Nonrelativistic Field Operators in Code

A nonrelativistic field becomes a numerical operator after choosing modes and truncating their occupations or total particle number. Mode projection converts spatial integrals into coefficient tensors; occupation-basis algebra then converts those tensors into a many-body matrix. Both stages require separate convergence checks.

Helpful background. Bosonic bases and fermionic bases supply ladder actions. Nonrelativistic field theory explains the continuum interpretation and physical matching of contact couplings.

Projection onto orthonormal modes

For orthonormal modes $\phi_\alpha(x)$, approximate

$$\hat\Psi(x)=\sum_{\alpha=0}^{M-1}\phi_\alpha(x)a_\alpha.$$

With exact mode ladder algebra, the finite-mode equal-time field (anti)commutator is the projector kernel $K_M(x,x')=\sum_{\alpha<M}\phi_\alpha(x)\phi_\alpha(x')^*$, not the continuum delta distribution. For bosons an additional occupation cutoff modifies even this identity through the top-level commutator defect. A comparison to a delta function at a single point is therefore not an appropriate finite-mode algebra test.

A self-adjoint one-body operator $h$ and a real symmetric, spin-independent pair potential $V(x,x')=V(x',x)$ give

$$\begin{aligned} H&=\sum_{\alpha\beta}h_{\alpha\beta}a_\alpha^\dagger a_\beta +\frac12\sum_{\alpha\beta\gamma\delta} V_{\alpha\beta\gamma\delta} a_\alpha^\dagger a_\beta^\dagger a_\gamma a_\delta,\\ h_{\alpha\beta}&=\int dx\,\phi_\alpha^*(x)h\phi_\beta(x),\\ V_{\alpha\beta\gamma\delta} &=\int dx\,dx'\,\phi_\alpha^*(x)\phi_\beta^*(x') V(x,x')\phi_\gamma(x')\phi_\delta(x). \end{aligned}$$

The operator order matches the integral definition. Other index conventions are valid, but an interaction tensor from another convention must be permuted before use.

For fermions, apply operators right to left with parity signs. If tensor elements have already been antisymmetrized, the prefactor and summation convention change; do not antisymmetrize a second time.

Contact interactions and lattice normalization

For bosons with a formal contact interaction $g\delta(x-x')$,

$$V_{\alpha\beta\gamma\delta} =g\int dx\,\phi_\alpha^*\phi_\beta^*\phi_\gamma\phi_\delta.$$

In a real-space cell basis with cell volume $\Delta V$, define $a_j\approx\sqrt{\Delta V}\,\Psi(x_j)$. Then the discretized contact term is

$$H_{\rm int}\approx\frac{g}{2\Delta V}\sum_j n_j(n_j-1).$$

The factor $1/\Delta V$ follows from four field factors and one integration weight. Omitting it makes the effective interaction vanish or change incorrectly under grid refinement.

A contact coupling in two or three dimensions generally requires a cutoff-dependent prescription tied to a physical scattering observable. Holding a bare lattice coupling fixed while taking a continuum limit is not automatically a renormalized calculation. The field representation itself does not solve that physical matching problem.

Single-mode benchmark

For one normalized mode and bosonic contact interaction, let $U=g\int dx\,|\phi(x)|^4$. The projected Hamiltonian is

$$H=\epsilon n+\frac U2n(n-1).$$

The two-particle interaction energy is $U$, not $2U$ or $U/2$. This tests the interaction prefactor and ladder factors. For identical spinless fermions in a single mode, occupations are only zero and one and this local contact term vanishes.

For multiple modes, compare matrix elements obtained from occupation rules with direct low-particle-number wavefunction integrals. This provides an independent check of index permutations.

Momentum conservation as an integral test

For one-dimensional periodic plane waves $\phi_m(x)=e^{2\pi imx/L}/\sqrt L$ with $-K\leq m\leq K$, a contact interaction gives

$$V_{\alpha\beta\gamma\delta} =\frac gL\,\delta_{m_\alpha+m_\beta,m_\gamma+m_\delta}.$$

The Kronecker delta follows from the integral of the net Fourier phase; the coefficient $g/L$ follows from four factors $L^{-1/2}$ and one integral of length $L$. This tests both normalization and momentum conservation, including matrix elements that must vanish.

Uniform quadrature with $Q$ points instead identifies Fourier indices modulo $Q$. Since the largest possible mismatch is $4K$, choosing $Q>4K$ integrates every such four-mode product exactly in this example. At $Q=4K$ the extreme mismatch aliases to zero: a forbidden scattering term acquires coefficient $g/L$. Increasing occupation cutoffs or tightening a many-body eigensolver cannot remove that integral error. The bound here is for these finite plane-wave products; other modes and interactions need their own quadrature study.

Convergence and diagnostics

Check numerical orthonormality of modes under the quadrature used for integrals. A nonorthogonal basis requires an overlap metric and a consistent transformation; ordinary canonical ladder algebra assumes orthonormal modes.

Increase mode count separately from the occupation cutoff and quadrature order. Monitor particle-number conservation, Hermiticity, expected exchange symmetries of the interaction tensor, and convergence of target energies or correlations.

Projecting first and multiplying projected operators need not equal projecting their full product. Missing intermediate high-energy modes can induce effective interactions. State whether the calculation uses a bare projected Hamiltonian or a matched effective one.

Exercise

For a constant normalized mode in a box of volume $\mathcal V$, find $U$ and the interaction energy of $N$ bosons.

Solution

$\phi=1/\sqrt{\mathcal V}$ gives $U=g/\mathcal V$. Therefore $E_{\rm int}=gN(N-1)/(2\mathcal V)$. The $N(N-1)$ factor counts distinct particle pairs and provides a normalization benchmark for mode integrals and operator assembly.

References

• Pethick, C. J., and Smith, H., Bose–Einstein Condensation in Dilute Gases, 2nd ed., Cambridge University Press (2008), doi:10.1017/CBO9780511802850 — field operators and contact interactions.
• Weinberg, P., and Bukov, M., “QuSpin. Part II: bosons, fermions and higher spins,” SciPost Physics 7, 020 (2019), arXiv:1804.06782 — finite occupation representations.