Recovery of Dirac Equations from Their Solutions

TL;DR

This paper demonstrates that in certain quantum field theory contexts, a static Bose field can be uniquely reconstructed as a functional of the Dirac field, providing a rigorous method for recovering the field from solutions.

Contribution

It introduces a rigorous method to recover static Bose fields as functionals of Dirac fields in quantum field theory, extending previous trivial cases.

Findings

Bose field $\\mathcal{B}$ is a functional of Dirac field $\psi$ for strictly canonical solutions.

Verification methods for static non-quantized Bose fields are detailed and rigorous.

The approach applies to all static, non-quantized Bose fields, not just trivial cases.

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.