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.

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 M ln( ) 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 M ln( ) for all matrices M in the fundamental representations of the non-compact groups SL SL 2, , 2,   ( ) ( ), and SO(1, 2). The construction for SL 2,  ( ) 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,  ( ) and SL 2,  ( ). Therefore we revisit the maximal neighbourhoods SL 2, ) which can be covered through single exponentials X exp( ) with X sl 2,  Î ( ) or X sl 2,  Î ( ), respectively, to clarify why M sl ln 2 , Ï ( ) ( ) or M sl ln 2 , Ï ( ) ( ) outside of the corresponding domains 1  or 1,   . On the other hand, for the Lorentz groups SO(1, 2) and SO(1, 3), we confirm through construction of the logarithm ln L ( ) that every transformation L in the connectivity component  +  of the identity element can be represented in the form X exp( ) with X so 1, 2 Î ( ) or X so 1, 3 Î ( ), 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,  ( ): The logarithms ln L ( ) in  +  correspond to logarithms on the first sheet of the covering map SL 2,    +  ( ) , which is contained in 1,   . 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.

Introduction
Physicists are well versed in the use of Lie groups and Lie algebras for the description of continuous symmetries in nature, and indeed some of the most extensive and finest textbooks on applications of Lie groups have been written by physicists, see e.g. [1][2][3]. It is also well known that at least in a neighbourhood 1  of the identity element, we can represent every linear symmetry transformation through a matrix see e.g.Proposition 1.6, Chapter II, in [4], and the properties of the matrices M in the Lie group G can be inferred from the properties of the matrices X in the corresponding Lie algebra g. The Lie algebra elements have an expansion in a finite basis X i (we use summation convention), The use of matrix notation already indicates that we identify Lie groups and their Lie algebras with their defining matrix representations for simplicity. The commutation relations (3) are in one-to-one correspondence to the composition law in the group in a neighbourhood of the identity, through the iterative Baker-Campbell-Hausdorff procedure to combine two matrix exponentials into a series of higher order commutators, and both of these factors can be represented in terms of single matrix exponentials, using e.g.the general construction of matrix logarithms given in equations (21), (22) below. This tells us that we can write every matrix in the connected component of a matrix symmetry group as the product of at most two exponentials. Another argument based on the decomposition into compact and non-compact generators is reviewed in [5].
A famous example for an everyday use of the polar decomposition is the Lorentz group, for which every element (up to reflections) can be written as the product of a boost with velocity v u The rotation matrix is given by the Rodrigues formula The generator of the boost matrix is with the rapidity u, The generators of the Lorentz group satisfy commutation relations This implies in particular, and the Baker-Campbell-Hausdorff (BCH) formula (5) therefore implies that the exponents in (8) can be combined trivially into a single exponent corresponding to the Lie algebra element u K L j + · · , if u 0 j´= . However, the Baker-Campbell-Hausdorff formula also seems to suggest that we can always combine the Lie algebra elements u K · and L j · into a single Lie algebra element ) · ( ) · , or can we? The reach of the exponential map is a non-trivial question for non-compact groups, and groups where the identity-connected component cannot be covered by single Lie algebra exponentials include e.g. SL n,  ( )and GL n,  ( ). While the general representation (8) of proper orthochronous Lorentz transformations in terms of two Lie algebra exponentials is standard textbook knowledge in relativity, electrodynamics and quantum mechanics (see e.g. [7][8][9]), definitive statements about the question whether every proper orthochronous Lorentz transformation can be expressed in the form do not seem to have made it into the textbook literature, although experts on the Lorentz group know that this property holds [10][11][12]. Riesz proved it using the decomposition of Minkowski space into different invariant subspaces under Lorentz transformations. Furthermore, exponentials of general so(1, 3) elements have been evaluated in closed form (generalized Rodrigues formulae) both in the vector and the spinor representations of the Lorentz group [11][12][13][14], and both Zeni and Rodrigues, as well as Özdemir and Erdoğdu, have pointed out that these formulae can in principle be used to answer the question of Lie algebra coverage for  +  affirmatively by comparing the closed forms of (16) with equation (8) and demonstrating that the We would like to confirm this observation from a different instructive angle, because the question for the maximal reach of single Lie algebra exponentials in a non-compact group can also be phrased in terms of the general definition of the logarithm of a matrix.
