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For two free leptons, the uncoupled equations are first taken as Pauliequations (2.58), including external vector potentials. Using again the com-pact notationπi=πiσi, they may be written in forms which also compriseKG particles,p02iψas=Kiψas,Ki=m2i+π2i.(4.388)The first step is to combine the two equations into a single one whichcontains only a common time shift in the formp01+p02≡p0. Withp021−p022=p0(p01−p02), the difference of the equations gives(p01−p02)p0ψas= (K1−K2)ψas.(4.389)
4.11 Dirac Structures of Binary Bound States197An eigenvalueK0ofp0will be assumed. Asp01−p02commutes withK1andK2, a second application ofp01−p02gives(p01−p02)2ψas= (K0)−2(K1−K2)2ψas.(4.390)Next, consider the sum of the two equations (4.388),(p021+p022−K1−K2)ψas= 0.(4.391)Withp01=12K0+12(p01−p02) etc, this leads to[K02/2 + (K1−K2)2/2K02−K1−K2]ψas= 0.(4.392)In terms of the triangle functionλ(4.76), (K0)−2λ(K02, K1, K2)ψas= 0.In the “constraint Hamiltonian mechanics” which dates back to Dirac, thecoupling between the two particles is introduced already in equations (4.388),but practical success has been limited (Crater and Van Alstine 1994).For particles of equal spins,λmay be decomposed symmetrically in theindices 1 and 2:λ= (K02−K1−K2)2−4K1K2.(4.393)Withm22−m21=m+m−,λ= (K02−m21−m22)2−2π21(K02+m+m−)−2π22(K02−m+m−)−4m21m22+ (π21−π22)2.(4.394)ForA= 0, the transformation (4.291), (4.307) of variables,p1=plab+KE1/E,p2=−plab+KE2/E,E≡(K02−K2)1/2,(4.395)yields together withplab,x=px, plab,y=py, plab,z=γpz,λ=γ2[(E2−m21−m22)2−4m21m22−4E2(p2x+p2y+p2lab,z/γ2)].(4.396)The factorγ2in front of the square bracket corresponds to the separation ofone factorγfrom (4.308). The square bracket is factorized by means of theDirac matricesβ, γ5=βxandiγ5β=βyas follows:λ/γ2= (f0+fzβ+fxβx+fyβy)(f0−fzβ−fxβx−fyβy),(4.397)f0=E2−m21−m22,fz= 2m1m2,fx= 2Epσ1,fy= 0.(4.398)The second factor of (4.398) is precisely that of the leptonium equation(4.237). More general forms include a rotation by an arbitrary angleωDabouttheβ-axis in Dirac space (2.103),fx= 2Epσ1cosωD, fy= 2Epσ1sinωD, ora rotation (4.270) in the Pauli product space. For constantf0, one may alsotakefy= 2Epσ2sinωD, asσ1iσ1jand (σ1icosωD+iσ2isinωD)(σ1jcosωD−iσ2jsinωD) are identical in their symmetric tensor components. (In the quark
1984 Scattering and Bound Statesmodel,fyis not excluded a priori even whenf0isr-dependent. For smallωD, it would mainly affect the hyperfine structure of quarkonium.)The inclusion ofAis trivial in the 16-component equation (3.186) or(3.190), whereas (3.218) is still unclear. The first-order relativistic ZeemanshiftδEof the ground state has been calculated by Faustov (1970) and byGrotch and Hegstrom (1971). Besides the factor (1−α2Z/3) fromHZee(2.285)in the Pauli reduction of the GY-equation (4.378), these authors find essen-