"Dirac equation 3"

u1 = sqrt(E + m c^2) *
     (1, 0, c pz / (E + m c^2), (px + i py) c / (E + m c^2))

u2 = sqrt(E + m c^2) *
     (0, 1, (px - i py) c / (E + m c^2), -pz c / (E + m c^2))

v1 = sqrt(E + m c^2) *
     (pz c / (E + m c^2), (px + i py) c / (E + m c^2), 1, 0)

v2 = sqrt(E + m c^2) *
     ((px - i py) c / (E + m c^2), -pz c / (E + m c^2), 0, 1)

E = sqrt(px^2 c^2 + py^2 c^2 + pz^2 c^2 + m^2 c^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)

p = (E/c, px, py, pz)

pslash = dot(p,gmunu,gamma)

"Verify spinors"

check(dot(pslash,u1) == m c u1)
check(dot(pslash,u2) == m c u2)

check(dot(pslash,v1) == -m c v1)
check(dot(pslash,v2) == -m c v2)

"ok"

"Verify normalization"

check(dot(u1,conj(u1)) == 2 E)
check(dot(u2,conj(u2)) == 2 E)

check(dot(v1,conj(v1)) == 2 E)
check(dot(v2,conj(v2)) == 2 E)

"ok"

"Verify completeness"

adjoint(u) = dot(conj(u),gamma0)

check(outer(u1,adjoint(u1)) + outer(u2,adjoint(u2)) == pslash c + m c^2 I)
check(outer(v1,adjoint(v1)) + outer(v2,adjoint(v2)) == pslash c - m c^2 I)

"ok"