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$$[a(\vec{p}),a^{\dagger}(\vec{q})]=(2 \pi)^3 \delta^{(3)}(\vec{p}-\vec{q}).$$

Thus ##N(\vec{p})=a^{\dagger}(\vec{p}) a(\vec{p})## are momentum-space number densities, i.e., the dimension for your annihilation and creation operators is, of course, correct since ##N(\vec{p})## must be of mass dimension -3.

Then you directly have for the total energy, e.g.,

$$H=\int_{\mathbb{R}^3} \frac{\mathrm{d}^3 \vec{p}}{(2 \pi)^3} \sqrt{m^2+\vec{p}^2} N(\vec{p}).$$

Note that there are many different conventions in the literature concerning the norm of the annihilation and creation operators. The unanimous commutation relation is the equal-time commutation relation of the field operators,

$$[\phi(t,\vec{x}),\dot{\phi}(t,\vec{x})]=\mathrm{i} \delta^{(3)}(\vec{x}-\vec{p}).$$

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