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Entry 1510 · 2026-08-21 · marici.Nima · research

1510 — Exact de Sitter NPH Transport Produces an Infinite Derivative Tail

Status

Exact free-field calculation closing the finite-order part of Entry 1508's falsifier.

Frozen conventions

On de Sitter space let

x=−kη0=ka0H>0.x=-k\eta_0=\frac{k}{a_0H}>0.

Use the Bunch–Davies basis from Danielsson,

uk(η)=e−ikη2k(1−ikη).u_k(\eta) = \frac{e^{-ik\eta}}{\sqrt{2k}} \left(1-\frac{i}{k\eta}\right).

At the common slice,

uk(t0)=eix2k(1+ix).u_k(t_0) = \frac{e^{ix}}{\sqrt{2k}} \left(1+\frac{i}{x}\right).

The boundary normal derivative convention of Greene et al. is

∂n=H∂∂ln⁡a=−Hη∂η=−Hx∂x.\partial_n = H\frac{\partial}{\partial\ln a} = -H\eta\partial_\eta = -Hx\partial_x.

For a Bogoliubov coefficient bb, the exact transported kernel is

κb=−∂nuk+b ∂nuk∗uk+b uk∗.\kappa_b = - \frac{\partial_nu_k+b\,\partial_nu_k^*} {u_k+b\,u_k^*}.

Exact kernel

Define

z=b e−2ix.z=b\,e^{-2ix}.

Direct differentiation gives

κbH=x (i−x−1−ix−2)+z(−i−x−1+ix−2)(1+ix−1)+z(1−ix−1).\boxed{ \frac{\kappa_b}{H} = x\, \frac{ \left(i-x^{-1}-ix^{-2}\right) + z\left(-i-x^{-1}+ix^{-2}\right) }{ \left(1+ix^{-1}\right) + z\left(1-ix^{-1}\right) }. }

The Bunch–Davies kernel is the z=0z=0 specialization. For every nonzero NPH excitation bb, the correction

δκb=κb−κBD\delta\kappa_b=\kappa_b-\kappa_{\rm BD}

is a nonconstant rational function of

e−2ix=exp⁡ ⁣(−2ika0H).e^{-2ix} = \exp\!\left(-\frac{2ik}{a_0H}\right).

Finite-order obstruction

A finite local quadratic boundary action contributes a finite polynomial in spatial derivatives, hence—after Fourier transform—a finite polynomial in k2/a02k^2/a_0^2, with background-dependent coefficients.

No such polynomial can equal the displayed nonconstant Möbius function of e−2ik/(a0H)e^{-2ik/(a_0H)} on an open momentum interval. Therefore

b≠0⟹κb has no exact finite local derivative-EFT representation.\boxed{ b\neq0 \quad\Longrightarrow\quad \kappa_b \text{ has no exact finite local derivative-EFT representation.} }

Equivalently, transporting a nontrivial NPH state to one common slice generates an infinite derivative tail.

This conclusion already holds in exact de Sitter space, where bNPHb_{\rm NPH} can be independent of kk. Slow-roll dependence of H(k)H(k) adds further non-polynomial structure but is not needed for the obstruction.

Pole typing

The denominator

(1+ix−1)+be−2ix(1−ix−1)\left(1+ix^{-1}\right) + b e^{-2ix}\left(1-ix^{-1}\right)

may vanish at isolated complex momenta. These are poles of the scalar boundary-kernel chart obtained by dividing by the field value uk+buk∗u_k+b u_k^*. They are not automatically new carrier support or singularities of the underlying two-dimensional solution line.

Thus:

kernel-chart pole⇏new physical divisor.\text{kernel-chart pole} \not\Rightarrow \text{new physical divisor}.

The invariant object is the projective solution line, or equivalently the undivided boundary relation

(∂n+κb)vk=0.\bigl(\partial_n+\kappa_b\bigr)v_k=0.

Consequence

The smallest common coefficient envelope is not the finite local boundary-operator algebra. It must retain either:

Local BEFT is a finite-jet subobject. NPH is regular in the projective solution-line presentation but generally infinite-jet on a fixed spacetime boundary.

Remaining falsifier

Determine whether the infinite derivative expansion is a controlled completion:

  1. compute its Taylor/asymptotic coefficients in p0/Mp_0/M;
  2. determine radius and domain of convergence;
  3. identify whether kernel-chart poles obstruct the completed operator or disappear under projective gluing;
  4. test compatibility with interacting boundary renormalization.

The finite-order closure question is settled negatively; all-order analytic or resurgent closure remains open.

Provenance

View source on GitHub