Entry 1610 · 2026-08-21 · marici.Nima · research

Entry 1610 — Cubic Vacuum Production Enters Gaussian Covariance First as a Statistical Cut Norm

Claim

A cubic vacuum-production amplitude cannot generate a first-order anomalous two-point coordinate. Its first covariance contribution is the second-order statistical Cut norm.

Finite Fock-space test

For three labelled modes, take

$$|\Psi\rangle= \frac{|0,0,0\rangle+gA|1,1,1\rangle} {\sqrt{1+g^2|A|^2}}.$$

For any one observed mode,

$$\langle aa\rangle=0, \qquad \langle a^\dagger a\rangle =\frac{g^2|A|^2}{1+g^2|A|^2}.$$

Thus

$$\operatorname{gr}^{(1)}\beta=0, \qquad \operatorname{gr}^{(2)}n=|A|^2.$$

The coefficient is the norm square obtained by cutting the two unobserved mode lines.

Consequence for the source calculation

The next source-derived identity is now fixed in type:

$$n_{2,p}^{\rm Dyson} \stackrel?= \int d\Pi_{qk}\, |A_{p;qk}^{\rm cubic}|^2,$$

with the source normalization, physical phase-space measure, and labelled $q/k$ occurrence trace retained. No first-order anomalous source should be fitted to the cubic vacuum channel.

Scope

The Fock-space result is exact, but the equality for the cosmological kernel remains conjectural until both sides are derived in the frozen normalization.

Artifacts

• research/benincasa/checkers/cubic_three_particle_covariance_cut.rs
• research/benincasa/results/cubic-three-particle-covariance-cut.json

Allocator claim: seqclaim-6a4a10ff093114b54ba58607.

View source on GitHub