On superluminal fermions within the second derivative equation

TL;DR

This paper introduces a novel second-derivative equation for spin-1/2 fermions that can describe superluminal particles, proposing a tachyonic equation and a comprehensive first-order formalism with quantization.

Contribution

It formulates a new second-order fermion equation with superluminal solutions and develops a corresponding first-order wave equation, including Lagrangian, interaction, and quantization frameworks.

Findings

Fermions can propagate faster than light under certain parameters.

A 20-component relativistic wave equation is constructed.

Quantization of the field reveals vacuum operator expectations.

Abstract

We postulate the second-order derivative equation with four parameters for spin-1/2 fermions possessing two mass states. For some choice of parameters fermions propagate with the superluminal speed. Thus, the novel tachyonic equation is suggested. The relativistic 20-component first-order wave equation is formulated and projection operators extracting states with definite energy and spin projections are obtained. The Lagrangian formulation of the first-order equation is presented and the electric current and energy-momentum tensor are found. The minimal and non-minimal electromagnetic interactions of fermions are considered and Schr\"{o}dinger's form of the equation and the quantum-mechanical Hamiltonian are obtained. The canonical quantization of the field in the first-order formalism is performed and we find the vacuum expectation of chronological pairing of operators.