### 6.2 Antiparticles

Both the Klein-Gordon and the Dirac
equation have a really nasty property. Since the relativistic
energy relation is quadratic, both equations have, for every
positive energy solution, a negative energy solution. We
don’t really wish to see such things, do we? Energies are
always positive and this is a real problem. The resolution is
surprisingly simple, but also very profound – It requires
us to look at the problem in a very diﬀerent light.

In ﬁgure
6.1
we have sketched the solutions
for the Dirac equation for a free particle. It has a positive
energy spectrum starting at
$m{c}^{2}$(you cannot have a particle at lower energy), but also a
negative energy spectrum below
$-m{c}^{2}$. The interpretation of the positive energy states is
natural – each state describes a particle moving at an
energy above
$m{c}^{2}$. Since we cannot have negative energy states, their
interpretation must be very diﬀerent. The solution is
simple: We assume that in an
empty vacuum all negative energy
states are ﬁlled (the “Dirac sea”).
Excitations relative to the vacuum can now be obtained by
adding particles at positive energies, or creating
holes at negative energies.
Creating a hole takes energy, so the hole states appear at
positive energies. They do have opposite charge to the particle
states, and thus would correspond to positrons! This shows a
great similarity to the behaviour of semiconductors, as you may
well know. The situation is explained in ﬁgure
6.2
.

Note that we have ignored the inﬁnite
charge of the vacuum (actually, we subtract it away assuming a
constant positive background charge.) Removing inﬁnities
from calculations is a frequent occurrence in relativistic
quantum theory (RQT). Many
unmeasurable quantities become
inﬁnite, and we are only interested in the ﬁnite
part remaining after removing the inﬁnities. This
process is part of what is called renormalisation, which is a
systematic procedure to extract ﬁnite information from
inﬁnite answers!