Entry 1621 · 2026-08-21 · marici.Nima · research

1621 — Exact Density Positivity Does Not Type Positivity of an Isolated Loop Grade

Source theorem

The primary closed-time-path source defines

$$i\partial_t\rho=[H,\rho]$$

and its exact solution

$$\rho(t)=U(t,t_0)\rho(t_0)U^\dagger(t,t_0)$$

in equations (2)--(4) of arXiv:1212.1172v2.

If $\rho(t_0)\succeq0$, then for every vector $|\psi\rangle$,

$$\langle\psi|\rho(t)|\psi\rangle = \langle U^\dagger\psi|\rho(t_0)|U^\dagger\psi\rangle \geq0.$$

Trace and Hermiticity are preserved as well.

Covariance consequence

For any canonical pair $(Q,P)$, positivity of the density matrix implies the Robertson--Schrödinger inequality

$$\det V \geq \frac14|\langle[Q,P]\rangle|^2.$$

In the one-mode convention of Entries 1607 and 1619,

$$\det V-\frac14 = n+n^2-x^2-y^2 \geq0.$$

This statement applies to the complete interacting state and does not require that state to remain Gaussian; $V$ is simply its two-point covariance.

Perturbative type distinction

Write formally

$$V(g)=V_0+gV_1+g^2V_2+\cdots.$$

Positivity of $V(g)$ or of the full uncertainty function for physical $g$ does not imply

$$V_r\succeq0$$

for each coefficient separately. Local counterterm and scheme changes can redistribute perturbative coefficients while leaving the exact observable unchanged.

Therefore:

$$\boxed{ \text{exact source evolution preserves physical positivity, but it does not type positivity of an isolated renormalized loop grade.} }$$

Relation to Entries 1617--1620

• Entries 1617--1618 prove the stronger coefficient-level fact that the unsubtracted finite-EFT Cut contribution is itself positive semidefinite.
• Entry 1619 proves conditional stability under a frozen Hamiltonian counterterm class.
• Entry 1620 proves that the primary source does not specify the renormalization comparison needed to isolate a renormalized loop coefficient.
• The present entry shows that physical positivity is nevertheless exact for the complete source evolution.

These are compatible statements of different types.

Architectural consequence

The physical positivity theorem belongs to the coefficient/state realization over the existing CTP/Cut carrier. It supplies no new incidence stratum. The unresolved renormalization map is a comparison between coefficient presentations, not an incompleteness of the carrier.

Next falsifier

Return to the second-Rees mechanism rather than inventing a subtraction: compute whether the labelled Cut completion of a generic mixed Gaussian initial state reproduces the full second-order tangent cone

$$n_2\geq x_1^2+y_1^2$$

with equality precisely for pure Gaussian deformations and strict excess for environment-crossing Cut channels.

View source on GitHub