# Positive and negative energy modes of scalar field

If we have the normal KG scalar field expansion:

$$\hat{\phi}(x^{\mu}) = \int \frac{d^{3}p}{(2\pi)^{3}\omega(\mathbf{p})} \big( \hat{a}(p)e^{-ip_{\mu}x^{\mu}}+\hat{a}^{\dagger}(p)e^{ip_{\mu}x^{\mu}} \big)$$ With $\omega(\mathbf{p}) = \sqrt{|\mathbf{p}^{2}|+m^{2}}$

Then why do we associate positive energy states with $\hat{a}(p)e^{-ip_{\mu}x^{\mu}}$ and negative energy states with $\dagger{a}(p)e^{ip_{\mu}x^{\mu}}$?

For some reason I thought this was the wrong way round (just because of the sign of exponential, the fact $p_{0} = \omega(\mathbf{p}) = E_{\mathbf{p}}$, and using metric sign $(+,-,-,-)$?

Actually upon quantisation of the free Klein Gordon field, all modes are evaluated on the positive energy solution $\omega(\mathbf p) = + p_0$. The terms 'positive energy solution' and 'negative energy solution' are artifacts of history and should not be taken literally. They come from noting that, by interpreting $\phi$ as a wave function, it satisfies the Schrodinger equation $$\mathrm i \hbar \frac{\partial}{\partial t} \phi = E \phi.$$
Let $\phi_{\pm} = \exp(\pm \mathrm i/ \hbar p_0t)$. Then the above implies $$\mathrm i \hbar \frac{\partial}{\partial t} \phi_{\pm} = \mp p_0 \phi_{\pm},$$ i.e $\phi_{\pm}$ gives either a positive or negative energy eigenvalue.
It's still common in the literature to refer to the plane wave solutions $\phi_{\pm}$ as positive and negative frequency modes, language originating from this historical development. It might be useful for you to check out the Feynman-Stueckelberg interpretation and how the $\hat a$ and $\hat a^{\dagger}$ become accredited as annihilation and creation operators for particle/antiparticle but, importantly, both carrying positive energy in this construction.