Uncertainty Relation in Quantum Mechanics with Quantum Group(2)

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

couldbeinsuchawaythatnotonlygravityisquantisedbutalsothatgravitywouldfeedbacktoquantumtheorybymodifyingthecanonicalcommutationrelations.Wewillhoweverforthepresentcon neourselvestothecaseofnonrelativis-ticquantummechanics.Thestudyofsomee ectsofnoncommutativegeometryinquantummechanicswasoutlinedin[5].Herewecoveramoregeneralcaseandgivedetailsandproofs.Ourresultswillsupporttheideathatnoncommutativegeometryhasindeedthepotentialtoregulariseultravioletandeveninfrareddivergenciesinquantum eldtheories.

1.1Heisenbergalgebra

InourapproachwegeneratetheHeisenbergalgebraofndegreesoffreedombymu-tuallyadjointoperatorsaranda r,(r=1,...,n).ThisproceedingwillautomaticallysupplyuswithaHilbert(Fock-)spacerepresentationoftheHeisenbergalgebra.Inusualquantummechanicsthisisofcourseequivalenttotheuseofthehermiteangeneratorsxrandpr,(whicharethewellknownlinearcombinationsoftheformerones)andtherepresentatione.g.ontheHilbertspaceofsquareintegrablefunctions.WewillusethequantumgroupSUq(n)asa’symmetry’groupfornontrivialcommutationrelationsi.e.aslinearquantumcanonicaltransformations.TechnicallytheHeisenbergalgebraisaFunSUq(n)-comodulealgebra[6].ArbitraryHamiltonianscanbestudiedwithinourframeworkandtheynotnecessarilyhavethissymmetry.ExplicitelythecommutationrelationsofthefollowinggeneralisedbosonicHeisen-bergalgebraareconservedundertheactionofthequantumgroupSUq(n):

aiaj qajai=0

a iaj qajai=0

aia j qajai=0forforfor

2i<ji>ji=j j<i(1)(2)(3)a jaj(4)aia i q2a iai=1+(q 1)

Hereirunsfrom1tonandqisreal.Theserelationsandtheirfermioniccounter-partwerederivedintheR-matrixapproachin[6].AsIlearnedlatertheyhad rstappearedinadi erentapproach[7].Theyarerelatedtothedi erentialcalculusonquantumplanes[8]andcanalsobeunderstoodasabraidedsemidirectproductconstruction[9].Comparealsowiththedi erentapproachese.g.in[10,11,12,13].Althoughquantumgroupsdoingeneralhavemorethanonefreeparameter,nofur-therparametersenterintheabovecommutationrelations[14,15].

1.2BargmannFockrepresentation

0|0 =1andai|0 =0fori=1,...,n

2AsusualtheFockspaceisconstructedfromavector|0 with