### A.35 Alternate Dirac equations

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

where was a vector with four components.

Now assume for a moment that is a state of definite momentum. Then the above equation can be rewritten in the form

The motivation for doing so is that the coefficients of the matrices are the components of the relativistic momentum four-vector, chapter 1.3.1.

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

Isn’t it beautifully concise? Isn’t it completely incomprehensible?

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:

In index notation, the parenthetical expression reads . So following Feynman

Now all that the typical physics book wants to add to that is a suitable non-SI system of units. If you use the electron mass as your unit of mass instead of the kg, as unit of velocity instead of m/s, and as your unit of angular momentum instead of kg m/s, you get

No outsider will ever guess what that stands for!