# Explicit form of Dirac field creation/annihilation operators?

The explicit form of the creation and annihilation operators for the complex scalar field seems to be shown in all QFT lecture notes, but not those for the Dirac field (instead they tend to only give the anticommutation relation).

What is the explicit form of the Dirac field creation/annihilation operators?

• What do you mean by explicit form? Dec 30, 2021 at 23:44
• Equivalent to something like our integral over the field operator and momentum density operator that we have for the scalar fields? Dec 30, 2021 at 23:49
• Mode expansion of Dirac fields is discussed in pretty much every QFT book/lecture note, just like that of the scalar field. For example, damtp.cam.ac.uk/user/tong/qft/five.pdf Equation (5.4) is what you are looking for. Dec 31, 2021 at 0:58
• Oh okay, so would finding the operators b and c there be as simple as inverting those two equations? Dec 31, 2021 at 1:06
• you mean like here? Dec 31, 2021 at 17:15

$$\psi(x) = \int \frac{d^3p}{(2\pi)^3}\frac{1}{\sqrt{2E_p}}\sum\limits_s[a^s_p u^2(p) e^{-ipx} + b^{s\dagger}_p v^s(p) e^{ipx}]$$
$$a^s_p = \frac{i}{2m}\int d^3 x \frac{\bar{u}(p)}{\sqrt{2E_p}} (e^{ipx} \partial_0 \psi - \psi\partial_0e^{ipx})$$