quantum mechanics - Operator norm of creation and annihilation operators

Are the creation and the annihilation operators $a(f)$ and $a^{\dagger}(f)$ for the bosonic Fock space bounded? What is their norm? So far I did not have found any note about this in the linked Wikipedia article.

Answer

OK, here you go:

No. Consider the one-particle sector, with states $|n\rangle$ having occupation number $n$. Wlog we can suppose that the states are normalized. Then $$a|n\rangle =\sqrt{n}|n−1\rangle,$$ so $$\|a|n\rangle\| = \sqrt n.$$ This means that $a$ is not bounded. The same goes for $a^\dagger$.