Abstract
The representation of the proper (restricted) Lorentz group by 2 × 2 unimodular complex matrices is extended by including also matrices obtained by multiplying (either on the left or on the right) unimodular complex matrices by a quaternionic square-root of -1, denoted byj, which anticommutes withi the complex square-root of -1. It is shown that the extended group thus obtained is a two-valued representation of the two pieces of the general Lorentz group with normal time. The other two pieces of the Lorentz group with time reversed are shown to be represented by the same matrices in a somewhat different role: the changed role is indicated by putting hats on the matrices. The proposed representation of the general Lorentz group has the nice property that the determinant of the 2 × 2 matrix in our representation has the same value as the determinant of the 4 × 4 real matrix representing the same Lorentz transformation.
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