Constructive Euclidean QFT and Cutoff Removal

Constructive QFT proves that a named regulated model has a cutoff-independent limit with specified observables and properties. This chapter follows the proof chain from a Gaussian random distribution through interacting stability, uniform correlation estimates, regulator removal, Osterwalder–Schrader reconstruction, and—only when separately proved—particle and scattering structure.

Helpful background. Regulators, cutoffs, and continuum limits supplies the physical limit problem; constructive and Borel-summability boundaries separates exact construction from resummed series; reflection positivity and transfer-matrix criteria supplies the lattice positivity test.

Enter the constructive proof chain

Begin by fixing the spacetime dimension, covariance, interaction, regulator family, and observable class. At fixed cutoff, verify normalization and stability. Next identify the estimate that is uniform in the ultraviolet or volume parameter. Only then ask whether the limiting Schwinger functions satisfy reflection positivity, regularity, covariance, and clustering. The map below makes those one-way dependencies explicit; its side branch records that scalar, fermionic, stochastic, and gauge constructions use different control mechanisms.

The constructive implication chain and its model-specific branch. Each arrow requires the hypothesis printed on it; the diagram is schematic and not to scale. Structured description and source data (JSON)

A route through the chapter

Read the pages in order when constructing a model; jump to the later pages when testing a specific existence claim.

1. The Constructive Program and Cutoff Removal states what a regulator-removal theorem must actually prove.
2. Gaussian Euclidean Fields as Measures constructs the free reference field on a distribution space.
3. Interacting Measures, Stability, and Wick Ordering makes local polynomial weights meaningful and normalizable.
4. Cluster Expansions and Correlation Inequalities develops two distinct routes to uniform correlation control.
5. Thermodynamic Limits, Correlation Decay, and Phase Control distinguishes compactness, boundary independence, and coexistence.
6. Constructing P(φ)₂ and φ⁴₃ Models records the dimension-specific scalar theorems and the four-dimensional boundary.
7. Constructive Fermionic and Yukawa Models replaces bosonic moment combinatorics by Grassmann determinants and multiscale bounds.
8. Constructive Scattering, Particle Structure, and Mass Gaps adds the spectral hypotheses needed beyond Euclidean existence.
9. Stochastic Quantization: Invariant Measures and Convergence Theorems constructs renormalized dynamics and relates stationary laws to Euclidean measures.
10. Three-Dimensional Yang–Mills–Higgs Scaling-Limit Theorems states the precise weak-coupling Gaussian gauge–Higgs result and its observable limits.
11. From Euclidean Measures to Relativistic Models checks the full OS-to-Wightman bridge.
12. Constructive Existence by Model, Dimension, and Observable compares exactly what representative theorems construct.

What each hypothesis licenses

The table is intentionally directional. A conclusion in one row becomes input to a later row only when its stated topology and observable class match.

Constructive hypotheses, licensed conclusions, and excluded converses

Object or stage	Essential hypotheses	Licensed conclusion	What does not follow
Gaussian random distribution	Continuous positive covariance on the declared test space; reflection-positive kernel when OS use is intended	A unique Gaussian law and Wick moments; free OS theory under the full conditions	Ordinary point values, an interaction, or reflection positivity from positive definiteness alone
Interacting finite-volume law	Defined Wick powers or determinant; lower stability bound; finite normalization	A probability law at the stated cutoff and volume	Uniform ultraviolet removal, thermodynamic uniqueness, or non-Gaussian continuum behavior
Cluster or correlation control	Convergent polymer norm, or a named order inequality with uniform moment bounds	Analyticity or monotonicity and controlled connected correlations within that domain	Behavior outside the norm, absence of phase transitions, or a critical limit
Regulator limit	Tightness plus uniqueness or a Cauchy estimate; counterterm tuning; uniform integrability	A limiting measure or Schwinger hierarchy in the stated topology	Independence of a different regulator path, reflection positivity, or interaction unless checked
Relativistic reconstruction	Symmetry, Euclidean covariance, OS positivity, growth and regularity; clustering for a unique vacuum	Positive-energy local Wightman theory, with vacuum properties covered by the inputs	A mass gap, isolated particle, scattering, or asymptotic completeness
Particle scattering	Reconstructed local theory, isolated mass shell, interpolating operators, separated velocity supports	Haag–Ruelle in/out states and wave operators for the covered particle species	Completeness of those states or absence of additional sectors and bound states

Structured comparison data (JSON)

Failure tests before enlarging a claim

The fastest way to test a constructive assertion is to drop one decisive hypothesis. Reverse the highest polynomial coefficient and normalization fails. Remove the mass and exponential polymer decay fails. Move from three to four dimensions and the counterterm/norm argument changes. Keep only two-point convergence and OS reconstruction of an interacting hierarchy fails. Ask a weak-coupling Higgs theorem for a confining Wilson-loop area law and both the regime and observable have changed.

Concrete obstructions to the most common enlarged constructive claims, paired with the strongest conclusion that remains valid. The diagnostic map is schematic and not to scale. Structured description and source data (JSON)

References

• Chatterjee, Sourav. “A Scaling Limit of SU(2) Lattice Yang–Mills–Higgs Theory.” Probability and Mathematical Physics 7 (2026): 339–381. DOI.
• Glimm, James, and Arthur Jaffe. Quantum Physics: A Functional Integral Point of View. 2nd ed. Springer, 1987. DOI.
• Osterwalder, Konrad, and Robert Schrader. “Axioms for Euclidean Green’s Functions.” Communications in Mathematical Physics 31 (1973): 83–112. DOI.
• Summers, Stephen J. “A Perspective on Constructive Quantum Field Theory.” arXiv:1203.3991, revised 2016. Open PDF.