- #1
toam
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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)
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...
\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...