Suppose M is an invertible square matrix which is related to its Jordan canonical form through Each of the smaller square matrices J n has the form l l l l l implies that none of the eigenvalues λ can vanish. We do not presume whether the matrix M is real or complex. However, the Jordan canonical form may require that we allow for complex eigenvalues λ n and complex transformation matrices V to ensure that the characteristic equation and it is possible to verify J X exp = ( )through direct calculation, see e.g.Appendix F in [9].
This indicates that we can always construct M ln( )(at least over ) for every element M of a matrix Lie group. How can it then be that we cannot represent every group element as a single exponential of a Lie algebra element for some groups like SL 2,  ( )? An explicit counterexample for SL 2,  ( ) is e.g.provided by the diagonal matrices a a diag , 1 [ ] with a 0 1 > >or a 1 < - [2], and Gilmore reviews the criterion Can the transformation with V in equation (22) introduce singularities in M ln( )if M is too far away from the identity element? Or can we have negative or complex eigenvalues? And can the imaginary parts of the logarithms of those eigenvalues survive the transformation with V in equation (22) if M is too far away from the identity element? What else could destroy the property M G M ln Î  Î ( ) g? And how does the Lorentz group behave under the exponential map of its Lie algebra?
We will study these questions from the point of view of the matrix logarithm. Specifically, we will revisit the group SL 2,  ( ) in section 2, the group SL 2,  ( ) in section 3, and the Lorentz group in section 4. The covering maps SL 2, ) and SL 2, ) onto the identity-connected components of SO(1, 2) and SO(1, 3) are then revisited in section 5 to clarify why the components n  +  are covered by single Lie algebra exponentials when their double covers do not share that property. Section 6 summarizes our conclusions.
2. The range of the exponential map in SL 2,  ( ) The group SL 2,  ( ) is the group of all real 2×2 matrices with unit determinant, The inverse matrix is of M M tr 2 1 -( ( ) ) then determine the elements a 3 and a ± of M ln( ). Here we used the inverse hyperbolic cosine to emphasize the connection between the results through x x arccosh i arccos =  . However, in the following we will use the representation through the logarithm for the inverse hyperbolic cosine function, p -= -+ - the relation between the matrix elements of M and the Lie algebra expansion coefficients a 3 and a  of M ln( )was invertible, whereas the Lie algebra expansion coefficients for ln 1 -( )are only restricted by the relation a a a n 2 1 To reveal what happens with the exponential map outside of the region 1  , we proceed in reverse direction and explicitly construct the logarithm M ln( )for the general SL 2,  ( ) matrix M (23) by following through with the general procedure through the canonical form The equations (42), (44) imply that we need to discuss the cases a d M M tr 2, tr 2 The inverse matrix V 1 contains the dual vectors v  as adjoint row vector. The orthogonality condition and the transformation equation into canonical form, Evaluation of the products then yields  Indeed, it is easily confirmed that the matrices We also note that we can express the results (51), (52) in the form ( ) ( ) (or more generally (2n+1)iπ). We then recover the result (38) in the following form, We can also go beyond the previous sl 2,  ( )constructions by using x n x n ln 2 1 i l n , The Baker-Campbell-Hausdorff series for this combination of exponentials apparently does not converge. We also note that equation ( , when we also have two-fold degeneracy of eigenvalues with only one corresponding eigenvector. However, in this case the calculation through the Jordan form recovers the result (53) which we already found from the limit of the non-singular cases.
Note that this is a star shaped neighbourhood of the identity element, as required by general Lie group properties.
, can also be inferred from Here s is the vector of Pauli matrices The functions x y X , ( )and x y Y , . This confirms that the only SL 2,

