Is the local linearity of space-time inherited from the linearity of probabilities?

TL;DR

This paper explores the idea that the linear structure of space-time could fundamentally stem from the linearity inherent in probability theory, linking physics and information theory.

Contribution

It proposes a novel perspective that the local linearity of space-time may be derived from the linearity of probabilities, supported by arguments from quantum information and group theory.

Findings

Supports the thesis with quantum information theory arguments

Connects space-time linearity to probability linearity

Aligns with recent ideas on emergent space-time in physics

Abstract

The appearance of linear spaces, describing physical quantities by vectors and tensors, is ubiquitous in all of physics, from classical mechanics to the modern notion of local Lorentz invariance. However, as natural as this seems to the physicist, most computer scientists would argue that something like a "local linear tangent space" is not very typical and in fact a quite surprising property of any conceivable world or algorithm. In this paper, we take the perspective of the computer scientist seriously, and ask whether there could be any inherently information-theoretic reason to expect this notion of linearity to appear in physics. We give a series of simple arguments, spanning quantum information theory, group representation theory, and renormalization in quantum gravity, that supports a surprising thesis: namely, that the local linearity of space-time might ultimately be a consequence of the linearity of probabilities. While our arguments involve a fair amount of speculation, they have the virtue of being independent of any detailed assumptions on quantum gravity, and they are in harmony with several independent recent ideas on emergent space-time in high-energy physics.