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

.