472 A Appendix
Recalling the definition of charge conjugation and parity operator CP (Sect. 6.8), if
.r;t/ represents the particle state, i
2
K is the corresponding antiparticle state.
Then, the operator turns, for example, the neutrino state (D spinor with positive
energy) with negative helicity, into an antineutrino state (D spinor with negative
energy) with positive helicity, see Fig. 6.3.
A.4.3 Properties of the Dirac Equation Solutions
The Dirac equation is a set of four equations, each of which must have a solution.
These solutions are normally written in the following form:
D ue
i.prEt/
D ue
ip
x
(A.44)
where the space-time dependence e
i.prEt/
e
ip
x
is separated from that
depending on the spin u. The four component spinors u are constants and correspond
to 4 1 matrices. There is a very useful representation, namely,
D
e
i.prEt/
(A.45)
where and are two component spinors. As an exercise (Problem 4.12), we
demonstrate that inserting (A.45) in the Dirac equation results in
D
p
E m
(A.46a)
D
p
E Cm
; (A.46b)
forming a homogeneous system which admits solution only if E
2
p
2
m
2
D 0.
We obtain again a quadratic relation, similar to that already obtained with the Klein–
Gordon equation whose solutions are
E D˙
p
p
2
C m
2
; (A.47)
including solutions with negative energy. Dirac suggested that all states with
negative energy are fully occupied. The creation of a e
C
e
pair is explained as
the transition (induced by a photon with energy E
>2m
e
c
2
)ofanelectronina
negative energy state into a positive energy state. The lack of an electron (a hole)in
this sea of particles with negative energy represents a positron.
An additional difficulty arises from the consideration that if all the negative
energy states are filled, the energy of the vacuum state is 1. Dirac solved this
issue, assuming that measurable quantities correspond to a variation with respect to
the vacuum state, in analogy with the classical potential energy. As the measured