̄hγμ∂
∂xμ
ψ+mcψ= 0TheDirac equation in the absence of EM fieldsis
(
γμ∂
∂xμ+
mc
̄h)
ψ= 0.ψis a4-component Dirac spinorand, like the spin states we are used to, represents a coordinate
different from the spatial ones.
Thegamma matricesare 4 by 4 matrices operating in this spinor space. Note that there are 4
matrices, one for each coordinate but that the row or column of the matrix does not correlate with
the coordinate.
γ 1 =
0 0 0 −i
0 0 −i 0
0 i 0 0
i 0 0 0
γ 2 =
0 0 0 − 1
0 0 1 0
0 1 0 0
−1 0 0 0
γ 3 =
0 0 −i 0
0 0 0 i
i 0 0 0
0 −i 0 0
γ 4 =
1 0 0 0
0 1 0 0
0 0 −1 0
0 0 0 − 1
Like the Pauli matrices, the gamma matrices form a vector, (this time a 4vector).
It is easy to see by inspection that theγmatrices areHermitian and traceless. A little compu-
tation will verify thatthey anticommuteas the Pauli matrices did.
{γμ,γν}= 2δμνSakurai shows that the anticommutation is all that is needed to determine the physics. That is, for
any set of 4 by 4 matrices that satisfy{γμ,γν}= 2δμν,
(
γμ∂
∂xμ+
mc
̄h)
ψ= 0will give the same physical result, although the representation ofψmay be different. This is
truly an amazing result.
There are a few other representations of the Dirac matrices thatare used. We will try hard to stick
with this one, the one originally proposed by Dirac.
It is interesting to note that theprimary physics input was the choice of the Schr ̈odinger-
Pauli Hamiltonian
[~p+
e
cA~(~r,t)]→~σ·[~p+e
cA~(~r,t)]