On Wave Equation Vector-Matrices and Their Spurs

If a system of tensor (or spinor) equations may also be written in the matrix form $p\psi ={\psi }^{\prime }$ where the column matrices $\psi$ and ${\psi }^{\prime }$ undergo the same transformation under a change of coordinates, then the square matrix $P$ is termed a vectrix. General considerations indicate that $P$ should obey invariant matrix identities. Of course, the matrix $P$ merely images an operation implicitly defined in the original system of equations in terms of a set of tensor parameters $p$. A natural method to seek identities is to iterate this operation in the original system with different values ascribed to the parameters. Of especial interest are the vectrices defined by wave equation systems such as those of Dirac and Proca. Here $p$ is to be interpreted as the four-vector of momentum and energy. By carrying out the iteration process, various identities are found. In particular, formulas are obtained for the spur of the product of an arbitrary number of Proca vectrices.