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

Abstract

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 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.