On Relativistic Wave Equations

The problem of the relativistic invariance of a first-order wave equation with matrix coefficients ${\beta }_k$ is examined. It is found that it is intimately connected with the structure of the enveloping algebra of the $\beta$-matrices. In particular the center of this algebra can contain only elements of a very restricted type. This fact provides a powerful means for the investigation of the abstract $\beta$-algebra, as is illustrated in the general $s$-dimensional case of the Dirac and the Duffin-Kemmer matrices. Relativistic invariance imposes severe restrictions on the spurs of the $\beta$-matrices and their multiple products. Conversely, these restrictions insure relativistic invariance. The theory is applied to the case of particles of higher spin.