New fundamental dynamical equation for higher derivative quantum field theories

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