Dimensional Transmutation and Dimensional Regulari

arXiv:hep-th/0003255v24Jan2001DimensionalTransmutationandDimensionalRegularizationinQuantumMechanicsI.GeneralTheoryHoracioE.CamblongDepartmentofPhysics,UniversityofSanFrancisco,SanFrancisco,California94117-1080andLuisN.Epele,HunerFanchiotti,andCarlosA.Garc´ıaCanal.LaboratoriodeF´ısicaTe´orica,DepartmentodeF´ısica,UniversidadNacionaldeLaPlata,C.C.67,1900LaPlata,ArgentinaDedicatedtoC.G.BolliniandJ.J.GiambiagiThisisthefirstinaseriesofpapersaddressingthephenomenonofdimensionaltransmutationinnonrelativisticquantummechanicswithintheframeworkofdimensionalregularization.Scale-invariantpotentialsareidentifiedandtheirgeneralpropertiesarederived.Astrategyfordimensionalrenormalizationofthesesystemsinthestrong-couplingregimeispresented,andtheemergenceofanenergyscaleisshown,bothforthebound-stateandscatteringsectors.Finally,dimensionaltransmutationisexplicitlyillustratedforthetwo-dimensionaldelta-functionpotential.I.INTRODUCTIONItiswellknownthat,forvariousmodelsofquantumfieldtheory,amassscaleemergesspontaneouslythroughtherenormalizationprocedure,evenwhentheoriginaltheoryhasnodimensionalparameters.Thisphenomenon,calleddimensionaltransmutation,wasfirstanalyzedinthe1973seminalworkofColemanandWeinberg[1],wherethescalarfieldofmasslessscalarelectrodynamicswasshowntodevelopanonzerobutarbitraryexpectationvalue;asaconsequence,theparticlesofthetheoryacquirenonzerophysicalmasses[2,3].Inshort,theColeman-WeinbergmechanisminducesradiativecorrectionstotheHiggspotential,therebysuggestingtherelevanceofdimensionaltransmutationforthegenerationofparticlemasses[3,4].Themaingoalofthispaperistopresentathoroughinvestigationofdimensionaltransmutationinnonrelativisticquantummechanics,withathreefoldpurposeinmind:(i)attheconceptuallevel,toshowthatquantumfieldtheoryisnotaprerequisiteforitsexistence;(ii)mathematically,tocharacterizetheclassofscale-invariantpotentials,aswellasthesubclassofpotentialsthatdisplaydimensionaltransmutation;and(iii)atthepracticallevel,todevelopusefultoolsforthetreatmentofacertainclassofsingularquantum-mechanicalpotentials.Inparticular,ourworkoffersadditionalinsightintotwoproblemsthathavebeenextensivelystudiedintheliter-ature:thetwo-dimensionaldelta-functionpotential[2,5,6]andtheinversesquarepotential[7–12].Parenthetically,thefamilyofdelta-functionpotentialsthatisdiscussedinthisarticleisactuallyincludedinalargerclassofsingularpotentialswhoseapparentphenomenologicalusefulnesshasbeenrecognizedforalongtime,sincetheintroductionofpseudopotentials[13]intheearlydaysofquantummechanics[14]andwithsubsequentapplicationsofthezero-rangepotentialinnuclearphysics[15],condensedmatterphysics[16],statisticalmechanics[17],atomicphysics[18],andparticlephysics[5].Likewise,theinversesquarepotentialisrelatedtothedipolepotential,whichhasfoundapplicationsinmolecularphysics[19].Eventhoughearlierresearchonthesubjecthadreliedsolelyontraditionalquantum-mechanicaltechniques,thissituationhaschangedinrecentyears,withtheintroductionofnumerousappli-cationsofquantumfield-theoretictoolsandrenormalizationtheorytothesameproblems.Specifically,amongthemanyapplicationsrelateddirectlyorindirectlytooursingularpotentials,thefollowingareworthmentioning:(i)themathematicalformulationofthetheoryofpseudopotentials[20],whichstartedwiththeworksofRef.[21]andincludesthetechniqueofself-adjointextensions[6,22,23];(ii)thestudyofthenonrelativisticlimitoftheφ4theory,withtheconcomitantquestionofitstriviality[24];(iii)thebasicconceptualunderstandingofquantumfieldtheorymecha-nismsinthesimplerframeworkofquantummechanics,includingstandardregularizationandrenormalizationofthesingularpotentialsmentionedabove[6,12,25–30],anomalies[6,26,29],renormalizationgroupanalysis[12,25,28,31–33],andeffectivefieldtheoryapproach[30,34,35];(iv)theanalysisof(2+1)-dimensionaltheories,includinggravity[36],aswellasChern-Simonstheory[37],theAharonov-Bohmeffect[23,38],andthedynamicsofanyons[39];(v)newapplicationsofcontactpotentialsincondensedmatterphysics,e.g.,forthequantumHalleffect[40];and(vi)themodernformulation,usingeffectivefieldtheory[41],ofthenucleon-nucleonpotential[42],whichhasledtoaplethora1ofquantum-mechanicalpseudopotentials[30,43].Ourpapernaturallyfollowsthetrendsetbythisextensivebibliog-raphy.Themainfocusofourworkwillbeontheconceptofdimension,atermthathasbeenextensivelyusedinthephysicsliteraturetodescribetwoconceptuallydistinctideas.Themeaningtowhichdimensionaltransmutationrefersisthatwhichrelatestomeasurementandwhichcharacterizestheclassofphysicalquantitiesthatexhibitacertaintypeofpower-lawbehavior(dimensionalhomogeneity)withrespecttoagivenchoiceoffundamentalquantities[44].Inwhatfollows,wewillusethetermdimensionalitytodenotetheexponentsoftheassociatedhomogeneousbehaviorforanygivenphysicalquantity[45].Atfirstsight,dimensionaltransmutationisparadoxical,becauseitproducesascaleinaproblemdevoidofdimensionalparameters,inapparentviolationofBuckingham’sΠtheoremoforthodoxdimensionalanalysis[44].However,thisparadoxcanbeultimatelyresolvedbyinvokingthedimensionalarbitrarinessintrinsicintherenormalizationframework[46].Theother