Nonrelativistic Lee model in three dimensional Rie

arXiv:0709.2632v1[hep-th]17Sep2007Non-relativisticLeeModelinThreeDimensionalRiemannianManifoldsFatihErman1,O.TeomanTurgut1,21DepartmentofPhysics,Bo˘gazi¸ciUniversity,Bebek,34342,˙Istanbul,Turkey2FezaG¨urseyInstitute,Kandilli,81220˙Istanbul,TurkeyElectronicmail:[e-mail],[e-mail]1,2008AbstractInthiswork,weconstructthenon-relativisticLeemodelonsomeclassofthreedimen-sionalRiemannianmanifoldsbyfollowinganovelapproachintroducedbyS.G.Rajeev[1].Thisapproachtogetherwiththehelpofheatkernelallowsustoperformtherenor-malizationnon-perturbativelyandexplicitly.Forcompleteness,weshowthatthegroundstateenergyisboundedfrombelowfordifferentclassesofmanifolds,usingtheupperboundestimatesontheheatkernel.Finally,weapplyakindofmeanfieldapproximationtothemodelforcompactandnon-compactmanifoldsseparatelyanddiscoverthatthegroundstateenergygrowslinearlywiththenumberofbosonsn.1IntroductionTheLeemodel,originallyintroducedin[2],isanexactlysolubleandrenormalizablemodelthatincludestheinteractionbetweenarelativisticbosonicfield(“pion”)andaheavysource(“nucleon”)withoneinternaldegreeoffreedom,whichhastwoeigenvaluesdistinguishing“proton”and“neutron”.Byheavy,wemeanthattherecoilofthesourceisneglected.Althoughthismodelisnotveryrealistic,itreflectsimportantfeaturesofnucleon-pionsystemandpresentsapowerfulaspectthatonecandotherenormalization1withouttheuseofperturbationtechniques.Moreover,thecompletenon-relativisticver-sionofthismodelthatdescribesoneheavyparticlesittingatsomefixedpointinteractingwithafieldofnon-relativisticbosonsisasimportantasitsrelativisticcounterpart.ItissimplerthanitsrelativisticversionbecauseitispossibletorenormalizetheHamiltonianofthesystemwithonlyanadditiverenormalizationofthemass(energy)differenceofthefermions.IthasbeenstudiedinatextbookbyHenleyandThirringforsmallnumberofbosonsfromthepointofviewofscatteringmatrix[3]andtherearefurtherattemptsintheliteraturefromseveralpointofviews[4].ItispossibletolookatthesameproblemfromthepointofviewoftheresolventoftheHamiltonianinaFockspaceformalismwitharbitrarynumberofbosons(infactthereisaconservedquantitywhichallowsustorestricttheproblemtothedirectsumofnandn+1bosonsectors).ThisisachievedinaveryinterestingunpublishedpaperbyS.G.Rajeev[1],inwhichanewnon-perturbativeformulationofrenormalizationhasbeenproposed.Wearenotgoingtoreviewtheideasdevelopedinthere.Instead,wesuggestthereadertoreadthroughthepaper[1]tomakethereadingofthispapereasier.Followingtheoriginalideasdevelopedin[1],wewishtoextendthenon-relativisticLeemodelontotheRiemannianmanifoldswiththehelpofheatkerneltechniques,hopingthatonemayunderstandthenatureofrenormalizationongeneralcurvedspacesbetter.Inthiswork,forthesakeofsimplicityweignorethemotionoftheheavyparticleandtakeitspositionaasagivenfixedpointonthemanifold.TheconstructionofthemodelissimplybasedonfindingtheresolventoftheregularizedHamiltonianHǫandshowthatawell-definitefinitelimitoftheresolventexistsasǫ→0+(calledrenormalization)withthehelpofheatkernel.Weprovethatthegroundstateenergyforafixednumberofbosonsisboundedfrombelow,usingthelowerboundestimatesofheatkernelforsomeclassofRiemannianmanifolds,e.g.,Cartan-Hadamardmanifolds(alsoexplicitlyH3),theminimalsubmanifoldsofR3andclosedcompactmanifoldswithnonnegativeRiccicurvature.Wealsostudythemodelinthemeanfieldapproximationforcompactandnon-compactmanifoldsseparatelyandprovethatthegroundstateenergygrowslinearlywiththenumberofbosonsforbothclassesofmanifolds.Thepaperisorganizedasfollows.Inthefirstpart,weconstructthemodelandshowthattherenormalizationcanbeaccomplishedonRiemannianmanifolds.Then,weprovethatthereexistsalowerboundonthegroundstateenergy.Finally,themodelisexaminedinthemeanfieldapproximation.22TheConstructionoftheModelWestartwiththeregularizedHamiltonianofthenon-relativisticLeemodelonathreedimensionalRiemannianmanifold(M,g)withacut-offǫ.Adoptingthenaturalunits(~=c=1),onecanwritedowntheregularizedHamiltonianonthelocalcoordinatesx=(x1,x2,x3)∈MHǫ=H0+HI,ǫ,(1)whereH0=ZMdgxφ†(x)−12m∇2g+mφ(x),(2)HI,ǫ=μ(ǫ)1−σ32+λZMdgxρǫ(x,a)φ(x)σ−+φ†(x)σ+.(3)Here,dgx=pdetgijdxisthevolumeelementand∇2gisLaplace-BeltramioperatororsimplyLaplacian,andφ†(x),φ(x)isthebosoniccreation-annihilationoperatorsdefinedonthemanifoldwiththemetricstructuregij.Sometimesweshallwriteφg(x)inordertospecifywhichmetricstructureitisassociatedwithbutfornowwesimplywritedownφ(x).Also,ρǫ(x,a)isafamilyoffunctionswhichconvergetotheDiracdeltafunctionδg(x,a)(withthenormalizationRMdgxδg(x)=1)aroundthepointaonMaswetakethelimitǫ→0+.ThePaulispinmatricesσ±=12(σ1±iσ2)andσ3areregardedasamatrixrepresentationofthefermioniccreationandannihilationoperatorsactingonC2andμ(ǫ)isabaremassdifferencebetweenthe“proton”and“neutron”statesofthetwostatesystem(“nucleon”).Itsexplicitformwillbedeterminedlateron.Althoughthenumberofbosonsisnotconservedinthemodel,onecanderivefromtheequationsofmotionthatthereexistsaconservedquantityQ=−1−σ32+ZMdgxφ†(x)φ(x)whichtakesonlypositiveintegervalues.IfQ=n∈Z+,wehavespin-upstate(“proton”)withnbosonsorspin-downstate(“neutron”)withn−1bosons.Wecanthinkofthelatterasab