Projected bilocal variation

Scope

Work in a specified finite containing spatial region $Q$, with one field $\Psi(t,x)$, fixed measure, and fixed channel projectors $P_a(x)$. The labels $x,y$ are two positions in the same physical space, not additional physical dimensions. Channel indices do not replace the required nested spatial embeddings; those maps must be supplied when applying this local model.

Let $\rho_a=\bar\Psi P_a\Psi$ be real, Grassmann-even scalar bilinears. Use projectors compatible with the chosen adjoint; in a Lorentzian Dirac convention this means $P_a^\dagger\gamma^0=\gamma^0P_a$. No positivity of $\bar\Psi P_a\Psi$ is assumed. Vary $\Psi$ and $\bar\Psi$ independently.

Variation

Absorb coupling coefficients and coordinate identifications into a real kernel with exchange symmetry

$$K_{ab}(x,y)=K_{ba}(y,x).$$

Write $\mathcal D=i\Gamma^A\nabla_A$ for the kinetic Dirac operator, where $A$ ranges over the one time coordinate and the spatial directions in $Q$, and $\nabla_A$ includes the chosen spin connection. Choose the interaction sign explicitly:

$$S=\int dt\int_Q\bar\Psi\,\mathcal D\Psi\,d\mu -\frac12\int dt\sum_{a,b}\iint_{Q\times Q} \rho_a(x)K_{ab}(x,y)\rho_b(y)\,d\mu_xd\mu_y.$$

At fixed projectors, kernel, and metric,

$$U_a(x)=\sum_b\int_Q K_{ab}(x,y)\rho_b(y)\,d\mu_y, \qquad \frac{\delta S_{\rm int}}{\delta\bar\Psi(x)} =-\sum_a U_a(x)P_a(x)\Psi(x).$$

To see the cancellation of the half, vary each density in the double integral and exchange $(a,x)$ with $(b,y)$ in the second contribution. The two contributions are identical. Thus the field equation is

$$\boxed{\left(\mathcal D-\sum_a U_aP_a\right)\Psi=0.}$$

With a plus interaction in the action the corresponding sign is plus. The projector remains attached to the varied field. The kernel occurs once inside $U_a$. If the kernel is not exchange-symmetric, only its symmetrized part contributes to the scalar-density functional.

At density level,

$$\frac{\delta U_a(x)}{\delta\rho_b(y)}=K_{ab}(x,y).$$

This is the reciprocal response. It does not require equal response coefficients after changing field normalizations or identifying different coordinate spaces.

Remaining physical requirements

The kinetic symbol includes the appropriate geometric connection; replacing covariant derivatives by partial derivatives on a curved sector needs a justified local approximation. Dynamical projectors, metric-dependent measures, and kernels produce additional variations, including contributions to stress-energy. They are held fixed only for the variation above.

For a local phase with generator $\hat Q$, the displayed scalar densities are invariant if $[P_a,\hat Q]=0$ and the adjoint transforms compatibly. More general charged bilocal factors need appropriate transport or invariant contractions. Gauge covariance is not implied by the double-integral notation.

The equation is a Dirac mean-field equation. Its reduction to a positive scalar Schrödinger density, the sign of self-binding, a causal completion of the instantaneous spatial kernel, and the physical mass operator remain separate obligations. In particular, the flat scalar oscillator closure cannot be obtained merely by relabelling $\bar\Psi\Psi$ as $|\Psi|^2$.

Verification and related

interaction_operator_checks.py compares the analytic variation with finite differences using complex fields, nontrivial projectors, unequal integration weights, and nonsymmetric kernels.

• Interaction operator tools — summarizes this action-level addition.
• Cross-sector moment kernel — supplies an explicit coordinate identification with its covariance law.
• Self-consistent functional — the separate positive scalar closure and its normalization assumptions.