I'm going through an introduction to many-body theory and I am getting tripped up on the formalism. I understand quantities such as $\hat {N} = \sum_{i}\hat{n}_{i}=\sum_{i}\hat{a}_{i}^{\dagger}\hat{a}_{i}=\int d^{3}x\psi^{\dagger}(x)\psi(x)$ but struggling with things interpreting things like the kinetic energy of the system
Specifically, how is it that one goes from the creation/annihilation formalism to the field operators? If you have a general many-body Hamiltonian, I can't really see the transformation (nor do I have a good intuition for it) between the creation and annihilation operators and the field operators formalism for the Hamiltonian. Why do we have the two-particle interaction "sandwiched" between the field operators, but the annihilation/creation operators do not follow the same pattern?
I am aware of basic quantum mechanics, commutation rules, as well as the Fourier transform. I need help developing an intuition for writing down a field operator Hamiltonian. When I read the field operator Hamiltonian, the story that I get is: There are some field operators that create and annihilate, and integrating over them with a energy density yields a total energy term.
But I get lost in the details. For instance, although it has been simplified by IBP above, the kinetic energy term acts on the annihilation operator before the creation operator acts on it. What is the meaning of the motif $H=\int d^{3}x\psi^{\dagger}(\mathbf{x})\hat{h}\psi(\mathbf{x})$?
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