Eigenmath Demo

-- Verify Casimir trick for Compton scattering (25-70 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))

u21 = (p2[1] + m, 0, p2[4], p2[2] + i p2[3])
u22 = (0, p2[1] + m, p2[2] - i p2[3], -p2[4])

u41 = (p4[1] + m, 0, p4[4], p4[2] + i p4[3])
u42 = (0, p4[1] + m, p4[2] - i p4[3], -p4[4])

u2 = (u21,u22)
u4 = (u41,u42)

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)

gammaT = transpose(gamma)
gammaL = transpose(dot(gmunu,gamma))

u4bar = dot(conj(u4),gamma0) -- adjoint of u4

q1 = p1 + p2
q2 = p2 - p3

qslash1 = dot(q1,gmunu,gamma)
qslash2 = dot(q2,gmunu,gamma)

C1 = 1/(s - m^2)
C2 = 1/(u - m^2)

"Sum over spin states"

S = 0

for(s2,1,2,for(s4,1,2,
  a1 = -C1 dot(u4bar[s4],gammaT,qslash1 + m I,gammaT,u2[s2]),
  a2 = -C2 dot(u4bar[s4],gammaT,qslash2 + m I,gammaT,u2[s2]),
  a11 = contract(dot(a1,gmunu,transpose(conj(a1)),gmunu)),
  a12 = contract(dot(a1,gmunu,conj(a2),gmunu)),
  a22 = contract(dot(a2,gmunu,transpose(conj(a2)),gmunu)),
  f = a11 + a12 + conj(a12) + a22,
  S = S + f
))

"Casimir trick"

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))

f = C1^2 f11 + C1 C2 f12 + C1 C2 conj(f12) + C2^2 f22
f

"Verify Casimir trick (1=ok)"

S/N == f