I have some problems about basic concepts of quantum field theory.

First let's look at Klein-Gordon field.

Klein-Gordon equation has two branches of solutions, one of which is positive frequency and the other is negative frequency. The solution can be written as $$\psi(x)=\int \frac{d^{3}p}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{p}}}(a_{p}e^{-ipx+i\omega t}+b_{p}e^{-ipx-i\omega t}).$$

Then we need to quantize above solution. In the exercise part of both books, Peskin and Srednicki, the complex scalar field operator is written as

$$\hat{\psi}(x)=\int \frac{d^{3}p}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{p}}}(\hat{a}_{p}e^{-ipx}+\hat{b}_{p}^{+}e^{ipx})\tag{1}$$

The question is:

Why equ(1) is not written as $$\hat{\psi}(x)=\int \frac{d^{3}p}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{p}}}(\hat{a}_{p}e^{-ipx}+\hat{b}_{p}e^{ipx})~?$$ Is this because when quantizing negative frequency, annihilation operator must be replaced by creation operator and $-p$ by $p$ in exponent?


1 Answer 1


Of course, you can give any name for coefficients in field expansion. But when you will analyze commutation relations:

$$ [\psi(x), \psi^{\dagger}(y)] \sim \hbar \delta(x-y) $$

For your choice of "names" you will obtain the another commutation relations:

$$ [b_p, b^\dagger_k] = \delta(p-k) $$

So you see, that $b_p$ is creation operator, $b^\dagger_k$ is annihilation operator. But usually physics choose reverse conventions. Due to these, in all books, peoples use (1).

Please, ask for clarification, if you need.

  • $\begingroup$ So in quantum field, commutation relaship of field operator and its cojugate momenta comes before that of annihilation operator and creation operator, though they're in fact equivalent? $\endgroup$
    – xiang sun
    Jul 27, 2020 at 15:38
  • $\begingroup$ @xiangsun, no, they are not equivalent. There are creation and annihilation operators. And for this operators there is canonical symbols: $b^\dagger$ and $b$. In your question, you suggest noncanonical symbols for these operators. $\endgroup$
    – Nikita
    Jul 27, 2020 at 15:51

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