components arise by multiplication of elements inSO(3,1) byP,T,PTwhere
P=
1 0 0 0
0 − 1 0 0
0 0 − 1 0
0 0 0 − 1
is called the “parity” transformation, reversing the orientation of the spatial
variables, and
T=
−1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
reverses the time orientation.
The Lorentz group has a subgroupSO(3) of transformations that just act
on the spatial components, given by matrices of the form
Λ =
1 0 0 0
0
0 R
0
whereRis inSO(3). For each pairj,kof spatial directions one has the usual
SO(2) subgroup of rotations in thejkplane, but now in addition for each pair
0 ,jof the time direction with a spatial direction, one hasSO(1,1) subgroups
of matrices of transformations called “boosts” in thejdirection. For example,
forj= 1, one has the subgroup ofSO(3,1) of matrices of the form
Λ =
coshφ sinhφ 0 0
sinhφ coshφ 0 0
0 0 1 0
0 0 0 1
forφ∈R.
The Lorentz group is six dimensional. For a basis of its Lie algebra one can
take six matricesMμνforμ,ν∈ 0 , 1 , 2 ,3 andj < k. For the spatial indices,
these are
M 12 =
0 0 0 0
0 0 −1 0
0 1 0 0
0 0 0 0
, M^13 =
0 0 0 0
0 0 0 1
0 0 0 0
0 −1 0 0
, M^23 =
0 0 0 0
0 0 0 0
0 0 0 − 1
0 0 1 0
which correspond to the basis elements of the Lie algebra ofSO(3) that we first
saw in chapter 6. These can be renamed using the same names as earlier
l 1 =M 23 , l 2 =M 13 , l 3 =M 12