According to Born’s rule, the probability is obtained from the square of the wavefunction. More generally, this is written as the product of the complex conjugate of the wavefunction and the wavefunction itself,

to ensure the probability density is real and non-negative.

Now focus on the Dirac Adjoint, i.e. when spin is 1/2:

There are two critical pointes to further clarify: 1. why (γμ)†=γ0γμγ0; 2. why S(Λ)=exp(−4i​ωμν​σμν), what’s w and what’s sigma here?

First,

Second,