Operators of quantum theory of Dirac's free field

TL;DR

This paper generalizes the quantum operators for Dirac's free field, introducing new particle-antiparticle mixing operators and analyzing their physical implications, including the potential to measure wave-packets without Zitterbewegung.

Contribution

It develops a new framework associating integral operators in momentum and configuration space, enabling the derivation of novel particle-antiparticle mixing operators with physical significance.

Findings

Derived principal operators including coordinate and spin operators with particle-antiparticle mixing.

Identified oscillating terms causing Zitterbewegung in off-diagonal operators.

Showed that measuring these operators could produce wave-packets without Zitterbewegung.

Abstract

The Pryce (e) spin and position operators of the quantum theory of Dirac's free field were re-defined and studied recently with the help of a new spin symmetry and suitable spectral representations [I. I. Cot\u aescu, Eur. Phys. J. C (2022) 82:1073]. This approach is generalized here associating a pair of integral operators acting directly on particle and antiparticle wave spinors in momentum representation to any integral operator in configuration representation, acting on mode spinors. This framework allows an effective quantization procedure giving a large set of one-particle operators with physical meaning as the spin and orbital parts of the isometry generators, the Pauli-Lubanski and position operators or other spin-type operators proposed so far. A special attention is paid to the operators which mix the particle and antiparticle sectors whose off-diagonal associated operators have oscillating terms producing Zitterbevegung. The principal operators of this type including the usual coordinate operator are derived here for the first time. As an application, it is shown that an apparatus measuring these new observables may prepare and detect one-particle wave-packets moving uniformly without Zitterbewegung or spin dynamics, spreading in time normally as any other relativistic even non-relativistic wave-packet.