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Propagation of a relativistic particle in terms of the unitary irreducible representations

2022-08-09 来源:伴沃教育
2002 ebF 12 2v360111/0ht-pe:hviXraPropagationofarelativisticparticleintermsoftheunitaryirreducible

representationsoftheLorentzgroup

RudolfA.Frick∗

Institutf¨urTheoretischePhysik,Universit¨atK¨oln,D-50923Cologne,Germany

(February1,2008)

InageneralizedHeisenberg/Schr¨odingerpictureweuseaninvariantspace-timetransformationto

describethemotionofarelativisticparticle.Wediscusstherelationwiththerelativisticmechanicsandfindthatthepropagationoftheparticlemaybedefinedasspace-timetransitionbetweenstateswithequaleigenvaluesofthefirstandsecondCasimiroperatorsoftheLorentzalgebra.Inadditionweuseavectoronthelight-cone.Amassiverelativisticparticlewithspin0isconsidered.Wealsoconsiderthenonrelativisticlimit.

PACSnumbers:03.65.Pm,03.65.-w

I.INTRODUCTION

InthispaperwepresentanewmathematicalformalismfordescribingthemotionofarelativisticparticlewhichisbasedontheprincipalseriesoftheunitaryirreduciblerepresentationsoftheLorentzgroupandageneralizedHeisenberg/Schr¨odingerpicture.TheprincipalseriesoftherepresentationsoftheLorentzgrouphasalreadybeenusedbymanyauthorsinthetheoryofelementaryparticlesandrelativisticnuclearphysics(e.g.,Refs.[1–7]).Inourpreviouspapers[8–10]ithasbeenshownthattheserepresentationsmaybeusedinageneralizedHeisenberg/Schr¨odingerpictureinwhicheithertheanalogueofHeisenbergstatesortheanalogueofSchr¨odingeroperatorsareindependentofbothtimeandspacecoordinatest,x.Forthesestatestheremustbespace-timeindependentexpansion.IfatfirstweusethemomentumrepresentationintheexpansionoftheLorentzgroup,thenthestatesandoperatorsofthePoincar´ealgebracanbeconstructedinanotherspace-timeindependentrepresentation.In[8]thetransitionfromtheHeisenbergtotheSchr¨odingerpictureinquantummechanicsS(t)=exp(−itH)wasgeneralizedtotherelativisticinvarianttransformation(wechoosehereasystemofunitssuchthat¯h=1,c=1)

S(x):=exp[−i(tH−x·P)],

(1.1)

whereHandParetheHamiltonandmomentumoperatorsoftheparticleinthegeneralizedSchr¨odingerpicture.Throughthistransformationtheplanewaves∼exp[−ixp]appearindifferentrepresentationandcannotbeusedintheiroriginalsenseasthestationarystatesofaparticle.Thereisnoxrepresentation.Inthisapproachonemustfindanewmethodfordescribingthemotionoftheparticle.

InthepresentpaperwewillshowthatusingthetransformationS(x)makesitpossibletodescribethemotionofamassiverelativisticparticleintermsofthematrixelementsoftheLorentzgroup.FirstweintroduceintherelativisticmechanicstheanalogueoftheoperatorsoftheLorentzalgebrainthegeneralizedHeisenbergpictureandobtainequationswhichintheinvariantformexpressthemotionofaparticle.ThenweusetheseequationsandtheunitaryirreduciblerepresentationsoftheLorentzgrouptodeterminethetransitonamplitudesforthefreerelativisticparticle.Inparticularweconsideramassiveparticlewithspin0.InthenonrelativisticlimitweusetheexpansionoftheGalileogroup.

II.LORENTZALGEBRAINTHEGENERALIZEDHEISENBERGPICTURE.ANALOGUEIN

RELATIVISTICMECHANICS

InthegeneralizedHeisenberg/Schr¨odingerpicturetheanalogueofSchr¨odingeroperatorsofaparticlearedefinedasspace-timeindependentoperatorsindifferentrepresentations.m=mass,p0:=

󰀁Inthemomentumrepresentation(p=momentum,p,

J=−ip×∇0+m

p+s:=L(p)+s.

(2.1)

canbeviewedassuchoperators.UsingthetransformationS(x)weobtaintheoperatorsoftheLorentzalgebrainthegeneralizedHeisenbergpicture

N(x)=S−1(x)NS(x)=N+tP−xH,J(x)=S−1(x)JS(x)=J−x×P.

