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

allcandidatesforeigenvectorsarenormalisable.Thusthespectraoftheself-adjointextensionsofthepositionandthemomentumoperatorsarenolongercontinuous.Onlydiscrete(’lattices’of)eigenvaluescanoccur.

Ingeneralonewillhoweverrepresentthexandp,i.e.thefullHeisenbergalgebra,onacommondomainonwhichthexandparesymmetric.Thesymmetryisofcoursetoinsurethatallphysicalexpectationvaluesarereal.Weconcludedthatinthiscasethexandpcannothaveeigenvectors.Thenon-existenceofpositionormomentumeigenvectorsmeantofcoursethenon-existenceofabsoluteprecisioninpositionormomentummeasurements.Wefoundminimalnonzerouncertaintiesinthesemeasurements.Themaximalcommondomainonwhichthexandparesymmetricremainstobedetermined.

Itshouldbeinterestingtoexaminewhetherthisquantummechanicalformalismwith’builtin’minimaluncertaintiesinthexandpcan ndapplicationsine ectivetheories,whereminimalnonzerouncertaintiesinpositionormomentummeasure-mentsappearnaturally,e.g.insolidstateornuclearphysics.

Thepresentpaperalsosupportstheideathatnoncommtutivegeometrymathe-maticshasthepotentialtoregularisethesmall xi.e.theultraviolet,aswellasthesmall p,i.e.theinfraredbehaviourofquantum eldtheories.

Acknowledgements

IwouldliketothankJ.MickelsonandJ.Wessfortheirinterestandusefulcritisims.References

[1]V.G.Drinfel’d,inProc.ICMBerkeley,AMS,Vol.1(1986)798-820

[2]L.D.Faddeev,N.Yu.Reshetikhin,L.A.Takhtajan,Alg.Anal.1,1,(1989)178

[3]S.Majid,Int.J.Mod.Phys.A.Vol.5,No1,(1990)1-91

[4]A.Connes,Publ.I.H.E.S.62(1986)257

[5]A.Kempf,toappearinProc.XXIIDGMConference,Sept.’93,Ixtapa,Mexiko

[6]A.Kempf,Lett.Math.Phys.26:(1992)1-12

[7]W.Pusz,S.Woronowicz,Rep.Math.Phys.27(1989)231

[8]J.Wess,B.Zumino,Nucl.Phys.Proc.Suppl.18B(1991)302

[9]S.Majid,J.Math.Phys.34(10)(1993)4843-4856

[10]A.Macfarlane,J.Phys.A22(1989)4581

[11]L.Biedenharn,J.Phys.A22(1989)L873

14