Uncertainty Relation in Quantum Mechanics with Quantum Group

We study the commutation relations, uncertainty relations and spectra of position and momentum operators within the framework of quantum group % symmetric Heisenberg algebras and their (Bargmann-) Fock representations. As an effect of the underlying noncom

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aDAMPT/93-65andhep-th/9311147UncertaintyRelationinQuantumMechanicswithQuantumGroupSymmetryAchimKempf DepartmentofAppliedMathematics&TheoreticalPhysicsUniversityofCambridge,CambridgeCB39EW,U.K.Email:a.kempf@damtp.cambridge.ac.ukNovember1993AbstractWestudythecommutationrelations,uncertaintyrelationsandspectraofpositionandmomentumoperatorswithintheframeworkofquantumgroupsymmetricHeisenbergalgebrasandtheir(Bargmann-)Fockrepresentations.Asane ectoftheunderlyingnoncommutativegeometry,alengthandamo-mentumscaleappear,leadingtotheexistenceofminimalnonzerouncertaintiesinthepositionsandmomenta.Theusualquantummechanicalbehaviourisre-coveredasalimitingcasefornottoosmallandnottoolargedistancesandmomenta.1Introduction

ThequantumgroupSUq(n)thatwewillusehereisageneralisationoftheLiegroupSUq(n).OnerecoverstheLiegroupwhentherealparameterqtendsto1.Tech-nicallywearedealingwithanon-commutativenon-cocommutativequasitriangularHopfalgebra.Itsdualisanoncommutativegeneralisationofthefunctionalgebraonthegroupmanifold.Wecanthusconsiderthequantumgroupasanexampleofnoncommutativegeometry[1,2,3,4].

Itistemptingtoexamine,whethernoncommutativegeometry,whenintroducedintoquantumtheory,regularisesitsshortdistancebehaviour.Theideais,nottobreaksymmetriesbythisregularisingprocedure,butto(quantumgroup-)generalisetheminstead.Onemaythenevenspeculateaboutgravityenteringthepicture.This