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I am currently preparing myself for my Bachelor thesis in local quantum field theory. I was encouraged by my advisor to read the books of M. Reed and Simon because of my lag of functional analysis experience but I have quite often problems understand the “obvious” conclusions.

For example:

Why is the creation operator not a densely defined operator? And how do I proof that formally correctly? I am not asking for a step by step solution but I have absolutely no idea how to start.

Reed, Simon define the creation operator as:

$$(a^\dagger(p)(\psi))^{(n)}(k_1,...k_n) = \frac{1}{\sqrt{n}} \sum_{l=1}^n \delta(p-k_l) \psi^{(n-1)}(k_1,...,k_{l-1},k_k,...,k_n)$$

as the adjoint of the annihilation operator :

$$(a(p)\psi)^{(n)}(k_1,...k_n) = \sqrt{n+1} \psi^{(n+1)}(p,k_1,...k_n)$$

with $$\psi$$ a Schwartz function.

I really appreciate any hints.