quantum field theory - Annihilation and Creation Operators in QFT

I have following question about creation and annihilation operators in QFT: The Klein-Gordon field is introduced as continuous interference of plane waves $\mathrm{e}^{i(\omega_kt-\vec{k}\cdot\vec{x})}$ with positive energy (resp $\mathrm{e}^{-i(\omega_kt-\vec{k}\cdot\vec{x})}$ with negative energy):

$$\varphi(\vec{x},t) = \int\frac{d^Dk}{\sqrt{(2\pi)^D2\omega_k}}\left[a(\vec{k})\mathrm{e}^{-i(\omega_kt-\vec{k}\cdot\vec{x})} + b^\dagger(\vec{k})\mathrm{e}^{i(\omega_kt-\vec{k}\cdot\vec{x})}\right].$$

Then, when we quantizing the coefficients $b_{\mathbf{p}}^{\dagger}$ and $a_{\mathbf{p}}$ we get

$$\varphi(\vec{x},t) = \int\frac{d^Dk}{\sqrt{(2\pi)^D2\omega_k}}\left[\hat{a}(\vec{k})\mathrm{e}^{-i(\omega_kt-\vec{k}\cdot\vec{x})} + \hat{b}^\dagger(\vec{k})\mathrm{e}^{i(\omega_kt-\vec{k}\cdot\vec{x})}\right].$$

with the annihilation operator $\hat{b}_{\mathbf{p}}^{\dagger}$ and creation operator $\hat{a}_{\mathbf{p}}$ ???

Previously I asked a question concerning distinguishing the annihilation and creation operators in this expression here: Creation and Annihilation Operators in QFT

and got indeed good answers.

But the point of this question is the following: Earlier I asked the same question my prof and he gave seemingly a more easy/intuitive answer that $\hat{b}_{\mathbf{p}}^{\dagger}$ must correspond to the exponential with positive energy:

He used the argument that $\hat{b}_{\mathbf{p}}^{\dagger}$ must be annihilation since the inititial state must have positive energy.

Can anybody decrypt how to interpret this argument? I don't understand exactly this line of thought. What is here the initial state? The exponential with positive energy?

Does he mean an evaluation argument like $= e^{\text{blabla}}$ as neccessary condition?