Math 2.0

Quantum theory Definition

Constructive quantum field theory

Builds on Osterwalder–Schrader reconstruction theoremWightman axiomsTempered distributionGaussian measureGaussian free field

Definition

Let d≥2d \ge 2 and let 𝒮′(ℝd)\mathcal{S}'(\R^d) be the space of tempered distributions on ℝd\R^d. A construction of a Euclidean scalar quantum field theory in dd dimensions is a Borel probability measure μ\mu on 𝒮′(ℝd)\mathcal{S}'(\R^d) with these properties.

  1. Its moments, the Schwinger functions Sn(f1,…,fn)=∫ϕ(f1)⋯ϕ(fn) dμ(ϕ)S_n(f_1, \dots, f_n) = \int \phi(f_1) \cdots \phi(f_n) \, d\mu(\phi), exist and satisfy the Osterwalder–Schrader axioms.
  2. The measure μ\mu is not Gaussian, so the theory is interacting.

By the Osterwalder–Schrader reconstruction theorem, such a μ\mu gives a field theory on Minkowski space satisfying the Wightman axioms. Constructive quantum field theory is the study of such constructions, usually obtained as limits of models with an ultraviolet cutoff and a finite volume.

Remarks

Example: the P(ϕ)2P(\phi)_2 models in d=2d = 2, where the interaction is a polynomial PP bounded below. Their measures are limits of measures with density proportional to exp⁡(−∫Λ:P(ϕ(x)): dx)\exp\left(-\int_\Lambda {:}P(\phi(x)){:}\, dx\right) relative to the Gaussian free field, as the volume Λ\Lambda grows. These limits exist, for instance at weak coupling by cluster expansions. They satisfy the axioms, and they are not Gaussian when PP has degree at least 44.

Non-example: the Gaussian free field of mass m>0m > 0 satisfies clause 1, but it fails clause 2. It gives the free field, which describes particles that do not interact.

The same aims can be set in the Minkowski or the algebraic setting, using the Wightman or the Haag–Kastler axioms directly. Fermionic models and gauge theories need other function spaces or Grassmann integrals, but the goal is the same.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Analytic continuation, concept, no evidence levelAnalytic continuationconceptEuclidean isometry, concept, no evidence levelEuclidean isometryconceptGaussian free field, concept, no evidence levelGaussian free fieldconceptGaussian measure, concept, no evidence levelGaussian measureconceptSchwartz space, concept, no evidence levelSchwartz spaceconceptTempered distribution, concept, no evidence levelTempered distributionconceptTranslation, concept, no evidence levelTranslationconceptWick rotation, concept, no evidence levelWick rotationconceptWightman axioms, concept, no evidence levelWightman axiomsconceptEuclidean group, object, no evidence levelEuclidean groupobjectOsterwalder–Schrader reconstruction theorem, statement, no evidence levelOsterwalder–Schrader …statementConstructive quantum field theory, definition, no evidence levelConstructive quantum …definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Analytic continuation concept
  2. Euclidean isometry concept
  3. Gaussian free field concept
  4. Gaussian measure concept
  5. Schwartz space concept
  6. Tempered distribution concept
  7. Translation concept
  8. Wick rotation concept
  9. Wightman axioms concept
  10. Euclidean group object
  11. Osterwalder–Schrader reconstruction theorem statement
  12. Constructive quantum field theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Constructive_quantum_field_theory
About this page
TagHTQ3VSA160
URIm2://HTQ3VSA160
Kinddefinition
WritingW0 stub
MSC81T08
DefinesConstructive quantum field theory
UsesOsterwalder–Schrader reconstruction theorem, Wightman axioms, Tempered distribution, Gaussian measure, Gaussian free field
Changed3 November 2026

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