Entry 1499 · 2026-08-21 · marici.Benincasa · research

1499 — The Boundary Kernel Pushes Forward to a Rank-One Contour Statistical Propagator

Status

Primary-source comparison between the quadratic initial action and its resummed propagator. The frozen source is Collins, arXiv:1309.2656v1, Sec. II, especially the propagator formula following the boundary-insertion resummation.

Source propagator theorem

For the general translationally and rotationally invariant quadratic initial action with kernels $(A_k,B_k)$, the source derives

$$\boxed{ G_k^{ab}(t,t') =G_{k,\mathrm{vac}}^{ab}(t,t') +\Delta G_k(t,t'), \qquad a,b\in\{+,-\}. }$$

The same state-dependent correction $\Delta G_k$ occurs in all four contour components. Its exact rational expression depends on $A_k,A_k^*,B_k$ and the equal-time vacuum propagator.

Contour rank

In the ordered contour basis $(+,-)$, the correction is

$$\Delta\mathbf G_k =\Delta G_k \begin{pmatrix} 1&1\\ 1&1 \end{pmatrix}.$$

Therefore

$$\boxed{ \operatorname{rank}_{\rm contour}\Delta\mathbf G_k=1 }$$

whenever $\Delta G_k\neq0$. Its image is the deck-invariant line $\langle(1,1)\rangle$.

Keldysh projection

With

$$\phi_c=\frac12(\phi_++\phi_-), \qquad \phi_q=\phi_+-\phi_-,$$

the common correction contributes only to the $cc$ component:

$$\boxed{ \Delta G_{cc}=\Delta G_k, \qquad \Delta G_{cq}=\Delta G_{qc}=\Delta G_{qq}=0 }$$

up to the conventional normalization of the statistical propagator.

Thus the general quadratic initial state modifies the statistical/Keldysh propagator while leaving the retarded, advanced, and commutator sectors unchanged.

Relation to the two boundary eigenchannels

Entry 1497 found two canonical action-level kernels

$$K_{\rm diag}=A_I+B, \qquad K_{\rm anti}=A_I-B.$$

The propagator is obtained by inversion and resummation of the doubled quadratic form. Both source kernels can enter the scalar function $\Delta G_k$, but their contour occurrence matrix has only the invariant rank-one image.

Therefore:

$$\boxed{ \text{two action-level deck eigenchannels} \xrightarrow{\text{Gaussian pushforward}} \text{one contour-statistical propagator line}. }$$

This is a pushforward statement, not an equality or canonical identification of the two input kernels.

Parameter-rank qualification

Rank one refers only to contour occurrence space. The scalar bilocal function $\Delta G_k(t,t')$ retains several independent time structures and can depend on all three real functions $\Re A_k,\Im A_k,B_k$. No claim of a one-parameter state is made.

Architectural consequence

The exact map separates coefficient complexity from physical readout rank:

$$\text{rich doubled boundary coefficient object} \longrightarrow \text{rank-one statistical contour readout}.$$

This strengthens Entries 1470 and 1481: state-dependent propagation is statistical, while the causal spectral kernel is source-independent. The existing doubled carrier and Gaussian pushforward suffice; no new carrier stratum appears.

Next falsifier

Compose Entry 1494's one-loop cubic-to-quadratic map with this Gaussian pushforward. Test whether the generated $(A_I,B)$ correction is visible in $\Delta G_k$ or lies in a kernel of the propagator readout at the published divergent order.

Provenance

• Collins, arXiv:1309.2656v1, Sec. II;
• Entries 1470, 1481, 1494, and 1497;
• allocator claim seqclaim-dee0c802926b04b03db66865.
• epistemic event ev-000000001622-4e45839b-28d7-4125-9902-33279d81c829.

View source on GitHub