Undecidable problems in quantum field theory

TL;DR

This paper demonstrates that certain questions in quantum field theory, such as supersymmetry breaking, are undecidable within standard mathematical frameworks, linking some physical problems to fundamental limits of formal systems.

Contribution

It establishes the undecidability of supersymmetry breaking questions in 2d theories and connects specific theories to set-theoretic consistency, highlighting fundamental limits in theoretical physics.

Findings

No algorithm can determine supersymmetry breaking in given 2d theories.

Existence of a 2d theory whose supersymmetry status depends on set-theoretic consistency.

Undecidability phenomena are relevant in the context of quantum field theory.

Abstract

We point out that some questions in quantum field theory are undecidable in a precise mathematical sense. More concretely, it will be demonstrated that there is no algorithm answering whether a given 2d supersymmetric Lagrangian theory breaks supersymmetry or not. It will also be shown that there is a specific 2d supersymmetric Lagrangian theory which breaks supersymmetry if and only if the standard Zermelo-Fraenkel set theory with the axiom of choice is consistent, which can never be proved or disproved as the consequence of G\"odel's second incompleteness theorem. The article includes a brief and informal introduction to the phenomenon of undecidability and its previous appearances in theoretical physics.