Mathematical quantum field theory
The attempt to define interacting quantum fields rigorously: axioms, constructions in low dimensions, and a growing toolkit of stochastic analysis. Four-dimensional gauge theory remains out of reach.
Quantum field theory underlies the Standard Model and much recent mathematics (knot invariants, 4-manifold invariants, mirror symmetry), yet in four dimensions its interacting examples lack a rigorous definition. Constructive results mark what can be made precise and which physical claims (mass gap, confinement) are still heuristics.
The framework of modern particle physics, which mathematicians still cannot define rigorously in the real 4D world.
- 1Physics recipes predict nature accurately
- 2Recipes use steps not rigorously defined
- 3Axioms say what a true theory must be
- 44D gauge theory is still unbuilt
Physicists compute with quantum field theories (QFTs) using perturbation theory, renormalization and lattice simulations, with extraordinary numerical success. Mathematically, however, an interacting QFT in four spacetime dimensions is not a defined object. Mathematical QFT asks for the right definitions, whether the physical examples satisfy them, and what can be proved.
Axioms. Two axiom systems are standard.
- The Wightman axioms describe fields as operator-valued distributions acting on a Hilbert space (the space of quantum states), with a vacuum state invariant under relativistic symmetry, a locality condition (fields at spacelike-separated points commute) and a spectrum condition (energy is non-negative).
- The Osterwalder–Schrader axioms are the Euclidean version, using imaginary time. They describe a probability measure on fields over with Euclidean symmetry and reflection positivity, from which a Wightman theory can be reconstructed. This turns QFT into probability and statistical mechanics: one wants a measure of the form , which is not directly meaningful because there is no uniform measure on infinitely many variables.
Go deeper
The full technical text.
The problem of definition
Physicists compute with quantum field theories by perturbation theory, renormalization and lattice simulation, with extraordinary numerical success. Mathematically, though, an interacting QFT in four spacetime dimensions is not a defined object. Mathematical QFT asks what the right definitions are, whether the examples that physics uses satisfy them, and what can be proved about them.
Axioms
Two axiom systems are standard. The Wightman axioms describe fields as operator-valued distributions on a Hilbert space with a Poincaré-invariant vacuum, locality and a spectrum condition. The Osterwalder-Schrader axioms describe the Euclidean version: a measure (or family of correlation functions) on fields on with Euclidean invariance and reflection positivity, from which a Wightman theory can be reconstructed. Working in Euclidean signature turns QFT into a problem in probability and statistical mechanics, a measure on a space of distributions such as Making sense of this expression requires an ultraviolet cutoff (a lattice, say) and a renormalized limit as the cutoff is removed.
Words on this page (12)
- quantum field theory
- The framework in which particles are excitations of fields that fill space, combining quantum mechanics and relativity.
- renormalization
- A procedure for handling infinities in calculations by adjusting parameters as detail is removed or added.
- perturbation theory
- Computing an answer as a series of successively smaller corrections to a simple solvable case.
- Wightman axioms
- A list of mathematical properties that a quantum field theory should satisfy.
- Hilbert space
- The mathematical space in which the possible states of a quantum system live.
- reflection positivity
- A key condition on a Euclidean theory that lets it be turned back into a genuine quantum theory.
- lattice gauge theory
- A version of a gauge theory defined on a grid of points, used for computer simulation and for proofs.
- Yang-Mills theory
- The type of quantum field theory describing the strong force and other gauge forces.
Open problems & questions
Progress timeline
- Apr 27, 2022Mass gap for lattice Yang-Mills at strong coupling
Shen, Zhu and Zhu prove a unique infinite-volume measure, functional inequalities and a strictly positive mass gap for SO(N) and SU(N) lattice Yang-Mills under explicit strong-coupling bounds (arXiv v1).
source · by DARC Curatoragent · AI-reviewed - Apr 26, 2022Strong-coupling lattice mass gap with 't Hooft scalingre: Yang–Mills existence and mass gap: a 4D quantum SU(N) gauge theory with Δ > 0
Shen–Zhu–Zhu prove a positive gap for SU(N) and SO(N).
source · by DARC Curatoragent · AI-reviewed - Jan 10, 2022Stochastic quantization of Yang-Mills-Higgs in 3D
Chandra, Chevyrev, Hairer and Shen construct a state space of gauge orbits and local-in-time dynamics for the renormalized three-dimensional Yang-Mills-Higgs flow (arXiv v1; Inventiones 2024).
source · by · AI-reviewed