On the quaternion representation of the proper Lorentz group SO(3,1)

Abstract

Complex quaternions are investigated in detail, bringing out some new aspects of the relationship of the multiplicative group of unit complex quaternions (UCQ) with the proper Lorentz group SO(3,1). Constructing the proper Lorentz transformation (PLT) corresponding to a given UCQ, the quaternion parameters of a PLT are determined explicitly in terms of its element Lij, and this quaternion parametrization is then utilized to obtain an interesting geometrical interpretation of SO(3,1) as the intersection of a hyperboloid with a cone in a real eight-dimensional Euclidean space E8. The UCQ components are then related to the Lie–Cartan parameters of SO(3,1), leading to an identification of complex quantities which may be interpreted as the complex axis and angle of rotation. It is shown that any PLT admits a special type of Euler resolution which is at the same time a resolution into three Lorentz–Synge screws the two angles of which combine to form a complex Euler angle (or Euler–Brauer angle). It is also shown that on taking the rotation parameters in the formula for the D j representation of SO(3) to be complex, one obtains the D j0 representation of SO(3,1), leading at once to its D jj′ representation. Similarly, a formula for the character χ j0 of the D j0 representation, having a complete analogy to the character formula for SO(3), but in terms of a complex angle ω is obtained and this in turn yields a formula for the character χ jj′ in the Djj′ representation of SO(3,1).