Two-component Dirac equation

TL;DR

The paper introduces a novel two-component approach to relativistic quantum dynamics using the Feshbach projection, providing insights into particle behavior, especially for small-mass particles like neutrinos, and refining the Pauli equation.

Contribution

It presents an alternative derivation of a two-component relativistic equation from the Dirac equation using the Feshbach projection technique, offering new perspectives on relativistic quantum physics.

Findings

Hermitian effective Hamiltonian for small-mass particles

No leakage between upper and lower spinors at leading order

Non-Hermitian correction improves the Pauli equation in weak relativistic regime

Abstract

We provide an alternative approach to relativistic dynamics based on the Feshbach projection technique. Instead of directly studying the Dirac equation, we derive a two-component equation for the upper spinor. This approach allows one to investigate the underlying physics in a different perspective. For particles with small mass such as the neutrino, the leading order equation has a Hermitian effective Hamiltonian, implying there is no leakage between the upper and lower spinors. In the weak relativistic regime, the leading order corresponds to a non-Hermitian correction to the Pauli equation, which takes into account the non-zero possibility of finding the lower-spinor state and offers a more precise description.