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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