# Many-body states that do not belong to Fock space?

This question may be related with other phys. stackexchange questions: (q1) and (q2).

Fock space is a direct sum of antisymmetrized tensor products of single-particle Hilbert space. In other words, an element of Fock space is a linear combination of Slater determinants with various number of particles.

However, there exists an element of, say, $N$-particle Hilbert space that cannot be expressed by a linear combination of the determinants unless the element consists of spin-orbitals.

According to (q1), Fock space is a special case of many particle Hilbert space, and elements of Fock space describe states in free quantum field theories. I agree with this statement, since a many-body wavefunction that can be expressed by Slater determinants (alternatively, spin-orbitals) implies that each particle has its own wavefunction and behaves independently, crudely speaking.

However, we can express the Coulomb repulsion (two-body interaction) in Fock space, which consists of two creation and two annihilation operators. Using position eigenstates in Fock space (e.g. particles at lattice points), we can even diagonalize it (though the diagonalization would need tremendous amount of calculation). Therefore, Fock space does not have any problem to describe the Coulomb interaction between two particles.

Then, what does the word "free" mean? Fock space would be sufficient to describe $N$-particles in a system with $M$-body interaction, which is far from "free" system. For instance, Fock space formalism has effectively explained superconductivity, antiferromagnetism, etc. induced by strong correlations between electrons...

Well, I am sure that some interaction may require more than just spin-orbitals. I want to know examples of such interaction.