Using the method of canonical quantization we can show that for a free scalar field we have: $$\phi(x)=\int d\tilde p (a(\vec p) e^{-ipx}+b^\dagger(\vec p) e^{ipx})$$ where $a(\vec p)$ and $b^\dagger(\vec p)$ are two operators. It turns out that $a(\vec p)$ and $b^\dagger(\vec p)$ are creation and annihilation operators, but every argument I have seen for this has been fairly hand-wavy.

So my question is: What is the easiest method to show that $a(\vec p)$ and $b^\dagger(\vec p)$ have to be creation and annihilation operators and can be nothing else?

The definition of creation and annihilation operators is the Fock definition, as given e.g. on page 12 here.


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