[k 1 ,k 2 ] =−l 3 , [k 3 ,k 1 ] =−l 2 , [k 2 ,k 3 ] =−l 1or
[lj,lk] =jklll, [kj,kk] =−jklll
and that the commutation relations between theljandkjare
[lj,kk] =jklklcorresponding to the fact that thekjtransform as a vector under spatial rota-
tions.
The Poincar ́e group is a semi-direct product group of the sort discussed in
chapter 18 and it can be represented as a group of 5 by 5 matrices in much the
same way as elements of the Euclidean groupE(3) could be represented by 4
by 4 matrices (see chapter 19). Writing out this isomorphism explicitly for a
basis of the Lie algebra, we have
l 1 ↔
0 0 0 0 0
0 0 0 0 0
0 0 0 −1 0
0 0 1 0 0
0 0 0 0 0
l 2 ↔
0 0 0 0 0
0 0 0 1 0
0 0 0 0 0
0 −1 0 0 0
0 0 0 0 0
l 3 ↔
0 0 0 0 0
0 0 −1 0 0
0 1 0 0 0
0 0 0 0 0
0 0 0 0 0
k 1 ↔
0 1 0 0 0
1 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
k 2 ↔
0 0 1 0 0
0 0 0 0 0
1 0 0 0 0
0 0 0 0 0
0 0 0 0 0
k 3 ↔
0 0 0 1 0
0 0 0 0 0
0 0 0 0 0
1 0 0 0 0
0 0 0 0 0
t 0 ↔
0 0 0 0 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
t 1 ↔
0 0 0 0 0
0 0 0 0 1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
t 2 ↔
0 0 0 0 0
0 0 0 0 0
0 0 0 0 1
0 0 0 0 0
0 0 0 0 0
t 3 ↔
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 1
0 0 0 0 0
(42.1)
We can use this explicit matrix representation to compute the commutators
of the infinitesimal translationstjwith the infinitesimal rotations and boosts
(lj,kj). t 0 commutes with thelj andt 1 ,t 2 ,t 3 transform as a vector under
rotations, For rotations one finds
[lj,tk] =jkltlFor boosts one has
[kj,t 0 ] =tj, [kj,tj] =t 0 , [kj,tk] = 0 ifj 6 =k, k 6 = 0 (42.2)