Eigenmath Demo

-- Verify Casimir trick for muon pair production

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

rho = sqrt(E^2 - M^2)

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

p3 = (E,
      rho expsin(theta) expcos(phi),
      rho expsin(theta) expsin(phi),
      rho expcos(theta))

p4 = (E,
      -rho expsin(theta) expcos(phi),
      -rho expsin(theta) expsin(phi),
      -rho expcos(theta))

u11 = (E + m, 0, p1[4], p1[2] + i p1[3])
u12 = (0, E + m, p1[2] - i p1[3], -p1[4])

v21 = (p2[4], p2[2] + i p2[3], E + m, 0)
v22 = (p2[2] - i p2[3], -p2[4], 0, E + m)

u31 = (E + M, 0, p3[4], p3[2] + i p3[3])
u32 = (0, E + M, p3[2] - i p3[3], -p3[4])

v41 = (p4[4], p4[2] + i p4[3], E + M, 0)
v42 = (p4[2] - i p4[3], -p4[4], 0, E + M)

u1 = (u11,u12)
v2 = (v21,v22)
u3 = (u31,u32)
v4 = (v41,v42)

N = (E + m)^2 (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))

v2bar = dot(conj(v2),gamma0) -- adjoint of v2
u3bar = dot(conj(u3),gamma0) -- adjoint of u3

"Sum over spin states"

S = 0

for(s1,1,2,for(s2,1,2,for(s3,1,2,for(s4,1,2,
  X21 = dot(v2bar[s2],gammaT,u1[s1]),
  X34 = dot(u3bar[s3],gammaL,v4[s4]),
  a = dot(X21,X34),  -- a is an amplitude
  f = a conj(a),     -- f is a probability density function
  S = S + f
))))

"Casimir trick"

pslash1 = dot(p1,gmunu,gamma)
pslash2 = dot(p2,gmunu,gamma)
pslash3 = dot(p3,gmunu,gamma)
pslash4 = dot(p4,gmunu,gamma)

X1 = pslash1 + m I
X2 = pslash2 - m I
X3 = pslash3 + M I
X4 = pslash4 - M I

T1 = contract(dot(X3,gammaT,X4,gammaT),1,4)
T2 = contract(dot(X2,gammaL,X1,gammaL),1,4)

T = contract(dot(T1,transpose(T2)))
T

"Verify Casimir trick (1=ok)"

1/N S == T