Question

Q3. In Dirac’s theory, the probability current density is defined by the relation j(r, t) = CΨ* αΨ ,
where Ψ is the four component wave vector. Write the relations for jx, jy, jz in terms of the
component of Ψ i.e.
J(r, t) = Ψ*α Ψ ; jx = C Ψ* αx Ψ

Answer 1

The answer on your question it is next formulas:

$j_1(x_\mu) = \Psi^*_0\Psi_3 + \Psi^*_1\Psi_2 + \Psi^*_2\Psi_1 + \Psi^*_3\Psi_0$

$j_2(x_\mu) = i\Psi^*_0\Psi_3 -i\Psi^*_1\Psi_2 + i\Psi^*_2\Psi_1 -i\Psi^*_3\Psi_0$

$j_3(x_\mu) = -\Psi^*_1\Psi_3 + \Psi^*_0\Psi_2 - \Psi^*_3\Psi_1 + \Psi^*_2\Psi_0$

where $\Psi_i^*$ -complex conjugated component of the 4-vector of Dirac. Note, that $\Psi_i = \Psi_i(x_\mu), \Psi^*_i = \Psi^*_i(x_\mu), i = 0..3$