3.5.17 Axiomatic Approaches
Over the past several decades, there have been many attempts to put Quantum Field Theory on firm mathematical footing by formulating a set of axioms for it. Proving the existence of a mass gap in Yang-Mills theory is linked to this issue, as we shall discuss it in section 9.
Wightman Axioms attempted to formalize the notion of operator-valued field within the context of functional analysis. It was possible to prove that any Quantum Field Theory satisfying these axioms satisfied certain general theorems, such as the spin-statistics theorem and the CPT theorem, but it proved extraordinarily difficult to show that any realistic field theory, including the Standard Model, satisfied these axioms. Most of the theories that could be treated with these analytic axioms were physically trivial, being restricted to low-dimensions and lacking interesting dynamics.
Another set of axioms based on topological ideas was proposed in the 1980s. All states of matter could be classified by geometry and the principle of broken symmetry, and General Relativity is based on the geometrical curvature of space and time, and also crystals, magnets and superconductors, all can be classified by the symmetries they break. However, in 1980 the quantum Hall effect provided the first example of a state of matter that has no spontaneous broken symmetry; its characterization is dependent on its topology and not on its geometry. The quantum Hall effect can be described by extending Quantum Field Theory into an effective topological Quantum Field Theory based on the Chern-Simons theory. The Standard Model allows for topological terms but is generally not formulated as a topological Quantum Field Theory. Topological Quantum Field Theory has also had broad impact in mathematics, with important applications in representation theory, algebraic topology, and differential geometry.