Show that the Dirac field satisfies the continuity equation


 * 1) Find the equations that describe the Dirac field: \(\rho = |\phi|^2\) \(\mathbf{j} = c\phi^{\dagger}\alpha\phi\)
 * 2) Substitute these into the continuity equation, \(\partial_t\rho + \nabla\cdot\mathbf{j} = 0\): \(\partial_t\rho + \nabla\cdot\mathbf{j} = \partial_t |\phi|^2 + \nabla c\phi^{\dagger}\alpha\phi\)
 * 3) Expand by the product rule: \(\partial_t\rho + \nabla\cdot\mathbf{j} = \phi^{\dagger}(\partial_t \phi)+ (\partial_t \phi^{\dagger}) \phi + c\phi^{\dagger}(\nabla\alpha\phi) + c(\nabla\phi^{\dagger}\alpha)\phi \)
 * 4) Simplify: \(\partial_t\rho + \nabla\cdot\mathbf{j} = \frac{\phi^{\dagger}}{i\hbar}\left(\partial_t \phi+ c\nabla\alpha\phi\right) + \left(-\partial_t  \phi^{\dagger}- c\nabla\phi^{\dagger}\alpha\right)\frac{\phi}{-i\hbar}\)
 * 5) Substitute in the Dirac equation \(i\hbar\partial_t\phi = i\hbar c \alpha \nabla\phi + \beta mc^2\phi\): \(\partial_t\rho + \nabla\cdot\mathbf{j} = \frac{\phi^{\dagger}}{i\hbar}\left(bmc^2\phi\right) - \left(bmc^2\phi^{\dagger}\right)\frac{\phi}{i\hbar} = 0\)