(ebook - physics) quantum field theory - an introduction to string theory

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1、Lecture1QuantumFieldTheories:AnintroductionThestringtheoryisaspecialcaseofaquantumfieldtheory(QFT).AnyQFTdealswithsmoothmaps
ofRiemannianmanifolds,thedimensionofisthedimensionofthetheory.WealsohaveanactionfunctiondefinedonthesetMap ofsmoothmaps.AQFTstudiesintegrals$%

2、 )(+*'&&-,(1.1)Map!#"(+*Here&-,standsforsomemeasureonthespaceofpaths,.isaparameter(usually%verysmall,Planckconstant)and
Map/ 021isaninsertionfunction.Thenumber65798/:shouldbeinterpretedastheprobabilityamplitudeofthecontribution;
<=ofthemap43totheinteg

3、ral.Theintegral>0?A@$ED&(1.2)Map4BCiscalledthepartitionfunctionofthetheory.InarelativisticQFT,thespacehasaLorentzianmetricofsignature#HIKJKJJ4/H GF.Thefirstcoordinateisreservedfortime,therestareforspace.Inthiscase,theintegral(1.1)isreplacedwith>@% G(;*J65798/:M&&

4、N,(1.3)Map437LLetusstartwitha
PO-dimensionaltheory.Inthiscaseisapoint,so&isapoint
<TU1QSRandisascalarfunction.TheMinkowskipartitionfunctionofthetheoryisanintegral>V@DJWB98/:Q(1.4)37LFollowingtheHarvardlecturesofC.Vafain1999,letusconsiderthefollowingexample:

5、12LECTURE1.QUANTUMFIELDTHEORIES:ANINTRODUCTIONExample1.1.RecalltheintegralexpressionfortheX-function:=[e[@@dcDDJXY'Z(1.5)^])_]f]Gg/_]4`Kba4`Kbaih kjThisintegralisconvergentforReMZObutcanbemeromorphicallyextendedtothewholeplanewithpolesatZlRSmn.Wehaveo@@@tsuHdoB poB

6、oqJcXY'ZZBXYMZXYXr@Bysubstitutingsvin(1.5),weobtaintheGaussintegral:]]Gw[uXY@@zyDJ(1.6)vv`8]g4x/aih[g`vAlthoughinthesubstitutionaboveisapositiverealnumber,onecanshowthatvvformula(1.6)makesense,asaRiemannintegral,foranycomplexwithRe 0{.OvWhenRe EjthisiseasytoseeusingtheHankelr

7、epresentationofXY'ZasaOcontourintegralinthecomplexplane.Whenvisapureimaginary,itismoredelicateandwereferto[Kratzer-Franz],1.6.1.2.@

8、uTakingv,wecanuse@D~}D]~aihtodefineaprobabilitymeasureon1.ItiscalledtheGaussianmeasure.Letuscomputetheintegral[[>@@D}DJWBWWB9Qh)7LL6[[@o7

9、Here.Wehave[[>@uDJ9)FQiQbQg[pO