A Variance Inequality for Meromorphic Measurement Functions under Exterior Probability

TL;DR

This paper introduces a variance inequality for measuring unbounded system attributes near singularities using exterior probabilities, revealing a fundamental limit akin to quantum uncertainty principles.

Contribution

It develops a novel variance inequality framework for meromorphic measurement functions under exterior probabilities, linking complex analysis with quantum uncertainty concepts.

Findings

Variance decomposes into two parts: one decreasing, one increasing with scale.

A minimum variance limit exists regardless of measurement precision.

Framework supports non-deterministic uncertainty at quantum scales.

Abstract

The problem of measuring an unbounded system attribute near a singularity has been discussed. Lenses have been introduced as formal objects to study increasingly precise measurements around the singularity and a specific family of lenses called Exterior probabilities have been investigated. It has been shown that under such probabilities, measurement variance of a measurable function around a 1st order pole on a complex manifold, consists of two separable parts - one that decreases with diminishing scale of the lenses, and the other that increases. It has been discussed how this framework can lend mathematical support to ideas of non-deterministic uncertainty prevalent at a quantum scale. In fact, the aforementioned variance decomposition allows for a minimum possible variance for such a system irrespective of how close the measurements are. This inequality is structurally similar to Heisenberg uncertainty relationship if one considers energy/momentum to be a meromorphic function of a complex spacetime.