Number operator in quantum field theory?

Question

The number operator, counting the number of quanta is defined as follows:

$$ N = \int \frac{d^3 p}{(2\pi)^3} \hphantom{ii} a^{\dagger}_pa_p$$

with the momentum eigenstates being defined as $\lvert p_1, p_2, ...p_n \rangle = a^{\dagger}_{p_1}a^{\dagger}_{p_2}...a^{\dagger}_{p_n}\lvert0\rangle$.

The claim is that $N \lvert p_1, p_2, ...p_n \rangle = n \lvert p_1, p_2, ...p_n \rangle$.

Can anyone show this explicitly? I have no idea what the action of $a^{\dagger}_pa_p$ is on a multi-particle state such as $\lvert p_1, p_2, ...p_n \rangle$.

Answer 1

It all boils down to the fact that $[a_p,a_q^\dagger] = \delta_{p,q}1$. Consider as an example $|2_p\rangle = a_p^\dagger a_p^\dagger|0\rangle$. Then the operator $a_p^\dagger a_p$ on $|2\rangle$ gives $$\begin{align} a_p^\dagger a_p|2_p\rangle &= a_p^\dagger a_p a_p^\dagger a_p^\dagger|0\rangle\\ &=a_p^\dagger[a_p,a_p^\dagger] a_p^\dagger|0\rangle + a_p^\dagger a_p^\dagger a_p a_p^\dagger|0\rangle\\ &= |2_p\rangle + a_p^\dagger a_p^\dagger a_p a_p^\dagger|0\rangle\\ &= |2_p\rangle + a_p^\dagger a_p^\dagger [a_p, a_p^\dagger]|0\rangle +a_p^\dagger a_p^\dagger a_p^\dagger a_p|0\rangle\\ &= |2_p\rangle + |2_p\rangle + 0\\ &= 2|2_p\rangle \end{align}$$ If $q\neq p$, then $a_p^\dagger a_p$ commutes with $a_q^\dagger$ and therefore it goes straight to $|0\rangle$ in the expression of $|1_q\rangle = a_q^\dagger|0\rangle$, so it gives 0, because there are no particle of momentum $p$ in $|1_q\rangle$.

Answer 2

The operator $a_p$ is an annihilator: on a state with momentum $p$ it gives the vacuum, $$ a_p \left| p\right> = \left|0\right>,$$ but on any other state it gives zero. (N.B. the vacuum state is quite different from the number zero!)

The operator $a_p^\dagger$ is a creator: on a state without a particle with momentum $p$, such as the vacuum, it creates one: $$ a_p^\dagger \left| 0 \right> = \left| p \right>.$$ What the creation operator does if there's already a particle with $p$ depends on what type of particles you're talking about. For fermions you're not allowed to have two particles in the same state, so $a_p^\dagger\left|p\right>=0$ (though you might properly need to specify some other quantum numbers, too). For bosons you may have multiple particles in the same state e.g. in lasers or condensates, so $a_p^\dagger \left|p\right>$ would not vanish.

This means that the product operator $a_p^\dagger a_p$ that it vanishes when acting on states which contain no particles with momentum $p$, $$ a_p^\dagger a_p \left| q \right> = 0, $$ but has eigenvalue 1 on states which contain particles with momentum $p$, $$ a_p^\dagger a_p \left| p \right> = \left| p \right>. $$ That's essentially what a Dirac delta function does. Your number operator, $$ N = \int \frac{d^3 p}{(2\pi)^3} \, a^{\dagger}_pa_p,$$ integrates over all momenta; it will leave the state unchanged, but pull out one term for every particle in the state.

You can be more explicit using commutators, as shown by Phoenix87.