Isotopic Space, Complex Conjugation, and U3 Symmetry for Particles

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Abstract
The inherent U4 symmetry over spinor spaces of the regular decomposition indicated previously is discussed from the viewpoint of a single spinor space. It is noted that this symmetry does not yet include isotopic spin. Isotopic space is introduced by considering that the space‐time vectors and its vector Clifford algebra are imbedded in the algebra of a six‐dimensional Euclidean space. Three of the latter's dimensions are identified with ordinary space, and three are identified with isotopic space. The ``time vector'' corresponds to the pseudoscalar element in isospace, and the imaginary unit is expressed as a linear combination of the bivectors in isospace. The relation among complex conjugation, time inversion, and space inversion is thereby clarified. Some comments on the application of these ideas to particle symmetry discussions are made. In particular one obtains as the first extension of the SU2 symmetry of isotopic spin, a U3 symmetry for particles.

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