-Structures
Maugin:NonlinearWavesinElasticCrystals
DassiosandKleinman:LowFrequencyScattering
Ambrosio,Fusco,andPallara:FunctionsofBoundedVariationandFreeDiscontinuityProblems
SlavyanovandLay:SpecialFunctions:AUnifiedTheoryBasedonSingularities
Joyce:CompactManifoldswithSpecialHolonomy
CarboneandSemmes:AGraphicApologyforSymmetryandImplicitness
Boos:ClassicalandModernMethodsinSummability
HigsonandRoe:AnalyticK-Homology
Semmes:SomeNovelTypesofFractalGeometry
IwaniecandMartin:GeometricFunctionTheoryandNonlinearAnalysis
JohnsonandLapidus:TheFeynmanIntegralandFeynman’sOperationalCalculus,paperback
LyonsandQian:SystemControlandRoughPaths
Ranicki:AlgebraicandGeometricSurgery
Ehrenpreis:TheUniversalityoftheRadonTransform
LennoxandRobinson:TheTheoryofInfiniteSolubleGroups
Ivanov:TheFourthJankoGroup
Huybrechts:Fourier-MukaiTransformsinAlgebraicGeometry
Hida:HilbertModularFormsandIwasawaTheory
BoffiandBuchsbaum:ThreadingHomologythroughAlgebra
Vazquez:ThePorousMediumEquation
Benzoni-GavageandSerre:Multi-DimensionalHyperbolicPartialDifferentialEquations
Calegari:FoliationsandtheGeometryof3-Manifolds
BoyerandGalicki:SasakianGeometry
Choquet-Bruhat:GeneralRelativityandtheEinsteinEquations
IgnaczakandOstoja-Starzewski:ThermoelasticitywithFiniteWaveSpeeds
Scott:TracesandDeterminantsofPseudodifferentialOperators
FranchiandLeJan:HyperbolicDynamicsandBrownianMotion:AnIntroduction
Jain,Srivastava,andTuganbaev:CyclicModulesandtheStructureofRings
Ringström:OntheTopologyandFutureStabilityoftheUniverse
Johnson,Lapidus,andNielsen:Feynman’sOperationalCalculusandBeyond:NoncommutativityandTime-Ordering
Feynman’s Operational
Calculus and Beyond
Noncommutativity and Time-Ordering
Gerald W. Johnson
Michel L. Lapidus
and
Lance Nielsen
3
Feynman’sOperationalCalculusandBeyond.FirstEdition.GeraldW.Johnson,MichelL.LapidusandLanceNielsen.
©GeraldW.Johnson,MichelL.LapidusandLanceNielsen2015.Publishedin2015byOxfordUniversityPress.
3
GreatClarendonStreet,Oxford,OX26DP,
UnitedKingdom
OxfordUniversityPressisadepartmentoftheUniversityofOxford.
ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship,
andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof
OxfordUniversityPressintheUKandincertainothercountries
©GeraldW.Johnson,MichelL.LapidusandLanceNielsen2015
Themoralrightsoftheauthorshavebeenasserted
FirstEditionpublishedin2015
Impression:1
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin
aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe
priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted
bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics
rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe
aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe
addressabove
Youmustnotcirculatethisworkinanyotherform
andyoumustimposethissameconditiononanyacquirer
PublishedintheUnitedStatesofAmericabyOxfordUniversityPress
198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica
BritishLibraryCataloguinginPublicationData
Dataavailable
LibraryofCongressControlNumber:2015931993
ISBN978–0–19–870249–8
PrintedinGreatBritainby
ClaysLtd,StIvesplc
LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand
forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials
containedinanythirdpartywebsitereferencedinthiswork.
DEDICATIONS
ToGeraldJohnson(Jerry),mywonderfulcoauthor,long-timefriendandcollaborator,forall
hisuniquequalities,includinghishumor,incredibleintegrity,loyalty,generosityand
perspicacity.
Tohiswife,Joan,hislife-longcompanionandbestfriend,withgratitudeforherfriendship
andforallshedoesforJerry.
Tomyownwifeandcompanion,Odile,theloveofmylife,withoutwhomIcannotlive.
TothememoryofRichardFeynmanandMarkKac,friendsandcolleagues,withwhomwe
sharedmanythoughtsandjoyfulmoments.
MichelL.Lapidus
IfirstmetJerryJohnsonafteraskingsomefacultyattheUniversityofNebraska,Lincoln,if
therewasanyoneinthedepartmentofmathematicswhoworkedwithpathintegrals.Iwastold
totalktoDr.Johnson,whowasfinishingclassinsuch-and-sucharoom.Ifoundthisroomand,
whileJerrywaserasingthechalkboard,Iaskedifheknewofagoodbookconcerningpath
integrals.Theresponsewas,withasmile,“I’mwritingone.”Ofcourse,thebookinquestionis
thatwrittenbyJerryandMichel,i.e.,TheFeynmanIntegralandFeynman’sOperational
Calculus[114].Thishappenstance,sometwodecadesago,startedmedownthepaththatI
continuetofollowtoday.
