Scalar QFT Fock Space

Question

I want to demostrate the following relation of the normal ordered product:

$\Omega\equiv:\exp{\left(-\int d^3k~a^{\dagger}(k)a(k)\right)}:=|0\rangle\langle0|.$

I proved the commutation relation $[\Omega,a^{\dagger}(p)]=-a^{\dagger}(p)\Omega$ and I must use the identity of the Fock space

$1=|0 \rangle\langle 0|+\sum_{n=1}^{\infty}|p_1,p_2,...,p_n\rangle\langle p_1,p_2,...,p_n|.$

Any ideas how to proceed to demostrate the first relation? I proved that commutation relation because it's usefull, but I don't know how to use it!

P.S.: $p$ and $k$ denotes momentum variables, $|0\rangle$ it's the vacuum state and $|p_1,p_2,...,p_n\rangle$ it's the $n$ particle state where the $i$-particle have $p_i$ momentum.

Answer 1

Would it help if I pointed out that your commutation relation shows that $a(p) \Omega = 0 = \Omega a^{\dagger}(p)$? You should be able to use this to argue that the only non-zero matrix element of $\Omega$ is $\langle 0 |\Omega|0\rangle$.

Answer 2

You can see $a(k)=a^{\dagger}(-k)$,so all the states with k are cancelled when you do the integral. The only thing left is the $|0 \rangle$ state.