Relativistic ring extension of the field of complex numbers

Abstract

Over the past sixty years many attempts have been made at modifying quantum mechanics by extending the field of complex numbers, but none has led to a structural unification of quantum mechanics with relativity. A solution nevertheless exists. It is based on a ring, new to mathematics, which we call the quantal ring. It is manifestly Lorentz covariant, CPT invariant, contains the field of complex numbers as a substructure, and is free of properties that have no physical interpretation. While the systematic derivation of this ring from a comparative abstract analysis of classical and quantum mechanics, as well as the proof of its uniqueness and the construction of a generalized Hilbert space over it, are being finalized for publication, we derive it in this note by a simple alternative construction.