This question is actually shockingly easy. It’s one of those questions whose statement is so short and is literally just “prove this identity” that you assume it’s going to take pages of obtuse and demoralising mathematical trickery. Not this time!
The question demands that we prove the Gordon identity, and assures us that this identity will be used in Chapter 6. It goes as follows:
I’m not going to explicitly go through the calculation of this because it’s trivial once you know the steps and have the necessary information about gamma matrices and whatnot.
- Start by writing . This is a trick that dawned on me exactly half way through the solution (when I realised I was clearly missing a term) but the factor of a half in the RHS of the identity should give it away immediately, really.
- What are ? We can write down a plane wave solution to the Dirac equation as (we can do this since the field also satisfies the Klein-Gordon equation), so is a column vector with a constraint obtained from plugging this into the Dirac equation,
- It’s not terribly important because it doesn’t actually impact the solution, but I feel it relevant to point out that what we have here is a four-dimensional matrix (not a scalar) acting on a column vector. Beside the term is a sneaky invisible , since is still a four-dimensional matrix, being just a number (the component of the momentum four-vector).
- Use the above to write and substitute this in.
- Recalling that , combine this with the above to write . (It is necessary to recall that and to get here.)
- After rewriting and in this way, we end up having some terms like . This can be rewritten as using (hint: adding together) the following properties of gamma matrices:
- Noting that is antisymmetric, we get… the result, as desired.
- Just a note: remember that is just a number and commutes with everything, so it can be neglected during gamma matrix manipulations.
I hope I have not just committed that wonderful crime students complain about,
“The result obviously follows…”
But… the result does follow, quite obviously!