Uncertainty Relation in Quantum Mechanics with Quantum Group(14)

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

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