Define Up dirac spinor:

(uup[px_, py_, pz_, 
            e_] = (Sqrt[e + m]) {1, 0, 
              (pz/(e + m)), ((px + I py)/(e + m))}) // 
      MatrixForm

Define gamma matrices:

MatrixForm[
      gamma0 = {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, -1, 0}, {0, 0, 
            0, -1}}]

MatrixForm[
      gamma1 = {{0, 0, 0, 1}, {0, 0, 1, 0}, {0, -1, 0, 0}, {-1, 0, 0, 
            0}}]

MatrixForm[
      gamma2 = {{0, 0, 0, -I}, {0, 0, I, 0}, {0, I, 0, 0}, {-I, 0, 0, 
            0}}]

MatrixForm[
      gamma3 = {{0, 0, 1, 0}, {0, 0, 0, -1}, {-1, 0, 0, 0}, {0, 1, 0, 
            0}}]

Define current vector:

current[a_, 
        b_] := {(a /. I -> -I) . gamma0 . gamma0 . 
          b, (a /. I -> -I) . gamma0 . gamma1 . 
          b, (a /. I -> -I) . gamma0 . gamma2 . 
          b, (a /. I -> -I) . gamma0 . gamma3 . b}

Make Lorentz vector from stationary spin up electron:

uup0 = uup[0, 0, 0, m]

vector  = current[uup0, uup0]

Make Lorentz transform in z direction:

beta = p/E
gamma  =  E/m
(lorentz  =  {{gamma, 0, 0, beta gamma}, {0, 1, 0, 0}, {0, 0, 1, 
              0}, {beta gamma, 0, 0, gamma}}) // MatrixForm

Transform vector:

lorentz . vector

Define moving Dirac state:

uupp = uup[0, 0, p, e]

Get vector of this state:

cuu = current[uupp, uupp]

Simplify[cuu /. p^2 -> e^2 - m^2]

Axial vector:

(gamma5 = I gamma0 . gamma1 . gamma2 . gamma3) // 
      MatrixForm

axial = current[uup0 . gamma5, uup0]

lorentz . axial

acuu = 
      Simplify[current[uupp . gamma5, uupp] /. 
          p^2 -> e^2 - m^2]