We start with a set of orthonormal single-particle states \( \{ \phi_i(x) \} \). (Note: this requirement, and others, can be relaxed, but leads to a more involved formalism.) Any orthonormal set will do.

To each single-particle state \( \phi_i(x) \) we associate a creation operator \( \hat{a}^\dagger_i \) and an annihilation operator \( \hat{a}_i \).

When acting on the vacuum state \( | 0 \rangle \), the creation operator \( \hat{a}^\dagger_i \) causes a particle to occupy the single-particle state \( \phi_i(x) \): $$ \phi_i(x) \rightarrow \hat{a}^\dagger_i |0 \rangle $$