Duffin-Kemmer algebra as a ring and its representations

Abstract

Moreover Kemmer [6] and Fujiwara [3] considered an algebra generated from npU (p = 1,2,..., n). The former author discussed on the representations of the algebra using “linearlized n-dimensional wave equations”, and obtained the decomposition scheme of the representations and results corresponding to Theorems 1 and 14 in this paper. The latter author first introduced the notation 4 (E + ~~7,) (p = 1,2,..., n, E, = &1) which is written e in this paper, and obtained the results corresponding to Theorems 1,2,6 and 9. He also intuitively constructed the linearly independent elements of this algebra. Besides in the recent paper [8], Tokuoka has attempted to resolve the algebra of n = 4 into subalgebras. Further references for the applications or discussions on this algebra in physics are given, for example, in [3] and [8], and Boerner has referred a little to this algebra in his work [l]. However, it seems that the complete analysis of this algebra using the general theory of rings has never been done, so we now clarify the structure of this algebra as a ring and discuss the regular representations of the algebra. Then we can understand Tokuoka’s treatment more simply as is shown in Appendix 1. First we define the algebra and some notations. Let K be a field of complex numbers and let ‘S& be an algebra over K.