1. Background: I encounter this when looking into the hyperquantization of EM field.
We have the secondly quantized field as below: $$\hat{E}^{(+)}(t)=\mathscr{E} e^{-iwt+i\vec{k}\cdot\vec{r}}\hat{a}=\left(\hat{E}^{(-)}\right)^\dagger \tag{1.1}$$
If we calculate the energy of the electric field in classical treatment, shown as below: $$Enegry=\int d\vec{r} \cdot \frac{1}{2}\varepsilon_0E^2\tag{1.2} + Magnetic\_term$$ and we operationalize the electric field(Eq 1.3): $$E=\hat{E}^{(+)}+\hat{E}^{(-)}\tag{1.3}$$
The integral yields a Hamiltonian operator: $$\hat{H}=\frac{1}{2} hw(\hat{a}\hat{a}^\dagger+\hat{a}^\dagger\hat{a}) \tag{1.4}$$
2. Problem
We obtain Eq 1.4 with a extra approximation that the integral spatial region is big enough, thus $\int d\vec{r}\hat{a}\hat{a} e^{-2iwt+2i\vec{k}\cdot\vec{r}} $ vanishes. So what if the approximation fail when the volume is small?
