The Algebra of Quantions

    There is an essential difference between the concept of structural unification and the non-structural solutions provided by quantum field theory.

    As suggested by the meaning of “structural”, the corresponding unification refers to the ideal of a mathematically sound fusion of relativity and quantum mechanics into a single theory. By contrast, non-structural unification refers to the solving of problems formulated in one domain (relativistic fields) with methods adapted from a different discipline (quantum mechanics). Characterized by the term “quantization”, this approach gives excellent results in electrodynamics, but does not work at all in gravitation. For this reason and for purely mathematical ones, quantum field theory is not a definitive theory. Like the “old quantum mechanics” (1900-1925), it is an intermediate theory. The question is, Intermediate to what?

    According to a point of view dominant in our days, the next theory is to be a complete merging of relativity and quantum mechanics which is expected to take place at the Planck scale, be it along the developing ideas of string theory, or of other current approaches to the problem of unification. If this belief is correct, the enormous energy gap between the energy scale accessible to current accelerators and the inaccessible Planck scale will have to be bridged by a single theoretical vault.

    According to a point of view abandoned long ago as fruitless, but revisited between these covers, the next theory ought to be a structurally relativistic quantum mechanics -- a theory in which there is no need for a mixed approach to problems, nor for the grafting of one theory onto another. If this belief is correct, there exists a fusion of relativity and quantum mechanics which is as complete as Maxwell’s of electricity and magnetism: it applies to all phenomena which are both quantum mechanical and relativistic. It also naturally specializes to one of these theories if the phenomena characteristic of the other theory can be neglected. Since the envisaged fusion does not seem to be characterized by any particular scale of its own, it does not call for the unsupported spanning of a large gap of experimentally inaccessible energies.

    As in the unification problem faced by Maxwell, the theories to be merged are known. Consequently, if approached as a game -- the “winning solution” being the unique unifying mathematical structure if it exists -- the fusion of relativity and quantum mechanics is not a problem in physics but an exercise in abstract mathematics.

    At this point, the reader may wonder whether this is only the description of an old but perforce discarded ideal or of a concrete mathematical theory which is being developed in the present book. It is the latter -- with the understanding that only a small part of all results obtained in the last eighty years and encapsulated today in the Standard Model has been reproduced so far by the methods developed in this first volume on the subject. Yet, the results that have been obtained seem very encouraging.

    Even though work in progress is not running into difficulties, only the future will tell whether the program of structural unification can be brought to completion. Not that the author has any doubts about it, but this is a book about a new unifying mathematical structure and its properties, not about personal beliefs.

    The structural unification investigated on these pages is based on an extension of the field of complex numbers to an associative algebra first listed in 1882 by Peirce under the symbolic name g4 in his catalogue of such algebras. This object has since been ignored -- probably because it has no obvious distinguishing properties. We shall see, however, that it is uniquely distinguished: in addition to an algebraic structure (quantum mechanics), it contains a metric structure (Minkowski’s), and a differential structure (Schrödinger’s and Dirac’s equations), but this becomes evident only once its regular representations have been computed.

    The algebra g4 was independently discovered in a different formulation by Edmonds in 1972 as a structure obviously related to physics. Edmonds fittingly named its elements “nature’s natural numbers”. He then immediately attempted to apply this algebra to field theory -- but in a representation which does not lead to unification. Specifically, it contains neither a built-in derivation operator nor relativistic spin. Very vaguely, it is to the needed regular representation what Pauli’s theory of spin is to Dirac’s theory.