Undecidable problems in 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 Gödel's second incompleteness theorem. The article includes a brief and informal introduction to the phenomenon of undecidability and its previous appearances in theoretical physics.

A 15-minute video presentation of the content is available at https://www.youtube.com/watch?v=H548i3dnsWE . v3: a nice reference listing undecidable problems in mathematics (arXiv:1204.0299) is added

References in corpus (11)

• Undecidability of the Spectral Gap (short version)
• A Mechanised Proof of Gödel's Incompleteness Theorems using Nominal Isabelle
• Undecidability of the Spectral Gap in One Dimension
• On the computation of non-perturbative effective potentials in the string theory landscape -- IIB/F-theory perspective
• Undecidability of the fate of relaxation in one-dimensional quantum systems
• Undecidability in quantum thermalization
• A Machine-Assisted Proof of Gödel's Incompleteness Theorems for the Theory of Hereditarily Finite Sets
• Uncomputability of Phase Diagrams
• Uncomputability and complexity of quantum control
• Kahler Moduli Stabilization and the Propagation of Decidability
• Uncomputably Complex Renormalisation Group Flows

Cited by in corpus (4)

• Undecidability in Physics: a Review
• Can quantum gravity be both consistent and complete?
• Gödel's undecidability theorems and the search for a theory of everything
• Undecidability of the spectral gap in rotationally symmetric Hamiltonians