of time”, instead of reversing the sign of the energy. Let us see how this would work in practice.
Creating a particle with positive energy would then be mapped into annihilating an anti-particle
with positive energy. Analogously, annihilating a particle with positive energy maps to creating an
anti-particle with positive energy. Thus, if the solutionu(k) multiplies an annihilation operator for
a particle, thenv(k) should multiply a creation operator for an anti-particle.Thus, the Dirac field
decomposes as follows,
ψ(x) =∑k,s{
us(k)bs(k)e−ik·x+vs(k)d†s(k)e+ik·x}ψ†(x) =∑k,s{
u†s(k)b†s(k)e+ik·x+v†s(k)ds(k)e−ik·x}
(23.5)Here, we have expressed the superposition inkas a sum, but an integral overkshould be used
instead onR^3. The physical interpretation of the operators is as follows,
bs(k) annihilation operator for a particle
d†s(k) creation operator for an anti-particle
b†s(k) creation operator for a particle
ds(k) annihilation operator for an anti-particle (23.6)The Dirac spinor representation being complex, there is a priori no reality relation between the
oscillatorsbs(k) andds(k). We shall see later on that a Lorentz invariant reality (or Majorana)
condition may be imposed consistently under certain assumptions, and in that restricted case we
haveds(k) =bs(k), so that the corresponding Majorana fermion particle coincides with its anti-
particle (just as the photon did).
From the above construction, we can immediately draw a conclusion on the electric charge
assignments of the particle and anti-particle. Recall thatthe electric charge is the N ̈other charge
associated to phase rotation symmetry of the Dirac field,
ψ(x)→ψ′(x) =eiθψ(x) (23.7)for constant phasesθ. We conclude that the electric charges ofbs(k) andd†s(k) must be the same.
This means that the change in charge due to annihilating a particle must be the same as the
electric charge change due to creating an anti-particle. Thus, the electric charges of particle and
anti-particle must be opposite. As a corollary, only electrically neutral particle can be their own
anti-particles.
Dirac at first did not know how to interpret the extra “negative energy solutions”, and neither
did anybody else for some time. In fact, Dirac originally proposed that the proton would be this
missing anti-particle to the electron, mainly because the electric charges of the proton and electron
are exactly opposite. This is actually incorrect, since we can clearly see from the Dirac equation
and its solutions that mass of the particle and anti-particle must be the same, while the masses of
the proton and the electron differ by a factor of 2000.