Complexity vs Energy: Theory of Computation and Theoretical Physics

TL;DR

This survey explores a rigorous analogy between computational complexity and physical energy, highlighting three mathematical contexts where concepts from computability theory inform and mirror formalisms in physics.

Contribution

It introduces three precise mathematical frameworks linking complexity in computer science with energy in physics, advancing the understanding of their deep analogy.

Findings

Mathematical contexts inspired by statistical physics and quantum field theory.

Formal connections between (un)computability and physical formalisms.

Enhanced understanding of the complexity-energy analogy through rigorous models.

Abstract

This paper is a survey dedicated to the analogy between the notions of {\it complexity} in theoretical computer science and {\it energy} in physics. This analogy is not metaphorical: I describe three precise mathematical contexts, suggested recently, in which mathematics related to (un)computability is inspired by and to a degree reproduces formalisms of statistical physics and quantum field theory.