Question

. 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 is in the next formula:

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

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

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

Note$\Psi^*_i$ is the complex conjugated component of the 4 vector of Dirac.

Notably;

$\Psi_i=\Psi_i(x\mu),\Psi^*_i=\Psi^*_i(x_\mu), i=0..3$