1.3. MATHEMATICAL PROLOGUE: SOME LIE GROUPS AND LIE ALGEBRAS 9
The group O(1, 3) has four connected components, which are determined by
the values of two homomorphisms from O(1, 3) onto the two-element group {±1}:
A det A and A sgn(A0)
0
(the sign of the (0, 0) entry of A).
The kernel of the first of these is the special Lorentz group SO(1, 3), and the kernel
of the second one is the orthochronous Lorentz group O↑(1, 3), the subgroup of
O(1, 3) that preserves the direction of time. The intersection
SO↑(1,
3) = SO(1, 3)
O↑(1,
3),
sometimes called the restricted Lorentz group or proper Lorentz group, is the con-
nected component of the identity in O(1, 3). The linear isometry group or orthog-
onal group O(3) of
R3
sits inside
O↑(1,
3) as the subgroup that fixes the point
(1, 0, 0, 0), and the rotation group SO(3) is O(3)
SO↑(1,
3).
The Lie algebra so(1, 3) of the Lorentz group consists of the 4 × 4 real matrices
X that satisfy
X†g
+ gX = 0.
A convenient basis for it may be described as follows. Let eμν be the matrix whose
(μ, ν) entry is 1 and whose other entries are 0, and define
Xjk = ekj ejk, Xk0 = −X0k = ek0 + e0k (j, k = 1, 2, 3).
Then {Xμν : μ ν} is a basis for so(1, 3), and of course one can replace any Xμν
by Xνμ = −Xμν . The commutation relations are given by
(1.8)
[Xμν,Xρσ] = 0 if {μ, ν} = {ρ, σ} or {μ, ν} {ρ, σ} = ∅,
[Xμν,Xνρ] = gννXμρ.
The Lie algebra so(3) of the spatial rotation group is the span of {Xjk : j, k 0}.
In detail, the relation between these basis elements and the Lie group is as
follows. If (j, k, l) is a cyclic permutation of (1, 2, 3), exp(sXjk) is the rotation
through angle s about the
xl-axis
(counterclockwise, as viewed from the positive
xl-axis).
On the other hand, exp(sX0k) is a so-called boost along the
xk-axis
that
changes from the initial reference frame “at rest” to one moving with velocity tanh s
along the
xk-axis.
Thus, for example,
exp(sX23) =
I 0
0 R(s)
, R(s) =
cos s sin s
sin s cos s
,
and
exp(sX01) =
B(s) 0
0 I
, B(s) =
cosh s sinh s
sinh s cosh s
.
Orbits and invariant measures. We need to consider the geometry of the
action of O(1, 3) on
R4.
Actually, there are two natural actions of O(1, 3) on
R4:
the identity representation of O(1, 3) and its contragredient A
A†−1.
Strictly
speaking, the latter is the action of O(1, 3) on the dual space
(R4)
; physically,
it is the action on momentum space rather than position space. Since the map
A
A†−1
is an automorphism of O(1, 3), however, the orbits in
R4
under the two
actions are identical, and we need not distinguish them. In what follows we shall
think of
R4
as momentum space, since this is the context in which the orbits usually
appear naturally.
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