Clifford algebra: Notes on the spinor metric and Lorentz, Poincaré, and conformal groups

Abstract

A particular normalization for the set of basis elements {Γi} of the complex Clifford algebras C(p,q) is motivated and defined by demanding that the physical bispinor densities ρi=Ψ̄ΓiΨ be real. This condition, referred to here as Dirac normalization, also necessitates the introduction of the spinor metric γ, and the solution of the metric conditions is given for arbitrary (p,q); when N=p+q is even the metric is unique, and when N is odd there are two distinct metrics. Then the Dirac normalization preserving automorphism group of the basis is explored. This is also the group of transformations leaving the spinor metric invariant. In particular, the physically important cases of the Lorentz, Poincaré, and conformal groups are sought as subgroups of the automorphism group. As expected, it is found that the Lorentz group is always contained in the automorphism group. However, it is found that the Poincaré and conformal groups are contained only in the cases where N is even and q is odd. Furthermore, when N is odd these groups may be found in the full isomorphism group, but only for one of the two possible spinor metrics. Possible physical implications of these results are discussed.