Comments on Entire Functions of the Derivative Operator

TL;DR

The paper shows that treating $\mathcal{O}=\exp[T^2 \tfrac{d^2}{dt^2}]$ as a benign, positive-definite nonlocal operator is misleading: a kernel analysis reveals an infinite spectrum of solutions beyond the simple harmonic oscillator, with $\Omega$ satisfying $\Omega^2=\frac{1}{T^2}[x^2+N\,2\pi i]$ and both $\Re(\Omega)$ and $\Im(\Omega)$ generically nonzero. It proves that this vast solution set corresponds to the ability to prescribe arbitrary initial value data over any finite interval via the integral kernel representation $\mathcal{O} q(t)=\frac{1}{\sqrt{4\pi T^2}}\int_{t_1}^{\infty} dt' e^{-(t-t')^2/(4T^2)} q(t')$, by solving a finite linear system for the future data. Consequently, the operator is not positive-definite and exhibits Ostrogradskian-like instability, implying that nonlocal form factors in gravity that rely on entire functions of the derivative may be unphysical. The result cautions against adopting such nonlocal constructions in quantum gravity and argues for seeking genuinely viable, local or differently stabilized approaches.

Abstract

Many attempts to introduce fundamental nonlocality into quantum (or classical) field theory are based on the assumption that exponentials of the d'Alembertian are positive-definite, so that these operators can be employed without engendering the Ostrogradskian instability associated with higher derivative Lagrangians. {\bf This assumption is false.} Working in the simple context of a 1-dimensional, point particle $q(t)$, I demonstrate that the equation $\exp[T^2 \tfrac{d^2}{dt^2}] q(t) = 0$ has an infinite number of rapidly oscillating, exponentially rising and falling solutions. This infinite kernel is in one-to-one correspondence with the ability to specify ``initial value data'' {\it arbitrarily} over {\it any} finite interval $t_1 < t < t_2$.