Current Conservation

The hermitian conjugate wave function is $\psi^\dagger = (\psi_1^*,\psi_2^*,\psi_3^*,\psi_4^*)$. Multiplying the Dirac equation by the conjugate wave function from the left gives

$$i\hbar \psi^\dagger\frac{\partial\psi}{\partial t} = \frac{\... ...ac{\partial\psi}{\partial x^k} + mc^2\psi^\dagger\beta\psi .$$ (5.11)

Forming the Dirac equation for the conjugate wave function and multiplying by the wave function from the right gives

$$-i\hbar \frac{\partial\psi^\dagger}{\partial t} \psi = -\fra... ...er}{\partial x^k} \alpha_k\psi + mc^2\psi^\dagger\beta\psi ,$$ (5.12)

where $\alpha_i^\dagger = \alpha_i$ and $\beta^\dagger = \beta$. Subtracting (5.11) from (5.12) gives

$$i\hbar \frac{\partial}{\partial t} \psi^\dagger\psi = \sum_{... ...{i} \frac{\partial}{\partial x^k} (\psi^\dagger\alpha^k\psi) .$$ (5.13)

Writing the result as a continuity equation, $\frac{\partial\rho}{\partial t} + \vec{\nabla}\cdot\vec{j} = 0$, gives the probability density

$$\rho = \psi^\dagger\psi ,$$ (5.14)

and the probability current

$$j^k = c\psi^\dagger\alpha^k\psi ,$$ (5.15)

where $c\alpha^k$ is like a velocity operator.

Integrating over all space and using the divergence theorem gives

$$\int d^3x \frac{\partial\rho}{\partial t} + \int d^3x \vec{\nabla}\cdot\vec{j} = 0 ,$$

$$\frac{\partial}{\partial t} \int d^3x\rho + \int \vec{j}\cdot d\vec{S} = 0 ,$$

$$\frac{\partial}{\partial t} \int d^3x\psi^\dagger\psi = 0 .$$ (5.16)

We have now proven the first two conditions required of the Dirac equation but we still have to show (c$\rho,\vec{j}$) forms a 4-vector under a Lorentz transformation and that the Dirac equation is Lorentz covariant.