Yang–Mills existence and mass gap

A Millennium Prize Problem asking for a rigorous construction of quantum Yang–Mills theory on four-dimensional spacetime and proof that its spectrum has a positive mass gap.

Gauge theory: A physical theory whose description has local symmetries and associated gauge fields. Yang–Mills theory is a non-Abelian instance of this broader framework.
Yang–Mills equations: Nonlinear field equations governing Yang–Mills connections and their curvature. They define the classical dynamics from which the quantum theory is to be constructed.
Hermann Weyl: A German mathematician and physicist whose work helped establish gauge symmetry as a physical idea. His early gauge theory work helped shape the conceptual ancestry of Yang–Mills fields.
Constructive quantum field theory: A mathematical program that rigorously constructs quantum field theories and proves their defining properties. A solution must meet the program’s standards for existence and consistency.
Quantum electrodynamics: The quantum field theory of electromagnetic interactions, based on the abelian gauge group U(1). Its gauge structure differs from the non-Abelian theories at the heart of the problem.
Quantum field theory: A framework describing quantum fields and their excitations across spacetime. The existence claim concerns a quantum field theory, not merely classical equations.
Lie group: A group that is also a smooth manifold, allowing continuous symmetries to be described mathematically. The choice of a compact simple gauge group specifies the theory in the problem.
Chen-Ning Yang: A Chinese-American theoretical physicist who co-introduced non-Abelian gauge theory with Robert Mills. The problem takes its name from Yang’s foundational theory.
Osterwalder–Schrader axioms: Conditions on Euclidean correlation functions that support reconstruction of a relativistic quantum field theory. They provide a route from Euclidean Yang–Mills measures to a quantum theory.
Perturbation theory: A method that approximates solutions by expanding in a small parameter. Perturbative expansions do not establish nonperturbative existence or a positive spectral gap.

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