Solutions of Podolsky's Electrodynamics Equation in the First-Order Formalism

TL;DR

This paper reformulates Podolsky's generalized electrodynamics in a first-order formalism, deriving wave equations, algebraic properties, and solution projection operators, providing a comprehensive mathematical framework for the theory.

Contribution

The paper introduces a first-order formalism for Podolsky's electrodynamics, deriving wave equations, algebraic structures, and solution operators, which were not previously established.

Findings

Derived 20-dimensional relativistic wave equation

Proved matrices obey Petiau-Duffin-Kemmer algebra

Constructed projection operators and Hamiltonian

Abstract

The Podolsky generalized electrodynamics with higher derivatives is formulated in the first-order formalism. The first-order relativistic wave equation in the 20-dimensional matrix form is derived. We prove that the matrices of the equation obey the Petiau-Duffin-Kemmer algebra. The Hermitianizing matrix and Lagrangian in the first-order formalism are given. The projection operators extracting solutions of field equations for states with definite energy-momentum and spin projections are obtained, and we find the density matrix for the massive state. The $13\times 13$-matrix Schrodinger form of the equation is derived, and the Hamiltonian is obtained. Projection operators extracting the physical eigenvalues of the Hamiltonian are found.