Recovery of Dirac Equations from Their Solutions

Abstract

We deal with quantum field theory in the restriction to external Bose fields. Let $(i\gamma^\mu\partial_\mu - \mathcal{B})\psi=0$ be the Dirac equation. We prove that a non-quantized Bose field $\mathcal{B}$ is a functional of the Dirac field $\psi$, whenever this $\psi$ is strictly canonical. Performing the trivial verification for the $\mathcal{B} := m = $ constant which yields the free Dirac field, we also prepare the tedious verifications for all $\mathcal{B}$ which are non-quantized and static. Such verifications must not be confused, however, with the easy and rigorous proof of our formula, which is shown in detail.