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?

  • $\begingroup$ What do you mean by explicit form? $\endgroup$ Dec 30, 2021 at 23:44
  • $\begingroup$ Equivalent to something like our integral over the field operator and momentum density operator that we have for the scalar fields? $\endgroup$
    – Alex Gower
    Dec 30, 2021 at 23:49
  • $\begingroup$ 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. $\endgroup$
    – Meng Cheng
    Dec 31, 2021 at 0:58
  • $\begingroup$ Oh okay, so would finding the operators b and c there be as simple as inverting those two equations? $\endgroup$
    – Alex Gower
    Dec 31, 2021 at 1:06
  • $\begingroup$ you mean like here? $\endgroup$ Dec 31, 2021 at 17:15

1 Answer 1


You can recover them from the mode expansion of the field:

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

You can verify that:

$$a^s_p = \frac{i}{2m}\int d^3 x \frac{\bar{u}(p)}{\sqrt{2E_p}} (e^{ipx} \partial_0 \psi - \psi\partial_0e^{ipx})$$

The verification can be done by plugging in the mode expansion, using orthogonality relations (page 48 Peskin and Schroeder), and using Fourier Transform properties (to get delta functions).


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