# Non-relativistic limit of the Dirac equation

How to recover non-relativistic limit of the Dirac equation $$\left( i\gamma^{\mu}\mathcal{D}_{\mu} - m \right)\Psi(x) = 0$$ where $\mathcal{D}_{\mu} = \partial_{\mu} + iqA_{\mu}$. I do not assume that the vector potential $A_{\mu}$ is zero, but I have a particle in central potential and magnetic field. During derivation please highlight where does the non-relativistic assumptions is used and how. I have found some notes here, but I don't understand why can we Taylor expand this expression $$\frac{1}{E + m -qA^0}\vec{\sigma}\cdot(\vec{p} - q\vec{A}) \approx \frac{1}{E+m}\vec{\sigma}\cdot(\vec{p} - q\vec{A}) + \frac{1}{(E+m)^2}qA^0\vec{\sigma}\cdot\vec{p}$$ around $qA^0 = 0$ and neglect $\vec{A}$ in the second term.

• The reason for neglecting $\vec A$ is that the magnetic field $\nabla \times \vec A$ is a factor of $v/c$ smaller than the electric field $-\nabla A^0$ (for a static vector potential). See also the answer to Why are magnetic fields so much weaker than electric? – Praan Dec 1 '15 at 23:12