(2.2)(2.3)

Timeandspacecoordinatesequallyoccurintheseoperators.FromthispointofviewonecanseeN(x),J(x)asfieldoperatorswhichsatisfytheequations

∂Ni(x)

∂xj

=−Hδij,

∂Ji(x)

III.TRANSITIONAMPLITUDES

Inthequantumversiontheequations(2.12)correspondtothetransitionS(x2−x1)oftheparticlefromonestatetoanotherstatewithequaleigenvaluesoftheoperators

C1=N2−J2,

C2=N·J.

(3.1)

FortheprincipalseriestheeigenvaluesoftheoperatorsC1andC2are1+α2−λ2andαλ,(0≤α<∞,λ=−s,...,s),respectively.Therepresentations(α,λ)and(−α,−λ)areunitarilyequivalent.InthemomentumrepresentationsforamassiveparticlewithspinzeroweusetheeigenfunctionsoftheoperatorsC1

ξ(0)(p,α,n):=

1

p0

Ψ(0)(p)ξ∗(0)(p,α,n)

(3.4)

whereΨ(0)(p)andΨ(0)(α,n)arethestatefunctionsoftheparticlewithspinzeroinpandintheα,nrepresentation.TheHamiltonoperatorH(0)(α,n)andmomentumoperatorsP(0)(α,n)wereconstructedinRef.[3].TheoperatorsNintheα,nrepresentationhavetheform(Ref.[8–10])

N:=αn+(n×L−L×n)/2,

Fortheparticlewithspins

N:=αn+(n×J−J×n)/2

C1=1+α2−(s·n)2,

[C1,n]=0,

J:=L(n)+s,C2=αs·n,

(3.6)(3.7)(3.8)

L:=L(n).

(3.5)

[C2,n]=0.

andasacompletesetofcommutingoperatorsonecanselecttheinvariantsC1,C2andthevectorn.

Intherelativisticmechanicswemustfindthepropertyofthevectornalongthetrajectoryoftheparticle.Inaccordancewithformulas(3.5)intherelativisticmechanicsthequantityN,thefieldN(x)andtheinvariantC1(x)canbeexpressedintheform

N:=αn+n×L(n),

N(x):=α(x)n(x)+n(x)×L(x),

Usingequations(2.11)and(2.12)weobtain

α2(t1,x1)=α2(t2,x2),

n(t1,x1)=n(t2,x2).

(3.11)

C1=α2,C1(x)=α2(x).

(3.9)(3.10)

Let|n,λ,α>bethestateswithawell-definedvalueoftheoperatorC1,C2andthevectorn.TheninaccordancewithEqs.(2.12)and(3.11)forthetransitionamplitudeforamassiverelativisticparticlewehavetheexpressionintermsofthematrixelementsoftheunitaryirreduciblerepresentationsoftheLorentzgroup

K(x2;x1,α,λ,n):=<α,λ,n|S(x2−x1)|n,λ,α>α,λ,n=α′,λ′,n′.

Forexampleforaparticlewithspinzero

3

(3.12)

󰀂dp

K(x2;x1,n)=(

(2π)3

󰀂

dp

[(pn)/m]2

.(3.13)

Thistransitionamplitudecontainthevectorofthelight-conen.Applyingtheoperator[i(n0∂t+n∇x)/m]2wehavetherelationtotheFeynmanpropagator△+(x)ofthefreeKlein-Gordonequation

△+(x)=

−i

(2π)3

󰀂

dp

(2π)3/2

exp[−i(αn)·p/m].(3.16)

ThefunctionsΨ(p,αn)aretheeigenfunctionsoftheoperatorsqandq2.In[10])wasremarkedthatintheexpansionoftheGalileogroup

Ψ(αn)=

1

∂t

=Pi/m,

∂qi(t,x)/m

2πı(t−t0)

]

3/2

exp

ım(x2−x1)2

IV.CONCLUSION

WehaveshownthatthepropagationofarelativisticparticlemaybedescribedintermsofthetransformationS(x)=exp[−i(tH−x·P)]andthematrixelementsoftheunitaryirreduciblerepresentationsoftheLorentzgroup.WehaveconsideredtheoperatorsoftheLorentzalgebrainthegeneralizedHeisenbergpictureasfieldoperatorsandfoundthattheanalogueofthespace-timeindependentoperatorsintherelativisticmechanicsmustbeseparatedfromtheintegralsofmotion.Inthiscasetheconversiontothequantumversioncantakeplace.Thetransitionamplitudeforaparticlewithspinzerocontainavectorofthelight-conewhichappearsintheexpansionsoftheLorentzgroup.FinallywehaveshownthatinthenonrelativisticlimitthetransitionamplitudemaybealsoexpressedintermsofthetransformationS(x).

ACKNOWLEDGMENTS

ThisworkwassupportedbytheDeutscheForschungsgemeinschaft(No.FR1560/1-1).

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