New fundamental dynamical equation for higher derivative quantum field theories

Z E Musielak,J L Fry,G W Kanan

doi:10.12988/astp.2015.511

Abstract

In space-time with the Minkowski metric, the group of the metric is the inhomogeneous Lorentz group, which is also known as the Poincar e group. A dynamical equation is called fundamental if it is invariant with respect to the group of the metric, which consists of all transformations that leave the metric invariant. A method based on this group is used to derive two innite sets of Poincar e invariant linear dynamical equations for scalar and analytical functions that represent free, spin-zero, massive elementary particles. The sets contain two dierent types of fundamental higher derivative dynamical equations, which are used to construct higher derivative quantum eld theories. One of these sets contains the original Klein-Gordon equation and it is shown that all physically acceptable solutions to the higher derivative equations in this set are the same as the solutions to the Klein-Gordon equation. This means that none of the higher order equation in this set can be considered as new and that the Klein-Gordon is the only fundamental dynamical equation available to construct local quantum eld theories. However, for the other set, it is demonstrated that all physically acceptable solutions to the higher derivative equations are the same as the solutions to the lowest order fundamental equation in this set. As a result, this lowest order equation is the only new fundamental equation in the set, and it is used to construct higher derivative (non-local) quantum eld

Highlights

The measure of space-time introduced by Minkowski plays an important role in formulating relativistic field theories of elementary particles
In our previous work [10], we used the method to derive a Schrodinger-like equation [14] for a scalar wave function, and we demonstrated that this equation was the only Galilean invariant equation among an infinite number of dynamical equations that were formally obtained by the method [11]
We present the main results of our search for Poincare invariant dynamical equations using the eigenvalue equations as the starting point

Summary

Introduction

The measure of space-time introduced by Minkowski plays an important role in formulating relativistic field theories of elementary particles. The group of transformations leaving the Minkowski metric invariant is the inhomogeneous Lorentz group, which is called the Poincare group. Wigner [1] was the first who classified all irreducible representations (irreps) of the Poincare group and identified an elementary particle with an irreducible unitary representation of the group. By using this definition, Wigner was able to determine classes of elementary particles that may exist in Minkowski space-time [2,3]

Objectives

Discussion

Conclusion