On Relativistic Wave Equations

The problem of the relativistic invariance of a first-order wave equation with matrix coefficients βₖ is examined. It is found that it is intimately connected with the structure of the enveloping algebra of the β-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 β-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 β-matrices and their multiple products. Conversely, these restrictions insure relativistic invariance. The theory is applied to the case of particles of higher spin.