I have a Hamiltonian of the following general form, $$H=\int \frac{d^{3} k}{(2 \pi)^{3}}\left[A_{\vec{k}}(t) a_{\vec{k}}^{\dagger} a_{\vec{k}}+B_{\vec{k}}(t) b_{\vec{k}}^{\dagger} b_{\vec{k}}+C_{\vec{k}}(t) a_{\vec{k}} b_{-\vec{k}}+D_{\vec{k}}(t) b_{-\vec{k}}^{\dagger} a_{\vec{k}}^{\dagger}\right]$$ where $A$, $B$, $C$ and $D$ are some function of time and the bosonic creation and annihilation operators are not time dependent. Now how can I build a number operator from this? What are the properties I should check the number operator follows? For a Hamiltonian as above the number of particles should change with time, how do I see that from the constructed number operator?


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