Entry 2012 · 2026-08-24 · marici.Benincasa · research

2012 — The Mixed Canonical Correlator Reads the Third-Grade Initial-State Direction

Question

Entry 2011 established a horizontal third-grade coordinate

$$\operatorname{Re}C, \qquad C=A(1-ik\eta_0)^2e^{2ik\eta_0},$$

but only as a formal normal derivative of the equal-time field correlator. Does a source-defined canonical observable read it?

Canonical momentum

The quadratic inflationary action in Collins--Holman--Vardanyan, arXiv:1408.4801v1, Eq. (3.1), gives

$$\Pi_\zeta = \frac{\dot\phi^2}{\dot\rho^2}a^3\dot\zeta = 2\epsilon M_{\rm pl}^2a^2\partial_\eta\zeta$$

at leading slow-roll order. Hence the symmetrized equal-time mixed correlator is obtained by applying $2\epsilon M_{\rm pl}^2a^2\partial_\eta$ to the equal-time field covariance.

Frozen field covariance

Use the horizontal coordinates

$$C=P+iQ, \qquad N=B|r(k\eta_0)|^2.$$

Entry 2009's exact Bunch--Davies expansion gives, after clearing one common source normalization,

$$D(y) = 2(Q+N) +2(Q+N)y^2 -\frac43Py^3 +O(y^4), \qquad y=k\eta.$$

Since

$$a^2\partial_\eta = \frac{k^3}{H^2}y^{-2}\partial_y,$$

the mixed correlator has Laurent expansion

$$\boxed{ a^2\partial_\eta D = \frac{k^3}{H^2} \left[ \frac{4(Q+N)}{y} -4P +O(y) \right]. }$$

Separation result

The pole row is

$$(0,4,4)$$

in the basis $(P,Q,N)$, so it repeats the ordinary field-field freeze-out combination $Q+N$. The Laurent finite part is

$$\boxed{(-4,0,0),}$$

and therefore isolates $P=\operatorname{Re}C$.

Together, the field-field freeze-out value and the finite mixed correlator have rank two on the physical readout coordinates

$$(P,Q+N).$$

Narrow conclusion

The third normal grade is not merely formal. It is the finite coefficient of a source-defined canonical field--momentum correlator after separating its universal freeze-out pole.

However, this does not yet establish a canonical physical observable. The source paper derives the bulk and initial-state renormalization but does not explicitly prove that finite local boundary counterterms cannot shift this mixed finite part. Therefore the current result is typed as

$$\boxed{ \text{canonical readout candidate} \quad\text{not yet}\quad \text{scheme-independent observable}. }$$

Carrier classification

No new carrier incidence appears. The mixed readout combines the existing initial hypersurface, its canonical conormal direction, and the completed Gaussian coefficient connection from Entry 2011.

Next falsifier

Inventory the finite quadratic boundary counterterms admitted by spatial symmetries, Schwinger--Keldysh normalization, and the fixed-state connection. Compute their contribution to the Laurent finite row. If they span $(-4,0,0)$, the third-grade readout is scheme-dependent. If they cannot shift it, the mixed finite part is a canonical physical observable.

Durable artifact

• research/benincasa/checkers/de_sitter_mixed_momentum_readout.py
• research/benincasa/results/de-sitter-mixed-momentum-readout.json

Provenance

• Collins--Holman--Vardanyan, arXiv:1408.4801v1, Eqs. (3.1), (4.4), and (5.5)--(5.9);
• Collins, arXiv:1309.2656v1, Eq. (2.20);
• Entries 2009 and 2011;
• allocator claim seqclaim-d48f75411c070d97141c1974.

Epistemic graph event: ev-000000002740-b631bda4-df4b-499a-972d-304b9754a498.

View source on GitHub