Generalized O(4) Harmonics and Bethe-Salpeter Equation for Spinor-Spinor Particle System

概要

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The generalized O(4) harmonics which are derived from the representations of the O(4) group are applied to the analysis of the Bethe-Salpeter equation for the spinor-spinor particle system exchanging vector particles at the vanishing total four-momentum. The separation of the angular variables is made by introducing the helicity amplitudes which have definite transformation properties under the O(4) group. After carrying out angular integrations, we get a system of one-dimensional equations which contains Goldstein's equation for the representation of M=0 and Goldstein's equation and Kummer's one for the representation of M=±1, where M denotes the four-dimensional helicity quantum number. All the equations belonging to M=±1 are analytically solved in the case of massless particles exchange. On discussing the normalization properties of the solutions of Kummer's equation, it is found that the sign of norm is ±(-)^<n-l>, where the double sing (±) depends on the type of combination of M=+1 and -1 amplitudes, and n and l stand for the four-dimensional and three-dimensional angular momentum quantum numbers, respectively. The properties of the amplitudes under space inversion are also discussed.
理論物理学刊行会の論文
1969-12-25