Matrix logarithms and range of the exponential maps for the symmetry groups SL ( 2 , R ) , SL ( 2 , C ) , and the Lorentz group

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 ln ( M ̲ ) of a general matrix M ̲ 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 ln ( M ̲ ) for all matrices M ̲ in the fundamental representations of the non-compact groups SL ( 2 , R ) , SL ( 2 , C ) , and SO (1, 2). The construction for SL ( 2 , C ) 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 SL ( 2 , R ) and SL ( 2 , C ) . Therefore we revisit the maximal neighbourhoods 1 ⊂ SL ( 2 , R ) and 1 , C ⊂ SL ( 2 , C ) which can be covered through single exponentials exp ( X ̲ ) with X ̲ ∈ sl ( 2 , R ) or X ̲ ∈ sl ( 2 , C ) , respectively, to clarify why ln ( M ̲ ) ∉ sl ( 2 , R ) or ln ( M ̲ ) ∉ sl ( 2 , C ) outside of the corresponding domains 1 or 1 , C . On the other hand, for the Lorentz groups SO (1, 2) and SO (1, 3), we confirm through construction of the logarithm ln ( Λ ̲ ) that every transformation Λ ̲ in the connectivity component + ↑ of the identity element can be represented in the form exp ( X ̲ ) with X ̲ ∈ so ( 1 , 2 ) or X ̲ ∈ so ( 1 , 3 ) , 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 SL ( 2 , C ) : The logarithms ln ( Λ ̲ ) in + ↑ correspond to logarithms on the first sheet of the covering map SL ( 2 , C ) → + ↑ , which is contained in 1 , C . 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.