-- www.eigenmath.org/muon-production-1.txt -- Verify the Casimir trick for muon production. p = sqrt(E^2 - m^2) rho = sqrt(E^2 - M^2) p1 = (E, 0, 0, p) p2 = (E, 0, 0, -p) p3 = (E, rho expsin(theta) expcos(phi), rho expsin(theta) expsin(phi), rho expcos(theta)) p4 = (E, -rho expsin(theta) expcos(phi), -rho expsin(theta) expsin(phi), -rho expcos(theta)) u11 = (E + m, 0, p1[4], p1[2] + i p1[3]) u12 = (0, E + m, p1[2] - i p1[3], -p1[4]) v21 = (p2[4], p2[2] + i p2[3], E + m, 0) v22 = (p2[2] - i p2[3], -p2[4], 0, E + m) u31 = (E + M, 0, p3[4], p3[2] + i p3[3]) u32 = (0, E + M, p3[2] - i p3[3], -p3[4]) v41 = (p4[4], p4[2] + i p4[3], E + M, 0) v42 = (p4[2] - i p4[3], -p4[4], 0, E + M) u1 = (u11,u12) v2 = (v21,v22) u3 = (u31,u32) v4 = (v41,v42) N = (E + m)^2 (E + M)^2 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)) v2bar = dot(conj(v2),gamma0) -- adjoint of v2 u3bar = dot(conj(u3),gamma0) -- adjoint of u3 -- S is the sum over spin states S = 0 for(s1,1,2,for(s2,1,2,for(s3,1,2,for(s4,1,2, X21 = dot(v2bar[s2],gammaT,u1[s1]), X34 = dot(u3bar[s3],gammaL,v4[s4]), a = dot(X21,X34), -- a is an amplitude f = a conj(a), -- f is a probability density function S = S + f )))) -- 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(X3,gammaT,X4,gammaT),1,4) T2 = contract(dot(X2,gammaL,X1,gammaL),1,4) T = contract(dot(T1,transpose(T2))) -- print 1 if equal S == N T