"Annihilation 2"

"Verify Casimir trick"

E = sqrt(p^2 + m^2)

p1 = (E, 0, 0, p)
p2 = (E, 0, 0, -p)

p3 = (E,
      E sin(theta) cos(phi),
      E sin(theta) sin(phi),
      E cos(theta))

p4 = (E,
      -E sin(theta) cos(phi),
      -E sin(theta) sin(phi),
      -E cos(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))

P1 = dot(p1,gmunu,gamma) + m I
P2 = dot(p2,gmunu,gamma) - m I

Q1 = dot(p1 - p3, gmunu, gamma) + m I
Q2 = dot(p1 - p4, gmunu, gamma) + m I

T1 = contract(dot(Q1,gammaT,P2,gammaL),2,3)
T2 = contract(dot(Q1,gammaT,P1,gammaL),2,3)
f11 = contract(dot(T1,T2))

T = contract(dot(Q2,gammaT,P2,gammaT,Q1,gammaL),2,4)
f12 = contract(dot(T,P1,gammaL),2,3,1,2)

T1 = contract(dot(Q2,gammaT,P2,gammaL),2,3)
T2 = contract(dot(Q2,gammaT,P1,gammaL),2,3)
f22 = contract(dot(T1,T2))

s = dot(p1 + p2, gmunu, p1 + p2)
t = dot(p1 - p3, gmunu, p1 - p3)
u = dot(p1 - p4, gmunu, p1 - p4)

check(f11 == 8 t u - 24 m^2 t - 8 m^2 u - 8 m^4)
check(f12 == 8 m^2 s - 32 m^4)
check(f22 == 8 t u - 8 m^2 t - 24 m^2 u - 8 m^4)

"ok"