# Where this interpretation for the field modes comes from?

I'm reading the book "Modeling Black Hole Evaporation" by Alessandro Fabbri and Jose Navarro-Salas, and in section 3.3.2 they talk about wavepackets at $$\mathscr{I}^+$$.

It all starts like this: one considers solutions to the massless KG equation which vanish at the future horizon and have positive frequency $$\omega$$ at $$\mathscr{I}^+$$. This gives one orthonormal set $$\{p_\omega\}$$ with

$$p_\omega=\dfrac{1}{4\pi\sqrt{\omega}}\dfrac{e^{-i\omega u}}{r}.$$

This gives rise to the operators $$a_{\text{out}}(\omega),a_{\text{out}}^\dagger(\omega)$$ which create and annihilate quanta on the mode $$p_\omega$$, and to the associated number operators $$N^{\text{out}}(\omega)=a_{\text{out}}^\dagger(\omega)a_{\text{out}}(\omega)$$.

The authors them say:

Let us clarify the meaning of the quantity $$\langle \text{in}|N^\text{out}(\omega)|\text{in}\rangle.$$ It gives the mean particle number detected at $$\mathscr{I}^+$$ with a definite frequency $$\omega$$. The use of this type of states, with a definite frequency, implies absolute uncertainty in time. Therefore $$\langle \text{in}|N^\text{out}(\omega)|\text{in}\rangle$$ provides, indeed, the number of particles with frequency $$\omega$$ emitted at any time $$u$$. But we are mainly interested in evaluating the mean particle number produced at late retarded times $$u\to +\infty$$, when, in a realistic situation, the black hole has settled down to a stationary configuration. To properly evaluate the late time particle production one has to replace the plane wave type modes, which are completely delocalized, by wave packets.

One can introduce a complete orthonormal set of wave packet modes at $$\mathscr{I}^+$$, with discrete quantum numbers, as follows:

$$p_{jn}=\dfrac{1}{\sqrt{\epsilon}}\int_{j\epsilon}^{(j+1)\epsilon} d\omega e^{2\pi i\omega n/\epsilon}p_\omega$$

with integers $$j\geq 0,n$$. These wave packets are peaked about $$u=2\pi n/\epsilon$$ with width $$2\pi/\epsilon$$. Taking $$\epsilon$$ small ensures that the modes are narrowly centered around $$\omega \approx j\epsilon$$. Therefore, the computation of $$\langle\text{in}|N^{\text{out}}_{jn}|\text{in}\rangle$$, associated with the wavepacket, has a clear physical interpretation. It gives the counts of a particle detector sensitive only to frequencies within $$\epsilon$$ of $$\omega_{j}$$ which is turned on for a time interval $$2\pi/\epsilon$$ at time $$u=2\pi n/\epsilon$$.

Now, why $$p_{jn}$$ can be interpreted so that the associated number operator's $$N_{jn}^{\text{out}}$$ mean value gives the count of a particle detector sensitive only to frequencies within $$\epsilon$$ of $$\omega_{j}$$ which is turned on for a time $$2\pi/\epsilon$$ at $$u = 2\pi n/\epsilon$$?

I do understand that if $$p_\omega$$ is a state of definite frequency so that the frequency operator acts as $$\Omega p_\omega = \omega p_\omega$$, then if $$\epsilon$$ is small, $$p_{jn}$$ has mean frequency close to $$j\omega$$ and the probabilities are equaly distributed in the interval.

I don't understand the reasoning about time of detection which I wrote in bold. Why does that interpretation holds?

Notice that the same thing could be stated in flat spacetime replacing $$p_\omega$$ by the plane wave of positive frequency $$\omega$$ and replacing $$u$$ by the inertial time $$t$$.