Comments on Entire Functions of the Derivative Operator

TL;DR

This paper challenges the assumption that exponential operators of the d'Alembertian are positive-definite, showing they have an infinite kernel with oscillating solutions, impacting nonlocal quantum field theories.

Contribution

It demonstrates that exponential of the second derivative operator has an infinite kernel, contradicting common assumptions used in nonlocal field theory models.

Findings

Exponential of the second derivative operator has an infinite number of solutions.

These solutions include rapidly oscillating and exponentially rising/falling functions.

The kernel's size relates to arbitrary initial data specification over finite intervals.

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 $t_1 < t < t_2$.