Rigged Hilbert Space Resonances and Time Asymmetric Quantum

The Rigged Hilbert Space (RHS) theory of resonance scattering and decay is reviewed and contrasted with the standard Hilbert space (HS) theory of quantum mechanics. The main difference is in the choice of boundary conditions. Whereas the conventional theor

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aRiggedHilbertSpaceResonancesandTimeAsymmetricQuantumMechanics A.BohmandH.KaldassUniversityofTexasatAustinPhysicsDepartmentAustin,TX78712E-mail:bohm@physics.utexas.eduFebruary1,2008AbstractTheRiggedHilbertSpace(RHS)theoryofresonancescatteringanddecayisreviewedandcontrastedwiththestandardHilbertspace(HS)theoryofquantummechanics.Themaindi erenceisinthechoiceofboundaryconditions.Whereastheconventionaltheoryal-lowsforthein-statesφ+andtheout-states(observables)ψ oftheS-matrixelements(ψ ,φ+)=(ψout,Sφin)anyelementsoftheHSH,{ψ }={φ+}(=H),theRHS×theorychoosestheboundaryditions:φ+∈Φ H Φ ,ψ ∈Φ+ H Φ×con-+,whereΦ (Φ+)areHardyclassspacesassociatedtothelower(upper)half-

planeofthesecondsheetoftheanalyticallycontinuedS-matrix.Thiscanbephenomenologicallyjusti edbycausality.ThetwoRHS’sforstatesφ+andobservablesψ providenewvectorswhicharenotinH,e.g.theDirac-Lippmann-Schwingerkets|E± ∈Φ× (solutionsoftheLippmann-Schwingerequationwith±i respectively)andtheGamowvectors|ER iΓ/2± ∈Φ× .TheGamowvectors|ER iΓ/2 haveallthepropertiesthatoneheuristicallyneedsforquasistablestates.Inaddition,theygiverisetoasymmetrictimeevolutionexpressingirreversibilityonthemicrophysicallevel.

The Rigged Hilbert Space (RHS) theory of resonance scattering and decay is reviewed and contrasted with the standard Hilbert space (HS) theory of quantum mechanics. The main difference is in the choice of boundary conditions. Whereas the conventional theor

1Introduction

ResonancesanddecayingstatescanreallynotbeunderstoodasautonomouselementaryparticlesinHilbertspacequantummechanicsbecausetheHilbertspacemathematicsdoesnotallowstatevectorscharacterizedbybothanenergyER,andalifetimeτ(orawidthΓ= /τ).ThisisincontrasttothewayexperimentalistsanalyzetheirdataandlisttheirresultsasBreit-Wignerpeakvalueandwidth(ER,Γ)forresonances(largevaluesofΓ/ER)andas(Ed,τ)fordecayingstatesifthe(Breit-WignerorLorentzian)lineshapecannotberesolvedbutthedecayratecanbe ttedtoanexponential(smallvaluesof

The Rigged Hilbert Space (RHS) theory of resonance scattering and decay is reviewed and contrasted with the standard Hilbert space (HS) theory of quantum mechanics. The main difference is in the choice of boundary conditions. Whereas the conventional theor

Inanlaboratoryexperimentthestateofthequantumphysicalsystemispreparedbyapreparationapparatus,e.g.anaccelerator.ThestateW(orφ)isthusexperimentallyde ned(“determined”)bythepreparationap-paratus.Thequantumphysicalobservablesareobservedorregisteredbyaregistrationapparatus,e.g.adetector.TheobservableΛ(orψ)orAarethusexperimentallyde nedbytheregistrationapparatus.

Inexperimentswithquantumsystemsonemeasuresratiosoflargeinte-gersNi/NorN(t)/N,e.g.asratiosofdetectorcountsofthei-thdetectorNiandcountsofalldetectorsNorasratiosofdetectorcountsN(t)inthetimeintervalbetweent=0andt=tandin“all”timeN=N(∞).Thisratiooflargenumbersisinterpretedasprobability,e.g.asprobabilityP(Pi)forapropertyPi

Ni

≈P(Λ(t))(3b)N

wheretheobservablesPiorΛareexperimentallygiven(“de ned”)bythedetector.Foramoregeneralobservable

A=∞ iaiPi(4)

oneobtainstheaveragevalue(oftheeigenvaluesai)

nite i=1aiNi

∞ i=1(3c)P(Pi)=1.N≈

Thesymbol≈denotestheassociationoftheexperimentallymeasuredquan-tityonthelefthandsidewiththetheoreticallycalculatedquantityontherighthandside.

InquantumtheorytheprobabilityofanobservableΛinthestateWattimetiscalculatedas

P(t)=P(Λ(t))=Tr(Λ(t)W0)=Tr(Λ0W(t)).

3(5a)

The Rigged Hilbert Space (RHS) theory of resonance scattering and decay is reviewed and contrasted with the standard Hilbert space (HS) theory of quantum mechanics. The main difference is in the choice of boundary conditions. Whereas the conventional theor

Ifthestateispure,givenbythestatevectorφ,andiftheobservableisapropertygivenbytheone-dimensionalprojector|ψ ψ|,orgivenbythe“observable”vectorψ,then

P(t)=| ψ|φ(t) |2=| ψ(t)|φ |2.(5b)Thetracein(5)iscalculatedusinganybasissystemofthespaceΦ;eitheranydiscretebasis

Φ φ=|i i|φ ;(6a)

i

oranycontinuousbasis(Diracbasisvectorexpansion)

Φ φ=dλ|λ λ|φ (6b)

oranybasissystemconsistingofdiscreteandcontinuous(generalized)eigen-vectorsofacompletesystemofcommutingobservables.Thusthetraceisgivenbye.g.

Tr(ΛW)= i|ΛW|i orTr(ΛW)=dλ λ|ΛW|λ .(7a)

i

Forthespecialcase(5b)theprobabilityfortheobservablyψinthestateφisgivenby

2 ∞ 2(7b)Tr(|ψ ψ|φ φ|)=| ψ|φ |= ψ|i i|φ , i=0

orby

ifoneusesacontinuousbasisofDirackets|λ .

Thetimeevolutionin(5a)and(5b)(dynamicsofthequantumsystem) 2 = dλ ψ|λ λ|φ (7c)

4

The Rigged Hilbert Space (RHS) theory of resonance scattering and decay is reviewed and contrasted with the standard Hilbert space (HS) theory of quantum mechanics. The main difference is in the choice of boundary conditions. Whereas the conventional theor

isgivenbytheHamiltonianoperatorHofthesystem;eitheras

W

[H,W(t)](8a);i φ(t)

i

t= t= Hψ(t).(8d)(Heisenbergpicture)

Noneoftheaboveequationsismathematicallypreciseuntilwede nethespaceΦ,thekets|λ ortheintegrationdλin(6b)and(7c),andspecifytheinitial-boundaryconditionsφ0etc.fortheequations(8).Beforewechoosethesemathematicalde …… 此处隐藏：21355字，全部文档内容请下载后查看。喜欢就下载吧 ……