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We also find 1 massless spin-! 22) which corresponds, as usual, to a particle and its antiparticle. 23) 34 Tevian Dray and Corinne A. Manogue corresponds to a massless particle with the same helicity moving in the opposite direction, not to a different particle with the opposite helicity. Each of the above states may be multiplied (on the right) by an arbitrary complex number. 24) As with the other massless momentum space states, this describes a particle and an antiparticle. Alone among the particles, this one does not contain k and, hence, does not depend on the choice of identification of a particular quaternionic subalgebra 1Hl satisfying C C 1Hl C O.

I) +0. I) as often as possible. 4) is fulfilled. All these calculations of course do not make sense on the cone z2 = o. 8). l). Hence such limit transitions can proceed on any path which ends at z = 0, except that it must not touch the cone z2 = o. 9), its most general solution follows when we add any matrix H which fulfills the homogeneous Dirac equation fjJH = O. For choosing this H we need the Main Theorem (Singularity). 4) either more singular than t- 3 or less than 00 or lim zl-' H(z) = O.

Sect. A Vol. L (1945), 261-270. [4] P. Jordan, Uber die Multiplikation quantenmechanischer GraBen, Z. Phys. 80 (1933), 285-291 . [5] P. Jordan, J. von Neumann, and E. Wigner, On an algebraic generalization of the quantum mechanical formalism, Ann. Math. 35 (1934), 29--64. [6] Feza Cursey and Chia-Hsiung Tze, On the Role of Division, Jordan, and Related Algebms in Particle Physics, World Scientific, Singapore, 1996. [7] S. Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics, Cambridge University Press, Cambridge, 1995.

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2-affine complete algebras need not be affine complete by Aichinger E.


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