-- Verify formulas for Rutherford scattering E = sqrt(p^2 + m^2) p1 = (E, 0, 0, p) p2 = (E, p sin(theta) cos(phi), p sin(theta) sin(phi), p cos(theta)) u11 = (E + m, 0, p1[4], p1[2] + i p1[3]) / sqrt(E + m) u12 = (0, E + m, p1[2] - i p1[3], -p1[4]) / sqrt(E + m) u21 = (E + m, 0, p2[4], p2[2] + i p2[3]) / sqrt(E + m) u22 = (0, E + m, p2[2] - i p2[3], -p2[4]) / sqrt(E + m) 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) "Verify Casimir trick" u21bar = dot(conj(u22),gamma0) -- adjoint of u21 u22bar = dot(conj(u22),gamma0) -- adjoint of u22 M11 = dot(u21bar, gamma0, u11) M12 = dot(u21bar, gamma0, u12) M21 = dot(u22bar, gamma0, u11) M22 = dot(u22bar, gamma0, u12) M = 1/2 (conj(M11) M11 + conj(M12) M12 + conj(M21) M21 + conj(M22) M22) pslash1 = dot(p1,gmunu,gamma) pslash2 = dot(p2,gmunu,gamma) f = 1/2 contract(dot(pslash2 + m I, gamma0, pslash1 + m I, gamma0)) check(f == M) "ok" "Verify probability density" check(f == 2 (E^2 + m^2 + p^2 cos(theta))) q = p1 - p2 q4 = dot(q,gmunu,q)^2 check(q4 == 16 p^4 sin(theta/2)^4) check(q4 == 4 p^4 (cos(theta) - 1)^2) -- verify integral f = 1 / (cos(theta) - 1)^2 I = 1 / (cos(theta) - 1) check(f sin(theta) == d(I,theta)) -- verify cdf F = (I - eval(I,theta,a)) / (eval(I,theta,pi) - eval(I,theta,a)) check(F == 2 (cos(a) - cos(theta)) / ((1 + cos(a)) (1 - cos(theta)))) "ok" Run