-- Verify formulas for Rutherford scattering -- eigenmath.org/rutherford-scattering-1.txt E = sqrt(p^2 + m^2) p1 = (E, 0, 0, p) p2 = (E, p expsin(theta) expcos(phi), p expsin(theta) expsin(phi), p expcos(theta)) u11 = (E + m, 0, p1[4], p1[2] + i p1[3]) u12 = (0, E + m, p1[2] - i p1[3], -p1[4]) u21 = (E + m, 0, p2[4], p2[2] + i p2[3]) u22 = (0, E + m, p2[2] - i p2[3], -p2[4]) u1 = (u11,u12) u2 = (u21,u22) N = (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) pslash1 = dot(p1,gmunu,gamma) pslash2 = dot(p2,gmunu,gamma) u2bar = dot(conj(u2),gamma0) -- adjoint of u2 "Sum over spin states" S = 0 for(a,1,2,for(b,1,2, M = dot(u2bar[b],gamma0,u1[a]), -- M is an amplitude f = M conj(M), -- f = |M|^2 S = S + f )) S = S / N -- normalize S "Casimir trick" T = contract(dot(pslash2 + m I,gamma0,pslash1 + m I,gamma0)) T "Does S equal T?" test(S == T,"yes","no") "Verify additional formulas (1=ok)" T == 4 (E^2 + m^2 + p^2 expcos(theta)) q = p1 - p2 q4 = dot(q,gmunu,q)^2 q4 == 16 p^4 expsin(theta/2)^4 q4 == 4 p^4 (expcos(theta) - 1)^2
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