10 1. PROLOGUE
The orbits of the restricted Lorentz group and the orthochronous Lorentz group
are identical. Parametrized by a nonnegative real number m that generally has a
physical interpretation as a mass, they are as follows:
(1.9)
Xm
+
= p :
p2
=
m2,
p0 0 , Xm

= p :
p2
=
m2,
p0 0 ,
Ym = p :
p2
=
−m2
(m 0),
{0}.
(Recall that
p2
=
pμpμ.)
The orbits Xm
+
(and sometimes also Xm)

are known as
mass shells, and the orbits X0
+
and X0

are called the forward and backward light
cones. (Under the action of the full group O(1, 3), the two orbits Xm ± coalesce into
one.)
Observe that
(1.10) p Xm
+
⇐⇒ p = (ωp, p), where ωp = m2 + |p|2.
The symbol ωp will carry this meaning throughout the book.
Each of the orbits (1.9) has an
O↑(1,
3)-invariant measure, which is known on
abstract grounds to be unique up to scalar multiples. The invariant measure on the
mass shell Xm
+
with m 0 may be derived as follows. Let V =
m0
Xm
+
= {p :
p2
0, p0 0} be the region inside the forward light cone. Then V is
O↑(1,
3)-
invariant, and since | det T | = 1 for T
O↑(1,
3), the restriction of Lebesgue mea-
sure
d4p
to V is an
O↑(1,
3)-invariant measure on V ; hence so is
f(p2)d4p
for any
nonnegative continuous f with support in (0, ∞). One obtains the invariant mea-
sure on Xm
+
by letting f turn into a delta-function with pole at
m2: δ(p2

m2)d4p.
The result often appears in precisely this way in the physics literature, but in order
to avoid possible pitfalls in using delta-functions with nonlinear arguments it is best
to take a little more care.
To wit, consider the map φ : (0, ∞) ×
R3
V defined by
φ(y, p) = ( y + |p|2, p),
so that q2 = y when q = φ(y, p). φ is a diffeomorphism, and its Jacobian is
1/2 y + |p|2, so for f Cc(0, ∞) we have
f(p2)d4p
=
f(y) dy
d3p
2 y + |p|2
.
Now let f approach the delta-function with pole at y =
m2:
if we write points in
Xm + as (ωp, p) where ωp = m2 + |p|2, we obtain the invariant measure
(1.11)
d3p
2 m2 + |p|2
=
d3p
2ωp
.
A limiting argument then shows that d3p/2|p| is an invariant measure on X0
+.
The
invariant measure on Xm, of course, is also given by (1.11). We leave the calculation
of the invariant measure on Ym, for which we shall have no use, to the reader.
SL(2, C), SU(2), and the Pauli matrices. The three Pauli matrices are
(1.12) σ1 =
0 1
1 0
, σ2 =
0 −i
i 0
, σ3 =
1 0
0 −1
.
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