The range of the exponential map in the Lorentz group
We will see in section 5 that application of equation (51) to SL 2,  ( ) provides logarithms for SO(1, 3) through the spinor representations, and this construction provides another proof of the fact that the Lorentz group is covered by single Lie algebra exponentials. However, it is also instructive to examine logarithms in the vector representation. The calculation of matrix logarithms for general SO(1, 3) transformations in the vector representation is algebraically much more involved, but we will see from the BCH formula (and also from equation (119) below) that for a demonstration of complete coverage of SO(1, 3) in the vector representation, it is sufficient to calculate the logarithms for the subgroup SO (1, 2), where the calculation is not harder than in the spinor representations.
The BCH formula (5) and the commutator (15) It is a direct consequence of the commutation relations (14) that the higher order expansion coefficients are all determined through cross products of u and j, and for given magnitudes u | |and j | |, we face the greatest obstacle in combining the boost and the rotation of the general proper orthochronous Lorentz transformation (8) into a single exponential if the rapidity u and the rotation axis ĵ are perpendicular. This is the case which tells us whether the logarithm ln L ( )of a proper orthochronous Lorentz transformation is always an element in the Lie algebra of the Lorentz group, or whether there are exceptions. Furthermore, the commutation relations (14) imply the conjugation properties The eigenvalue condition is where we introduced We therefore find that ln L ( ) is always a Lie algebra element,  If γ and j | |are too large then c p > | | , see figure 3. Furthermore, contrary to the angle j, the relations (105) and (106) of L in terms χ are not periodic in χ. Therefore χ can only be interpreted as a rotation angle in the limit γ→1.
5. The ranges of the exponential maps in the covering groups SL 2,  ( ) and SL 2,  ( ) versus the ranges in the corresponding Lorentz groups We have reconfirmed through the matrix logarithms that the identity connected components of the Lorentz groups, i.e. the proper orthochronous Lorentz groups 3  +  and 4  +  in three or four dimensions, are covered by single Lie algebra exponentials, while their corresponding double covers SL 2,  ( ) and SL 2,  ( ) cannot be covered by single Lie algebra exponentials. How can that be? The point is that the surjective mappings from the covering groups onto the proper orthochronous Lorentz groups are not injective, and therefore we cannot infer that bothelements U and U of a covering group, which map into the same Lorentz transformation L in 3  +  or 4  +  , have logarithms in sl 2,  ( )or sl 2,  ( ), respectively. On the other hand, the fact that the proper orthochronous Lorentz groups are covered by single Lie algebra exponentials must also hold in the spinor representations, and therefore the maximal reaches SL 2, ) of the exponential maps in the covering groups must encompass complete sheets of the corresponding covering maps onto the proper orthochronous Lorentz groups. We will confirm this by calculating the matrix logarithm of the spinor representation of the Lorentz transformation (84) in {ct, x, y} subspace (with z z ¢ = ) in the (1/2,0) representation of SL 2,  ( ). However, the standard embedding of spinor representations of SO(1, 2) in the righthanded (1/2,0) representation or the left-handed (0,1/2) representation of SL 2,  ( ) does not respect SL 2,  ( ) since the rotation generator i 2 is complex in those representations. Therefore we will also confirm that SL 2, )encompasses a complete sheet of the covering map SL 2, ) in a separate calculation in section 5.3. 4  +  and SL 2,  ( ) revisited It is well known that SL 2,  ( ) is the universal double cover of the proper orthochronous Lorentz group 4  +  through the mappings

The relation between
and those two SL 2,  ( ) transformations yield the same Lorentz transformation L (114). We denote the range 0jπ as the first sheet of the covers (114), (115) and the range π<j2π as the second sheet.

The matrix logarithm in the (1/2,0) spinor representation
The surjective mappings (114), (115) are mappings between matrix elements, but they are not direct mappings between matrix logarithms in the two groups. Therefore it is not possible to draw a direct link between the facts that the exponential map so We can confirm this explicitly through a calculation of logarithms of SL 2,  ( ) matrices e.g.in the righthanded spinor representation (109) and comparison with our previous results in the vector representation. The right-handed spinor representation corresponding to the Lorentz transformation (84) is The corresponding eigenvalues κ + =1/κ − and eigenvectors are where U is the standard (1/2,0) representation matrix from equation (120). The logarithm of M can therefore be directly inferred from our previous result (124),