Higher Derivative Scalar Field Theory in the First Order Formalism

TL;DR

This paper develops a first order formalism for higher derivative scalar field theories, deriving wave equations, Lagrangians, and quantization methods for particles with two mass states.

Contribution

It introduces a novel first order formalism for higher derivative scalar fields, including wave equations, invariant forms, and quantization procedures for multiple mass states.

Findings

Derived a 10-dimensional matrix form of the wave equation.

Obtained the relativistically invariant bilinear form and Lagrangian.

Performed canonical quantization and found propagators.

Abstract

The scalar field theory with higher derivatives is considered in the first order formalism. The field equation of the forth order describes scalar particles possessing two mass states. The first order relativistic wave equation in the 10-dimensional matrix form is derived. We find the relativistically invariant bilinear form and corresponding Lagrangian. The canonical energy-momentum tensor and density of the electromagnetic current are obtained. Dynamical and non-dynamical components of the wave function are separated and the quantum-mechanical Hamiltonian is found. Projection operators extracting solutions of field equations for definite energy and different mass states of particles are obtained. The canonical quantization of scalar fields with two mass states is performed, and propagators are found in the formalism considered.