Quantum Mechanics for Mathematicians

and recall that these satisfy theso(3) commutation relations

[l 1 ,l 2 ] =l 3 , [l 2 ,l 3 ] =l 1 , [l 3 ,l 1 ] =l 2

and correspond to infinitesimal rotations about the three spatial axes.
Taking the first index 0, one gets three elements corresponding to infinitesi-
mal boosts in the three spatial directions

M 01 =









0 1 0 0

1 0 0 0

0 0 0 0

0 0 0 0







, M^02 =









0 0 1 0

0 0 0 0

1 0 0 0

0 0 0 0







, M^03 =









0 0 0 1

0 0 0 0

0 0 0 0

1 0 0 0









These can be renamed as

k 1 =M 01 , k 2 =M 02 , k 3 =M 03

One can easily calculate the commutation relations between thekjandlj, which
show that thekjtransform as a vector under infinitesimal rotations. For in-
stance, for infinitesimal rotations about thex 1 axis, one finds

[l 1 ,k 1 ] = 0, [l 1 ,k 2 ] =k 3 , [l 1 ,k 3 ] =−k 2 (40.1)

Commuting infinitesimal boosts, one gets infinitesimal spatial rotations

[k 1 ,k 2 ] =−l 3 , [k 3 ,k 1 ] =−l 2 , [k 2 ,k 3 ] =−l 1 (40.2)

Digression. A more conventional notation in physics is to useJj =ilj for
infinitesimal rotations, andKj=ikjfor infinitesimal boosts. The intention of
the different notation used here is to start with basis elements of the real Lie
algebraso(3,1), (theljandkj) which are purely real objects, before complexifying
and considering representations of the Lie algebra.

Taking the following complex linear combinations of theljandkj

Aj=

1

2

(lj+ikj), Bj=

1

2

(lj−ikj)

one finds
[A 1 ,A 2 ] =A 3 , [A 3 ,A 1 ] =A 2 , [A 2 ,A 3 ] =A 1

and
[B 1 ,B 2 ] =B 3 , [B 3 ,B 1 ] =B 2 , [B 2 ,B 3 ] =B 1
This construction of theAj,Bjrequires that we complexify (allow complex
linear combinations of basis elements) the Lie algebraso(3,1) ofSO(3,1) and
work with the complex Lie algebraso(3,1)⊗C. It shows that this Lie al-
gebra splits into a sum of two sub-Lie algebras, which are each copies of the
(complexified) Lie algebra ofSO(3),so(3)⊗C. Since

so(3)⊗C=su(2)⊗C=sl(2,C)

we have
so(3,1)⊗C=sl(2,C)⊕sl(2,C)

In section 40.4 we’ll see the origin of this phenomenon at the group level.