If you look in advanced books on quantum mechanics, you will likely find the Dirac equation written in a different form than given in chapter 12.12.
The Hamiltonian eigenvalue problem as given in that section was
Now assume for a moment that is a state of definite momentum.
Then the above equation can be rewritten in the form
It is easy to check that the only difference between the and matrices is that through get a minus sign in front of their bottom element. (Just multiply the original equation by and rearrange.)
The parenthetical expression above is essentially a four-vector dot
product between the gamma matrices and the momentum four-vector.
Especially if you give the dot product the wrong sign, as many
physicists do. In particular, in the index notation of chapter
1.2.5, the parenthetical expression is then
. Feynman hit upon the bright idea to
indicate dot products with matrices by a slash through the
name. So you are likely to find the above equation as
Also consider the case that is not an energy and momentum
eigenfunction. In that case, the equation of interest is found from
the usual quantum substitutions that becomes
and becomes
. So the rewritten Dirac
equation is then: