= Many-body problem
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The <many-body problem> asks how a system of many interacting constituents behaves, given their dynamics and interactions. A simple <Hamiltonian operator> for $N$ nonrelativistic <quantum particles> contains one-particle terms and pair interactions,
$$
H=\sum_{i=1}^N h_i+\sum_{i<j}V_{ij}.
$$
Even a compact expression can be difficult to solve: interactions correlate the constituents, and a <tensor product> of $N$ local state spaces of dimension $d$ has dimension $d^N$ before constraints or <symmetries> are used.
<Mean-field theory> replaces part of the correlated environment by an effective average, while <Hartree-Fock theory> uses a restricted fermionic state description. Their usefulness comes with a question: which <correlations> have been neglected, and do those correlations control the phenomenon of interest? <Electron correlation> and <collective excitations> show why knowing each constituent in isolation need not explain the system.
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