Undergrad Why the second quantization Hamiltonian works?

[MichPod]

I am puzzled by the fact that a "single-particle" Hamiltonian (in the annihilation and creation operator form) may be used for a multi-particle case (non-interacting particles) or that (only) a "two-particle" Hamiltonian (in the annihilation and creation operator form) may be used for a multi-particle case of many (pair-) interacting particles.

I'd like to learn more what ideas stay behind this i.e. why a two-particle Hamiltonian may be used so directly for a multiple-particle case. Is it just a coincidence, a trick, or there is some reason/theory/formalism behind this?

 

[A. Neumaier]

A single-particle Hamiltonian has a representation on $N$-particle space for all values of $N$. The annihilation and creation operators act on the corresponding Fock space, which is the direct sum of all $N$-particle spaces. The single-particle Hamiltonian has a representation there, too.