Linear Approximation for the Massless Lorentz Gauge Field

Abstract

In the gauge theory for inhomogeneous Lorentz transformations two kinds of gauge fields are introduced/),2) namely, the translation gauge field c/' and the Lorentz gauge field A km p. The Lagrangian density proposed by Hayashi2) is the most general one which is quadratic or less in the first-order derivatives of A km p and Ck • In a previous paper,3) which will be referred to as I hereafter, we used Hayashi's Lagrangian and examined the massive Lorentz gauge field A km P in the linear approximation under the condition that C k P and matter fields can be neglected. In this appr9ximation, there is no difference between Latin indices and Greek ones. The massive Lorentz gauge field was decomposed into six diagonalized fields, each of which obeys the Klein-Gordon equation and irreducibility conditions for spin. Other authors )-7) studied the Lorentz gauge field in the linear field approximation, but they restricted their discussions to the case of massive Lorentz gauge field. In this paper we investigate the massless Lorentz gauge field in the free field approximation. Massless particle can appear when some of the conditions *) a+(2a/ 3)=0, /3(2a/ 3)=0 or r+(3a/ 2)=0 are satisfied as it was discussed in I. However, general discussions for the massless case are not so straightforward as the massive one, because the rest frame does not exist for the former. Thus, as the first step we discuss the axial-vector component of the Lorentz gauge field,