Astimemovedon,Jerryconsentedtobemyadvisor,andIcontinuetobeverythankfulthatI
wasluckyenoughtohavehadsomeonewithJerry’swisdom,integrity,humorandpatienceasa
mentorandfriendduringmytimeasaPh.D.candidateandduringmyyearsinacademia.Itis
becauseofJerrythatIlearnedhowtodoandwritemathematicsand,wereitnotforhim,I
wouldnotbeacoauthorofthisvolume.Ithasbeenasingularprivilegetoworkonthisvolume
withJerryandalsowithMichel,bothmathematiciansofenormoustalentandindividualsIhold
inthehighestesteem.
Finally,I’dliketothankmydearestfriendAmyforhergood-humoredtolerancetomy
continualprotestationsthat“Ican’t,Ihavetoworkonthebook.”Amyandheryoungestson
Sam(aswellasthefamilydogLucky!)oftengavemeaneededescapefromthemanuscript
whenIcouldnotbeartolookatLATEXanylonger.
LanceNielsen

ACKNOWLEDGMENTS
MichelLapidus’sresearchdescribedinpartsofthisbookwaspartiallysupportedbythe
U.S.NationalScienceFoundation(NSF)undergrantsDMS-0707524andDMS-1107750
(aswellasbymanyearlierNSFresearchgrantssincethemid-1980s).
The second author would also like to express his gratitude to many research insti-
tutes at which he was a visiting professor while aspects of this research program were
developed,includingespeciallytheMathematicalSciencesResearchInstitute(MSRI)in
Berkeley,California,USA,theErwinSchrödingerInternationalInstituteofMathematical
Physics in Vienna, Austria, and the Institut des Hautes Etudes Scientifiques (IHES) in
Bures-sur-Yvette,Paris,France.
MichelLapidus
I would like to express my gratitude to the attendees—Jerry Johnson and Dave Skoug
among others—of the functional integration seminar at the University of Nebraska,
Lincoln, for the many opportunities to present and discuss my research, parts of which
appearinthisbook.Iwouldalsoliketothankthesecondauthor,MichelLapidus,forpro-
viding a portion of the funding for a trip to Riverside, California, in 2009, to begin the
processofwritingthisvolume.
LanceNielsen

PREFACE
Thisbookisaimedatprovidingacoherent,essentiallyself-contained,rigorousandcom-
prehensiveabstracttheoryofFeynman’soperationalcalculusfornoncommutingoperators.
AlthoughitisinspiredbyFeynman’soriginalheuristicsuggestionsandtime-orderingrules
inhisseminalpaper[58],aswillbemadeabundantlyclearintheintroduction(Chapter1)
andelsewhereinthetext,thetheorydevelopedinthisbookalsogoeswellbeyondthemin
anumberofdirectionswhichwerenotanticipatedinFeynman’swork.Hence,thesecond
partofthemaintitleofthisbook.
It may be helpful to the reader to situate the present research monograph relative to
a companion book [114], written by the first two named authors (Gerald Johnson and
MichelLapidus)andtitledTheFeynmanIntegralandFeynman’sOperationalCalculus.(Let
usreassurethereaderatoncethat[114]isnotaprerequisiteforthepresentbook,however,
aswillbediscussedinmoredetailfurtheroninthispreface.)Thelatternearly800-page
book[114]wasinitiallypublishedin2000byOxfordUniversityPress(withapaperback
editionin2002andanelectroniceditioninthelate2000s)inthesameseriesasthepresent
monograph.ItprovidesanumberofdifferentapproachestotheFeynmanpathintegral(or
“sumsoverhistories”),inboth“real”and“imaginary”time.
BeginningwithChapter14andendingwithChapter18,thesecondpartof[114](based,
in part, on [110–113] along with [137–143]) develops a rigorous theory of Feynman’s
operational calculus, using certain operator-valued Wiener and Feynman path integrals
(called“analytic-in-massFeynmanintegrals”)aswellasassociatedcommutativeBanach
algebrasoffunctionals,called“disentanglingalgebras,”andcorrespondingnoncommutat-
iveoperations(namely,anoncommutativeadditionandmultiplication)actingonthem.
Theresultingtime-indexedfamilyofdisentanglingalgebras,alongwiththeassociatednon-
commutativeoperations,providesarichalgebraic,analyticandcombinatorialstructurefor
thedevelopmentofaconcretetheoryofFeynman’soperationalcalculuswithinthecontext
ofFeynmanpathintegralsandrelatedpathorstochasticintegrals.
Ontheotherhand,Chapter19of[114](basedontheearlierjointworkoftheauthorsof
[114]withBrianDeFacioin[33,34])beginstobuildabridgebetweentheaboverigorous
concreteversionoftheoperationalcalculusandapossible,moregeneraloperationalcalcu-
lusvalidforabstractoperators(actingonBanachorHilbertspaces)notnecessarilyarising
viaWienerorFeynmanfunctionalsandassociatedpathintegrals.Theconnectionswitha
largeclassofassociatedevolutionequationsarealsostudiedinChapter19of[114].
Inasense,Chapters15–18togetherwith,specifically,Chapter19of[114]laythefoun-
dationsandprovideapossiblestartingpointforthedevelopmentofafullyrigorousand
moreabstracttheoryofFeynman’soperationalcalculus,whichistheobjectofthepresent
book.ThereaderfamiliarwithChapters15–19of[114]willrecognizesomeaspectsof,
andmotivationsfor,thetheorydevelopedinthepresentbook,butinessence(withthe