In order for a wave equation to satisfy the special relativistic requirement of Lorentz covariance, derivatives in space and time must all appear in the same order. The K-G equation illustrates that an expression which satisfies this condition but is non-linear in the space and time derivatives gives rise to anomalous results. Dirac set out to find an equation which was first order in space and time derivatives. The result of his efforts, the Dirac equation, is difficult to motivate and far more complicated that the non-relativistic analog. However, Dirac's wave equation for a single particle satisfies all the requirements of special relativity and quantum mechanics, and is able to predict the properties of one particle systems with remarkable accuracy.
Dirac's equation for an electron in field-free space is given by
where
and 
is simply the three component momentum. In order to determine the
nature of the three components of 
and 
, it is useful
to compare the modified equation
to the Klein-Gordon equation. Equations ( 1.3) and ( 1.6) are equivalent if we enforce the conditions
where 
represents 
for i=0 and
,
, and 
for i= 1-3,
respectively. In order for this set of four objects to fulfill these
anti-commutation relations, each 
must be, minimally,
four-dimensional.
One set of matrices which obey similar anti-commutation relations are the
Pauli spin matrices: 
. These are 2
2 matrices,
however, and there are only three of them, so they are not useful in their
usual form.
The anti-commutation conditions given in 1.7 are satisfied
if the 
's are defined as
where 
is a two by two identity matrix and 
are the Pauli
spin matrices. Though these four by four matrices do not represent the
only set of matrices which satisfy the anti-commutation relations,
any set which does may only differ by a similarity transform from this set.
Because the Dirac equation contains operators represented by four
dimensional matrices, the solutions {
} must
be represented by a four component vector
The interpretation of the components of this four-vector is not readily evident. Expressing the Dirac equation in full matrix form provides some elucidation
The non-relativistic electronic wave-function has two components
corresponding to the 
and 
components of spin angular
momentum. In the non-relativistic limit,
approaches mc, and the terms
which couple 
with 
drop out. What remains are four
eigenvector equations, with approximate eigenvalues of 
for
and 
, and 
for 
and 
.
and 
, then, may be interpreted as the 
and
components of positive energy, electron-like solutions, but the
solutions which are dominated by 
and 
do not posses a
readily evident interpretation.
The existence of these negative energy solutions troubled Dirac.
He felt that it was a ``great blemish'' on his theory that these apparently
unphysical solutions remained. Over the next few years, however, several
experimental investigations, most notably those of Carl
Anderson[1], gave evidence for the existence of positrons,
particles with the mass of an electron but a positive charge. Experimental
confirmation of the existence of these negative energy solutions was a great
triumph for Dirac's theory.
