-- Verify Casimir trick for Bhabha scattering p = sqrt(E^2 - m^2) p1 = (E, 0, 0, p) p2 = (E, 0, 0, -p) p3 = (E, p expsin(theta) expcos(phi), p expsin(theta) expsin(phi), p expcos(theta)) p4 = (E, -p expsin(theta) expcos(phi), -p expsin(theta) expsin(phi), -p expcos(theta)) v11 = (p1[4], p1[2] + i p1[3], E + m, 0) v12 = (p1[2] - i p1[3], -p1[4], 0, E + m) u21 = (E + m, 0, p2[4], p2[2] + i p2[3]) u22 = (0, E + m, p2[2] - i p2[3], -p2[4]) v31 = (p3[4], p3[2] + i p3[3], E + m, 0) v32 = (p3[2] - i p3[3], -p3[4], 0, E + m) u41 = (E + m, 0, p4[4], p4[2] + i p4[3]) u42 = (0, E + m, p4[2] - i p4[3], -p4[4]) v1 = (v11,v12) u2 = (u21,u22) v3 = (v31,v32) u4 = (u41,u42) N = (E + m)^4 I = ((1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)) gmunu = ((1,0,0,0),(0,-1,0,0),(0,0,-1,0),(0,0,0,-1)) gamma0 = ((1,0,0,0),(0,1,0,0),(0,0,-1,0),(0,0,0,-1)) gamma1 = ((0,0,0,1),(0,0,1,0),(0,-1,0,0),(-1,0,0,0)) gamma2 = ((0,0,0,-i),(0,0,i,0),(0,i,0,0),(-i,0,0,0)) gamma3 = ((0,0,1,0),(0,0,0,-1),(-1,0,0,0),(0,1,0,0)) gamma = (gamma0,gamma1,gamma2,gamma3) gammaT = transpose(gamma) gammaL = transpose(dot(gmunu,gamma)) v1bar = dot(conj(v1),gamma0) -- adjoint of v1 u4bar = dot(conj(u4),gamma0) -- adjoint of u4 "Sum over spin states" S = 0 for(s1,1,2,for(s2,1,2,for(s3,1,2,for(s4,1,2, X13 = dot(v1bar[s1],gammaT,v3[s3]), X42 = dot(u4bar[s4],gammaL,u2[s2]), X12 = dot(v1bar[s1],gammaT,u2[s2]), X43 = dot(u4bar[s4],gammaL,v3[s3]), a = -1/t dot(X13,X42) + 1/s dot(X12,X43), -- a is an amplitude f = a conj(a), -- f is a probability density function S = S + f )))) S "Casimir trick" pslash1 = dot(p1,gmunu,gamma) pslash2 = dot(p2,gmunu,gamma) pslash3 = dot(p3,gmunu,gamma) pslash4 = dot(p4,gmunu,gamma) X1 = pslash1 - m I X2 = pslash2 + m I X3 = pslash3 - m I X4 = pslash4 + m I T1 = contract(dot(X1,gammaT,X3,gammaT),1,4) T2 = contract(dot(X4,gammaL,X2,gammaL),1,4) f11 = contract(dot(T1,transpose(T2))) T = contract(dot(X1,gammaT,X2,gammaT,X4,gammaL,X3,gammaL),1,6) f12 = contract(contract(T,1,3)) T1 = contract(dot(X1,gammaT,X2,gammaT),1,4) T2 = contract(dot(X4,gammaL,X3,gammaL),1,4) f22 = contract(dot(T1,transpose(T2))) f = f11/t^2 - f12/(s t) - conj(f12)/(s t) + f22/s^2 f "Verify Casimir trick (1=ok)" S/N == f