Deriving the continuity equation from the Dirac equation (Relativistic Quantum)

[toam]

So I am trying to derive the continuity equation:

$$\frac{\partial}{\partial x^{\mu}}J^{\mu} = 0$$

From the Dirac equation:

$$i\gamma^{\mu} \frac{\partial}{\partial x^{\mu}}\Psi - \mu\Psi = 0$$

And its Hermitian adjoint:

$$i\frac{\partial}{\partial x^{\mu}}\overline{\Psi}\gamma^{\mu} - \mu\overline{\Psi} = 0$$

Where:

$$\overline{\Psi}=\Psi^{+}\gamma^{0}$$ (Dirac conjugate)

The Attempt at a Solution

By multiplying the Dirac equation on the right by $$\overline{\Psi}$$ and the adjoint on the right by $$\Psi$$ I get:

$$i(\frac{\partial}{\partial x^{\mu}}(\gamma^{\mu}\Psi)\overline{\Psi} + \frac{\partial}{\partial x^{\mu}}(\overline{\Psi})\gamma^{\mu}\Psi) - \mu(\Psi\overline{\Psi} - \overline{\Psi}\Psi)=0$$

The first term is basically what I am after (except I am not 100% sure I can simply apply the product rule - what is the correct order?) which means I shoudl expect the second term to go to zero:

$$\Psi\overline{\Psi} - \overline{\Psi}\Psi =0$$

But because $$\gamma^{0}$$ is a 4x4 matrix, $$\Psi$$ is a 4x1 and $$\overline{\Psi}$$ is a 1x4, I should also expect the second term to be multiplied by the 4x4 identity matrix (so that the subtraction makes sense). However the first term is NOT a constant multiplied by the identity so I don't see how this works.



Any help would be greatly appreciated...

 

[tiny-tim]

By multiplying the Dirac equation on the right by $$\overline{\Psi}$$ and the adjoint on the right by $$\Psi$$ …

Hi toam!

Don't you have to multiply one of them on the left?

 

[toam]

I tried that and got something else that didn't work. However I will try again because I was surprised that it didn't work so I may have made a mistake or missed something obvious...

 

[toam]

Ok so it turned out I had multiplied the wrong function on the left. It worked out quite simply when I fixed that. The lecture notes had erroneously shown both functions multiplied on the right.

Thanks, tiny-tim.