Uncertainty Relation in Quantum Mechanics with Quantum Group(12)

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

Theideaistocheckwhetherthe’Cayleytransform’canbeisometricallyextended.AninverseCayleytransformthenyieldsaself-adjointextensionofx .Tothisendwecalculatetheorthogonalcomplementofthespaces:

L±i,x :=(x ±i).Dx (64)

¯x =H,thesede ciencysubspacesL⊥ Sincex isclosed,symmetricandhasD±i,xcanbewrittenas (65)L⊥±i,x :=ker(x i).Dx

¯x =H.SinceHereweusedthatx =x whichholdsbecausex isclosedandD

thereisonlyoneviandonev ithedimensionsofthesespaces,i.e.thede ciencyindicesarebothequalto1.Wecanthusde nethefollowingone-parameterfamilyofself-adjointextensions:

xsa(φ).a:=i(b+U.b)foralla=b U.b(66)

withtheisometricoperatorUde nedon(x +i).Dx ⊕Cvias

U.v:=(x i)(x +i) 1.v

and v∈(x +i).Dx =L+i,x (67)

(68)U.αvi:=αeiφv i

Hereφisafreerealparameter,labelingtheself-adjointextensions.FortheeigenvaluesonecanstaywiththeextendedCayleytransformU,calculateitseigenvalues,andaninverseM¨obiustransformthenmapsthemontotheeigenvaluesofxsa(φ).

Theanalysisforpanalogouslyleadstoaone-parameterfamilyofself-adjointextensionspsa(ψ).Onemaynowbetemptedtotryto xthechoiceoftheextensionparametersφandψbyrequiringthatxsa(φ)andpsa(ψ)bede nedonthesamedomain.Onewouldthenliketodiagonalisexsa(φ)toobtainacoordinatespacerepresentationortodiagonalisepsa(ψ)toobtainamomentumspacerepresentation.However,weknowfromsection3thatxandpcannotbeextendedtoacommondomainonwhichtheyhaveeigenvectors.

4.3Thendimensionalcase

One ndsinthendimensionalcaseessentiallythesamesituation:

Wecalculatethematrixelementsofe.g.thepositionoperatorinj-directionxjintheorthonormalbasisofthevectors:

es1,...,sn:=s1 sn(a 1)·...·(an)[s1]q!·...·[sn]q!|0 (69)

12