are eigenvectors of
3
,
u
(
p
,
r
)
and
v(
p
,
r
)
will be eigenvectors of
3
.
In
Sect.9.4.1
of last chapter, a general method was applied to the construction
of the singleparticle quantum states

p
,
r
acted on by a unitary irreducible repre
sentation of the Lorentz group. The method consisted in first constructing the states
of the particle
 ¯
p
,
r
in some special frame
S
0
in which the momentum of the par
ticle is the standard one
¯
p
, and on which an irreducible representation
R
of the
little group
G
(
0
)
of
¯
p
acts (
¯
p
=
(
mc
,
0
)
and
G
(
0
)
=
SU
(
2
)
for massive particles,
while
¯
p
=
(
E
,
E
,
0
,
0
)/
c
and
G
(
0
)
is effectively SO
(
2
)
for massless particles). A
generic state

p
,
r
is then constructed by acting on
 ¯
p
,
r
by means of
U
(
p
)
, see
(
9.111
), that is the representative on the quantum states of the simple Lorentz boost
p
connecting
¯
p
to
p
:
p
=
p
¯
p
.
This suffices to define the representative
U
(
)
of a generic Lorentz transformation, see (
9.112
). In this section we have applied
this prescription to the construction of both the positive and negative energy eigen
states of the momentum operators. The role of

p
,
r
is now played by the spinors
u
(
p
,
r
), v(
p
,
r
)
, and that of
U
(
)
by the matrix
S
(
)
, as it follows by comparing
(
10.149
) with (
9.112
). It is instructive at this point to show that the expressions for
u
(
p
,
r
), v(
p
,
r
)
given in (
10.154
) or, equivalently, (
10.157
), for massive fermions,
could have been obtained from the corresponding spinors
u
(
0
,
r
), v(
0
,
r
)
in
S
0
using
the prescription (
9.111
), namely by acting on them through the Lorentz boost
S
(
p
)
:
u
(
p
,
r
)
=
S
(
p
)
u
(
0
,
r
)
;
v(
p
,
r
)
=
S
(
p
)v(
0
,
r
).
(10.163)
This is readily proven using the matrix form (
10.118
) of
S
(
p
)
derived in
Sect.10.4.4
and the definition of
u
(
0
,
r
), v(
0
,
r
)
in (
10.152
). The matrix product on the right hand
side of (
10.163
) should then be compared with the matrix form of
u
(
p
,
r
), v(
p
,
r
)
in (
10.157
).