The abstract complex Lorentz transformation group with real metric. II. The invariance group of the form ‖t‖2−∥x∥2

Abstract

Let TX=C×V∞ be equipped with the Hermitian form ‖t‖2−∥x∥2, where t∈C is a complex number and x∈V∞ is a vector in an abstract complex inner product space V∞. The abstract complex Lorentz group (with real metric) is the invariance group of this form. A novel formalism to deal with the abstract real Lorentz transformation group has recently been developed by Ungar [Am. J. Phys. 59, 824–834 (1991); Am. J. Phys. 60, 815–828 (1992)], allowing one to solve in an abstract context previously poorly understood problems in one time and three space dimensions. Intrigued by the success of the abstract real Lorentz group formalism, resulting in the understanding of Thomas gyration in its abstract context, a formalism to deal with the abstract complex Lorentz group is proposed in this article. The extension from the real to the complex Lorentz group is not trivial. Complex Lorentz groups involve two interacting gyrations, a complex-time gyration and a complex-space gyration, as opposed to the real case which involves a single gyration, that is, a real-space gyration called Thomas gyration. The proposed formalism allows one to manipulate the seemingly involved abstract complex Lorentz group in a way analogous to the way one commonly manipulates Galilei transformation groups. Thus, for instance, equipped with the proposed formalism, one can readily (i) compose abstract complex Lorentz transformations, and (ii) determine those which link any two given events by manipulations analogous to Galilei group manipulations. When the abstract complex inner product space V∞ associated with the underlying abstract complex Minkowski space is realized by a finite-dimensional complex Hilbert space of dimension n, the abstract complex Lorentz group studied in this article reduces to the group U(1,n) or SU(1,n), depending on whether unitary or special unitary transformations are being considered.