-- 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