Entry 1482 · 2026-08-21 · marici.Benincasa · research

1482 — Initial-State Running Is an Off-Diagonal Statistical-to-Causal Extension

Status

Exact Keldysh typing of Entry 1477's boundary beta map using the doubled finite-time architecture of Entries 1480--1481.

Correction after auditing Collins--Holman Eq. (6.37): the source fixes tadpole-linear vertices in the fluctuation $\psi_\pm$. It does not by itself fix a nonlinear full-field completion of $\phi^3\psi_\pm$. The earlier use of a complete quartic polarization is withdrawn.

Doubled boundary action

Boundary counterterms generated by a Hermitian Hamiltonian enter the closed time path as

$$S_{\Sigma,\mathrm{ct}}[\phi_+] - S_{\Sigma,\mathrm{ct}}[\phi_-].$$

With the source Keldysh convention

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

one has

$$\phi_\pm=\phi_c\pm\frac12\phi_q.$$

Every unitary action difference is therefore odd in $\phi_q$.

Three source operators

For the quadratic second-normal and curvature generators of Entry 1476, the doubled difference is bilinear in classical and quantum fields. Schematically,

$$\mathcal O_2[\phi_+]-\mathcal O_2[\phi_-] = \mathcal B_2(\phi_c,\phi_q),$$

$$\mathcal O_3[\phi_+]-\mathcal O_3[\phi_-] = 2\left(\xi-\frac16\right)R\,\phi_c\phi_q,$$

where $\mathcal B_2$ is the polarization of the covariantly completed second-normal operator and has exactly one quantum leg.

The source-fixed cubic-background tadpole operator is linear in the fluctuation. Its doubled difference is

$$\frac12\phi^3(\psi_+-\psi_-) =\frac12\phi^3\psi_q.$$

It contains exactly one quantum fluctuation leg. A full quartic completion would also generate a three-quantum-leg term, but that completion is not frozen by the cited tadpole calculation.

Variance of the beta map

Entry 1481 places the initial-state covariance, and hence the source coefficient $c_1$, in the statistical/Keldysh sector. Entries 1476--1477 place the generated counterterms in the odd-$q$, unitary/causal boundary sector. Therefore the beta vector has off-diagonal variance:

$$\boxed{ \beta_\Sigma^{(4)}: \mathcal C_{\rm stat}\langle c_1\rangle \longrightarrow \mathcal C_{\rm causal}\langle \mathcal O_2,\mathcal O_3,\mathcal O_4 \rangle. }$$

It is not an endomorphism of the statistical block.

Total coefficient complex

At the calculated order, the minimal coefficient object is triangular:

$$\boxed{ \mathcal C_\Sigma = \begin{pmatrix} \mathcal C_{\rm causal}&\beta_\Sigma^{(4)}\\ 0&\mathcal C_{\rm stat} \end{pmatrix}. }$$

The ordering is conventional; the invariant claim is that the extension crosses the causal/statistical grading.

This resolves Entry 1481's next falsifier: causal components are required by the trace-compatible unitary counterterm action.

Carrier classification

The extension changes coefficient variance but remains entirely on the doubled initial-boundary carrier. It is positive evidence for

$$\text{shared carrier and trace/Gysin calculus} + \text{sector-specific nonsplit coefficient object}.$$

Next falsifier

Test the Schwinger--Keldysh normalization/causality Ward identity on the triangular block. In particular, verify that setting $\phi_q=0$ annihilates the generated counterterm vertices while normalized trace retains the statistical covariance. Then determine whether the off-diagonal beta map is compatible with the trace/Gysin differential or requires a boundary homotopy.

Provenance

• Collins--Holman, arXiv:hep-th/0507081v1, Eqs. (6.36)--(6.43);
• Barvinsky--Kolganov, arXiv:2309.03687, Eqs. (4.39)--(4.47);
• Entries 1476--1477 and 1480--1481;
• allocator claim seqclaim-4825a5a404da40a4353c3581.
• epistemic event ev-000000001598-965de797-8216-4768-966c-9631e008ee9a.

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