Quaternions, Lorentz Group and the Dirac Theory(Particles and Fields)

概要

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It is shown that a subgroup of SL/(2,H), denoted Spin(2,H) in this paper, which is de-fined by two conditions in addition to unit quaternionic determinant, is locally isomorphic to the restricted Lorentz group, L^↑_+. On the basis of the Dirac theory using the spinor group Spin(2,H), in which the charge conjugation transformation becomes linear in the quarter-nionic Dirac spinor, it is shown that the Hermiticity requirement of the Dirac Lagrangian, together with the persistent presence of the Pauli-Giirsey SU(2) group, requires an additional imaginary unit (taken to be the ordinary one, i) that commutes with Hamilton's units, in the theory. A second quantization is performed with this i incorporated into the theory, and we recover the conventional Dirac theory with an automatic ''anti-symmetrization' of the field operators. It is also pointed out that we are naturally led to the scheme of complex quarter-nions, H^C, in which a space-time point is represented by a Hermitian quaternion, and that the isomorphism SL(1,H^C)/Z_2≌L^↑_+ is a direct consequence of the fact Spin(2,H)/Z_2≌L^↑_+. Using SL(1,H^C)≌SL(2,C), we make explicit the Weyl spinor indices of the spinor-quaternion, which is the Dirac spinor defined over H^C.
2007-03-25