The operator structures for the constants of the motion of the relativistic hydrogen atom are examined. ThoughJ 3 andJ · J are constants of the motion,J is not. Its replacement, $\tilde {\rm K}$ , is shown to emerge rather naturally in transforming the equation to spherical coordinates. The separation of variables is presented in hypercomplex number form. This leads to some interesting suggestions regarding the matter/antimatter operator for the Dirac equation.

A new four-component spin-1/2 wave equation for ordinary mass is discussed. It is shown that this equation has a conserved current not easily identified with a transition probability, only pure imaginary energy states, and is covariant. A tachyon-like Klein-Gordon equation is satisfied by this equation, but rest states are explicitly constructed.

A survey is presented of the essential principles for formulating relativistic wave equations in curved spacetime. The approach is relatively simple and avoids much of the philosophical debate about covariance principles, which is also indicated. Hypercomplex numbers provide a natural language for covariance symmetry and the two important kinds of covariant derivative.

A “mental” multiplication scheme is given for the super hypercomplex numbers, which extend the 16-element Dirac algebra to 32 elements by appending the complex octonions. This extends the 5-vectors of relativity to 9-vectors. The problems with nonassociativity, for the group structures and wave equation covariance, are explored.

The basic structure of a second quantized relativistic quantum theory is outlined. The vector space is over the ring of complex quaternions instead of the usual field of complex numbers. This is motivated by the simple quaternion structure of the Dirac equation.

A self-consistent relativistic formalism is presented which postulates that mass is the eigenvalue of a fifth momentum operator component. Lorentz covariance is generalized so that a systematic program for covariant wave equations can be formed. The fifth dimension is identified with cosmic time, resulting in a bias toward matter over antimatter for the universe. A distinction between μ ande also seems possible through the space-time extension.

Partons (quarks) are unobservable, it is suggested, because they have no well-defined rest-massconcept.