Solution to becker becker schwarz

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1、SolutionstoK.Becker,M.Becker,J.SchwarzStringTheoryAndM-theoryMikhailGoykhmangoykhman89@gmail.comAbstractDetailedsolutionsto154of182ofhomeworkproblemsinK.Becker,M.Becker,J.SchwarzStringTheoryAndM-theorytextbookarepresented.1IntroductionStudyingthecourseofStringTheo

2、ryandsolvingtheseproblemsIhaveextensivelyusedtext-booksofBBS[1];GSW[2];Polchinski[3];Kaku[4];DiFrancesco,etal.[5];Kiritsis[6];Hawking,Ellis[7];S.Weinberg[8],etc.SomeoftheproblemsgiveninBBSashomeworktaskareactuallywellknownstringtheoryfactswhicharedescribedinpapers

3、ortextbooks,especiallyinGSW.Referencestoequationsofsometextbookaregivenhereinformatbook(formula),e.g.BBS(12.140)givesmetricforanextremalblackD3-brane.Equationswithoutanyauthoracronyminfrontofitrefertothepresentpaper.ThisworkwasdonebymyselfwhileIwasfthyearundergra

4、duatestudentatMIPTonmyMasterProgramandwasworkinginBLTPJINRandcarriesnoendorsementfromK.Becker,M.BeckerorJ.H.Schwarz.2ThebosonicstringProblem2.1(i)Stringequationsofmotioninconformalgaugeofworld-sheetmetric( atmetricforworld-sheetwithnotopologicalobstructions)are@

5、2@2X=0:(2.1)@2@2Theseequationsaresatisedbythefollowingopenstringclassicalconguration01X=B;X=Bcoscos;2iX=Bsincos;X=0;i>2:ObviouslyNeumannboundaryconditions0X(=0;)=0(2.2)aresatisedtoo.3-velocityofsomepointonstringisequaltodXi1dXiiv==;dX0Bd1modulusofw

6、hichontheendsofthetreatedstringisevidentlyequalto1,andthereforetheendsofthisstringareindeedmovingwiththespeedoflight.(ii)AsananalogytothepointparticlewecanwriteD-momentumofpointsofstring(wecanuseNoethertheoremtoo,andbuildenergy-momentumtensor,fromwhichwecangetthis

7、-densityof4-momentum):P=T@X:(2.3)FromthisexpressionwecanndthetotalenergyofstringZ0E=dP0=BT:WecanuseNoethertheoremtoderivethedensityofaconservedangularmomentumtensorJ=TXX_XX_(2.4)anduseittoderivetotalangularmomentumoftheconsideredstring:ZZ1212J=djJ3

8、j=dJ=TB:2ObviouslyittakesplaceanequalityE2=2T:J(iii)InconformalgaugetheconstraintT=0,depictingequationsofmotionforworld-sheetmetric,mayberewrittenasX_