Exact correspondence between classical and Dirac-Pauli spinors in the weak-field limit of static and homogeneous electromagnetic fields

TL;DR

This paper rigorously proves that in the weak-field limit, the Dirac-Pauli spinor in the Foldy-Wouthuysen representation behaves exactly like a classical relativistic spinor, confirming long-standing speculation.

Contribution

It provides a rigorous proof that the FW transformation of the Dirac-Pauli Hamiltonian matches the classical relativistic spinor in the low-energy limit.

Findings

FW transformation agrees with classical Hamiltonian in weak fields

Quantum and classical spinors are equivalent in the low-energy limit

Method of direct perturbation theory confirms classical correspondence

Abstract

It has long been speculated that the Dirac or, more generally, the Dirac-Pauli spinor in the Foldy-Wouthuysen (FW) representation should behave like a classical relativistic spinor in the low-energy limit when the probability of particle-antiparticle pair creation and annihilation is negligible. In the weak-field limit of static and homogeneous electromagnetic fields, by applying the method of direct perturbation theory inductively on the orders of $1/c$ in the power series, we rigorously prove that it is indeed the case: the FW transformation of the Dirac-Pauli Hamiltonian is in full agreement with the classical counterpart, which is the sum of the orbital Hamiltonian for the Lorentz force equation and the spin Hamiltonian for the Thomas-Bargmann-Michel-Telegdi equation.