Is There an Error in the Third Dirac Current Equation?

[woodland]

I'm having a problem writing the third Dirac current eq.
$$1 = \int ψ^t \gamma^0 \gamma^2 ψ$$
which should come out as
$$1 = \int i ψ^0 ψ^3 - i ψ^1 ψ^2 + i ψ^2 ψ^1 - i ψ^3 ψ^0$$
By inspection the first and last terms add to zero and the second and third terms add to zero, so the integral cannot equal one. I checked my matrix math, calculated new gammas but nothing works. Am I making a typo error somewhere or what? Thanks

 

[Matterwave]

Shouldn't the transpose actually be an adjoint?

 

[woodland]

Thanks. I checked on the adjoint point. Adjoint means the conjugate transpose. I'm trying to use some curve fitting on the ψ's and this leads to problems. Such as the four terms in $$ψ^t γ^0 γ^0 ψ$$ would all ways be positive and this would make the first Dirac current always larger than the other three. I simulated this for 10000 tries and it was true every time. As such all four equations cannot be integrated to 1 simultaneously. This is if the adjoint is used. So checking around I cannot find anywhere that using it must be the case. It's just used. I remember my first college physics classes. A lot of the other students (myself included I suppose) had teachers who used conjugation just to get real answers rather than use another class time to explain what happens to the imaginary part. So I have to ask if the Dirac currents should be the adjoint where is the explanation for this need?

 

[woodland]

Sorry my last entry should have had the first Dirac eq as the adjoint $$ψ^* γ^0 γ^0 ψ $$
I redid the idea of is real(Dirac eq #1) <> real(Dirac eq 1 to 4) a million times with random choices for the complex (about 1 to 0) $$ψ^μ$$ and the real part of the first Dirac eq is always larger than the real part of any of the other three. I cannot see how the integral for all four can be made 1. Remember all four ψ's occur in any current so any choice of a coefficient of one applies to all. That's why I was researching the Dirac current not being needed to be adjoint. To be fair this needs to be continued on another thread. I'll check back though, maybe someone might have some ideas.