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In a one-dimensional identical-particle model, a contact interaction has two-body potential $U(x,x^{\prime })=g\delta (x-x^{\prime })$, using the Dirac delta distribution. For bosons, its second quantization is the displayed normally ordered Hamiltonian operator. Projection onto orthonormal modes gives $H_{\mathrm{int}}=(g/2){\sum }_{ijkl}{I}_{ijkl}{a}_i^{†}{a}_j^{†}{a}_k{a}_l$, where $I_{ijkl}=\int {\phi }_i^{\ast }{\phi }_j^{\ast }{\phi }_k{\phi }_{l}dx$. The overlap integrals determine which mode-changing processes survive. A finite-mode model describes that projection; discarded modes can still matter to the full physical system.

In second quantization, expand the annihilating quantized field in a complete orthonormal basis of one-particle states, $\hat{\psi }(x)={\sum }_{\alpha }{\phi }_{\alpha }(x)a_{\alpha }$. At equal times, the canonical commutation relation for bosons and the canonical anticommutation relations for fermions are, respectively,andHere $\delta$ is a Dirac delta distribution. Multiplying by the basis wave functions and integrating gives $[a_{\alpha },a_{\beta }^{†}]={\delta }_{\alpha \beta }$ for bosons, or $\{a_{\alpha },a_{\beta }^{†}\}={\delta }_{\alpha \beta }$ for fermions; the two-annihilator and two-creator brackets vanish in the corresponding convention.

The mode number operator and total particle number operator areThe bosonic occupation numbers are $0,1,2,\dots$. For fermions, $a_{\alpha }^2=0$ and the canonical anticommutation relations imply $n_{\alpha }^2=n_{\alpha }$, so the fermionic occupation numbers are only $0$ and $1$, as required by the Pauli exclusion principle.

Keeping only two modes later is a projection: the projected field bracket is ${\sum }_{j=1}^2{\phi }_j(x){\phi }_j^{\ast }(x^{\prime })$, rather than the full Dirac delta distribution.