75
3.4. The negative-energy solutions
and s labels the spin degree of freedom in some suitable way (e.g. the helicity eigenvalues). The complete plane-wave solution ψ for such a positive
4-momentum state is then
ψ = u(p, s)e
−ip+ ·x
(3.74)
μ
with p = (E, p).
+
3.4.2 Negative-energy spinors
Now we look for spinors appropriate to the solution
0
2
p = −(p + m
2 )
1/2
≡ −E < 0
(3.75)
(E is always defined to be positive). Consider first what are appropriate
solutions at rest. We have now
0
p = −m
p = 0
(3.76)
and
−m
(
φ
χ
)
=
(
m1
0
0
− m1
) (
φ
χ
)
(3.77)
leading to
φ = 0.
(3.78)
Thus the two independent negative-energy solutions at rest are just
ω(p
0 = −m, s) =
(
0
χ
s
)
.
(3.79)
The solution for finite momentum +p, i.e. for 4-momentum (−E, p), is then
(
)
−σ · p χ
s
0
ω(p = −E, p, s) = ( E + m )
(3.80)
χ
s
with χ
s† χ
s = 1. However, it is clearly much more in keeping with relativity
if, in addition to changing the sign of E, we also change the sign of p and
μ
consider solutions corresponding to negative 4-momentum (−E, −p) = −p + .
We therefore define
(
)
σ · p χ
1,2
0
( E + m
)
ω(p = −E, −p, s) ≡ ω
3,4 = N
.
(3.81)
χ
1,2
Adopting the same N as in (3.73) implies the same normalization (ω
† ω =
2E) for (3.81) as in (3.73); in this case the spinors are called v(p, s) where
(problem 3.8)
(
)
σ · p χ
s
m)
1/2 ( E + m )
v(p, s) = (E +
s = 1, 2.
(3.82)
χ
s