Matrix logarithms and range of the exponential maps for the symmetry groups , and the Lorentz group

Abstract

Physicists know that covering the continuously connected component of the Lorentz group can be achieved through two Lie algebra exponentials, whereas one exponential is sufficient for compact symmetry groups like SU(N) or SO(N). On the other hand, both the general Baker-Campbell-Hausdorff formula for the combination of matrix exponentials in a series of higher order commutators, and the possibility to define the logarithm of a general matrix through the Jordan normal form, seem to naively suggest that even for non-compact groups a single exponential should be sufficient. We provide explicit constructions of for all matrices in the fundamental representations of the non-compact groups , and SO(1, 2). The construction for also yields logarithms for SO(1, 3) through the spinor representations. However, it is well known that single Lie algebra exponentials are not sufficient to cover the Lie groups and . Therefore we revisit the maximal neighbourhoods and which can be covered through single exponentials with or , respectively, to clarify why or outside of the corresponding domains or . On the other hand, for the Lorentz groups SO(1, 2) and SO(1, 3), we confirm through construction of the logarithm that every transformation in the connectivity component of the identity element can be represented in the form with or , respectively. We also examine why the proper orthochronous Lorentz group can be covered by single Lie algebra exponentials, whereas this property does not hold for its covering group : The logarithms in correspond to logarithms on the first sheet of the covering map , which is contained in . The special linear groups and the Lorentz group therefore provide instructive examples for different global behaviour of non-compact Lie groups under the exponential map.