Eigenmath Demo

-- Verify Casimir trick for electron scattering

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

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

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

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

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

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

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

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

u1 = (u11,u12)
u2 = (u21,u22)
u3 = (u31,u32)
u4 = (u41,u42)

N = (E + m)^4

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))

u3bar = dot(conj(u3),gamma0) -- adjoint of u3
u4bar = dot(conj(u4),gamma0) -- adjoint of u4

"Sum over spin states"

a11 = 0
a12 = 0
a22 = 0

for(a,1,2,for(b,1,2,for(c,1,2,for(d,1,2,
  a1 = dot(dot(u3bar[c],gammaT,u1[a]),
           dot(u4bar[d],gammaL,u2[b])),
  a2 = dot(dot(u4bar[d],gammaT,u1[a]),
           dot(u3bar[c],gammaL,u2[b])),
  a11 = a11 + a1 conj(a1),
  a12 = a12 + a1 conj(a2),
  a22 = a22 + a2 conj(a2)
))))

a11
a12
a22

"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,X1,gammaT),1,4)
T2 = contract(dot(X4,gammaL,X2,gammaL),1,4)
f11 = contract(dot(T1,transpose(T2)))

T = contract(dot(X3,gammaT,X1,gammaT,X4,gammaL,X2,gammaL),1,6)
f12 = contract(contract(T,1,3))

T1 = contract(dot(X4,gammaT,X1,gammaT),1,4)
T2 = contract(dot(X3,gammaL,X2,gammaL),1,4)
f22 = contract(dot(T1,transpose(T2)))

f11
f12
f22

"Verfiy Casimir trick (1=ok)"

f11 == 1/N a11
f12 == 1/N a12
f22 == 1/N a22