The history of physics is replete with fascinating puzzles, the eventual solution of which has served to illuminate and enrich the entire enterprise. One of the best examples is the experimental observation that gravitational and inertial masses are identical. There is no reason, within the confines of Newtonian gravity or the Newtonian laws of motion, respectively, for this marvelous "coincidence." Why, then, is it true? Einstein's answer to this particular puzzle turned out to be the general theory of relativity.

In the field of elementary-particle physics no puzzle has been more resistant to an explanation, hence more promising, than the puzzle of families -- sometimes referred to as the "Family Problem": Why are "fundamental" fermions (quarks and leptons) grouped into families? Why are families "replicated" and why are there only three families?

On the face of it, all of the matter encountered in everyday life (i.e., matter composed of electrons, protons, neutrons), and even matter that is observed indirectly in the glow of distant galaxies, can be explained in terms of a single family of fundamental fermions. What, then, are the other families for?

Tantalizing partial answers and explanations of the Family Problem have been proposed. For example, if there were less than three families, there would be no natural basis for the violation of time-reversal invariance. Under these circumstances there could be no excess of matter over antimatter after the "big bang" and, therefore, no universe as we know it. In string theories having more than the three familiar spatial dimensions, family replication is the result of "compactification" of the "extra" space dimensions. The number of "handles" on the compactified space (a topological property) is supposed to determine the number of families. While this seems to "explain" family replication, there is no way at present in string theories to limit the number of families to three. Moreover, it may be possible to formulate string theories in 4-dimensional spacetime, in which case compactification is no longer an option. Finally, there are "theories" involving chaos and random dynamics at the Planck level that seem to require three families.

All of the foregoing theories are interesting, and some of them may even have correct aspects. But, it is also fair to say that these theories tend to raise as many questions as they answer. In the present book, the author takes a "minimalistic" approach to the Family Problem, by simply generalizing the familiar standard-model concept of scalar fermion-numbers.

It is well known that the dual requirements of gauge (phase) and Lorentz invariance in quantum field-theories leads to the existence of antiparticles and particle statistics. In particular, these theories lead to the concept of generally conserved scalar fermion-numbers (f=+1 for all fermions and f=-1 for all antifermions).

Now, because f applies to all fundamental fermions and antifermions, regardless of their flavor or possible internal "composition," flavors cannot be distinguished by the number f. But the situation is quite different for a matrix, which can introduce flavor degrees of freedom.

By an appropriate mathematical generalization of the concept of scalar fermion-numbers (i.e., replace the number f by a special 2x2 matrix F, where f and F are both square-roots of unity) one arrives at a new organizing principle for fundamental fermions. Using this principle, it is possible to describe individual flavors, flavor doublets, and families of fundamental fermions in the context of a new, internal non-Euclidean ("Lorentzian") 2-space.

Certain algebraic and geometric properties of this 2-space "automatically" explain both family replication and the number of families. Moreover, the new organizing principle results in entirely new ways of looking at the fermion sector of the world. For example, the foregoing 2-space description automatically "unifies," or links three apparently unrelated quantum "dichotomies," namely, the matter-antimatter, "up"-"down" and quark-lepton dichotomies. In essence, these dichotomies are different geometric aspects of the 2-space.

Gerald L. Fitzpatrick has a background in geophysics and physics. For 30 years has has specialized in applied physics, doing government- and industry-sponsored research in linear and nonlinear physical acoustics, optical- and acoustic-holography, laser vibrometry and magneto-optic/eddy-current imaging. He has published over 35 research papers and holds a number of U.S. and foreign patents. During the same time period he immersed himself in the study of elementary particles in an attempt to address the rhetorical question posed by I. I. Rabi in 1947 when confronted with the discovery of the muon: 'Who ordered that?' This book represents the author's best "answer" to Rabi's question.

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