-- Verify formulas for Compton scattering (run time 20-40 seconds) E = sqrt(omega^2 + m^2) p1 = (omega, 0, 0, omega) p2 = (E, 0, 0, -omega) p3 = (omega, omega expsin(theta) expcos(phi), omega expsin(theta) expsin(phi), omega expcos(theta)) p4 = (E, -omega expsin(theta) expcos(phi), -omega expsin(theta) expsin(phi), -omega expcos(theta)) 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)) q1 = p1 + p2 q2 = p4 - p1 qslash1 = dot(q1,gmunu,gamma) qslash2 = dot(q2,gmunu,gamma) pslash2 = dot(p2,gmunu,gamma) pslash4 = dot(p4,gmunu,gamma) P2 = pslash2 + m I P4 = pslash4 + m I Q1 = qslash1 + m I Q2 = qslash2 + m I T = dot(P2,gammaT,Q1,gammaT,P4,gammaL,Q1,gammaL) f11 = contract(contract(contract(T,3,4),2,3)) T = dot(P2,gammaT,Q2,gammaT,P4,gammaL,Q1,gammaL) f12 = contract(contract(contract(T,3,5),2,3)) T = dot(P2,gammaT,Q2,gammaT,P4,gammaL,Q2,gammaL) f22 = contract(contract(contract(T,3,4),2,3)) "Verify momentum formulas (1=ok)" f11 == 32 dot(p1,gmunu,p2) dot(p1,gmunu,p4) + 64 m^2 dot(p1,gmunu,p2) - 32 m^2 dot(p1,gmunu,p3) - 32 m^2 dot(p1,gmunu,p4) + 32 m^4 f12 == 16 m^2 dot(p1,gmunu,p2) - 16 m^2 dot(p1,gmunu,p4) + 32 m^4 f22 == 32 dot(p1,gmunu,p2) dot(p1,gmunu,p4) + 32 m^2 dot(p1,gmunu,p2) - 32 m^2 dot(p1,gmunu,p3) - 64 m^2 dot(p1,gmunu,p4) + 32 m^4 "Verify Mandelstam formulas (1=ok)" s = dot(p1 + p2,gmunu,p1 + p2) t = dot(p1 - p3,gmunu,p1 - p3) u = dot(p1 - p4,gmunu,p1 - p4) f11 == -8 s u + 24 s m^2 + 8 u m^2 + 8 m^4 f12 == 8 s m^2 + 8 u m^2 + 16 m^4 f22 == -8 s u + 8 s m^2 + 24 u m^2 + 8 m^4 m = 0 s == 4 omega^2 u == -2 omega^2 (expcos(theta) + 1) "Verify probability density (1=ok)" d11 = (s - m^2)^2 d12 = (s - m^2) (u - m^2) d22 = (u - m^2)^2 A = 1/4 (f11/d11 + 2 f12/d12 + f22/d22) b = expcos(theta) + 1 B = 2 (b/2 + 2/b) A == B
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