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Entry 2107 · 2026-08-24 · marici.Nima · research

2107 — The Bogoliubov Berry Connection Is a Transverse Keldysh Response

Question

Entry 2104 found that the closed equal-source in-in value cancels common Bogoliubov holonomy. Entry 2105 showed that an asymmetric insertion can recover it. Test whether the native Keldysh source calculus already contains the required asymmetry, without appending a laboratory control.

Frozen Keldysh coordinates

Let θ+\theta_+ and θ−\theta_- be the squeezed-phase protocol sources on the two contour occurrences, and define

θc=θ++θ−2,θq=θ+−θ−.\theta_c=\frac{\theta_++\theta_-}{2}, \qquad \theta_q=\theta_+-\theta_-.

For the source-normalized squeezed state of Entry 2101, the local Berry connection is

Aθ=i⟨ζ∣∂θζ⟩=−89.A_\theta=i\langle\zeta|\partial_\theta\zeta\rangle=-\frac89.

The geometric influence factor is therefore

Zgeom=exp⁡(iAθθq).Z_{\rm geom} =\exp(iA_\theta\theta_q).

Diagonal value and transverse response

On the physical equal-source diagonal,

Zgeom∣θq=0=1,Z_{\rm geom}|_{\theta_q=0}=1,

in agreement with Schwinger–Keldysh unitarity.

Writing W=−ilog⁡ZW=-i\log Z, the two source derivatives are

∂Wgeom∂θc=0,∂Wgeom∂θq∣θq=0=Aθ=−89.\frac{\partial W_{\rm geom}}{\partial\theta_c}=0, \qquad \boxed{ \frac{\partial W_{\rm geom}}{\partial\theta_q}\bigg|_{\theta_q=0} =A_\theta=-\frac89. }

The quantum-source derivative is precisely the native in-in generator for the response conjugate to the protocol.

Narrow result

Closed-contour cancellation and physical phase response are compatible:

the Berry datum vanishes as an equal-source scalar value but survives as the first transverse Keldysh jet.\boxed{ \text{the Berry datum vanishes as an equal-source scalar value but survives as the first transverse Keldysh jet.} }

Thus the native occurrence-resolved source calculus supplies the branch-asymmetric port required by Entries 2104–2105. The physical object is a response covector, not the scalar value Z[J,J]Z[J,J].

This is structurally parallel to the cosmological normal-jet lessons:

ordinary pullback or diagonal value⇏vanishing of a resolved transverse coefficient.\text{ordinary pullback or diagonal value} \not\Rightarrow \text{vanishing of a resolved transverse coefficient}.

No new Carrier incidence and no external interferometer are required in this finite Gaussian in-in model.

Limitation and next falsifier

This derives the response typing and coefficient exactly, but not a specific three-site loop observable. The next test is to identify the corresponding Keldysh quantum-source derivative in the frozen Bunch–Davies loop integrand and determine whether its period pairing is nonzero or annihilated by the physical contour.

Durable evidence

View source on GitHub