Constructive Euclidean QFT and Cutoff Removal
Constructive QFT turns a regulated model into a theory with controlled limiting observables. The chapter follows the estimates needed to remove cutoffs, reconstruct a relativistic model, and establish any additional particle properties.
Enter this chapter
Section titled “Enter this chapter”Start with the regulated measure and identify which estimate survives each removed regulator. Keep ultraviolet and infinite-volume limits distinct. 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.
Topics
Section titled “Topics”- The Constructive Program and Cutoff Removal — States the objects, limits, and convergence claims in a construction.
- Gaussian Euclidean Fields as Measures — Builds Gaussian fields as random distributions.
- Interacting Measures, Stability, and Wick Ordering — Controls normalization and large-field tails of interacting measures.
- Cluster Expansions and Correlation Inequalities — Obtains uniform estimates through expansions or positivity inequalities.
- Thermodynamic Limits, Correlation Decay, and Phase Control — Separates existence of infinite-volume phases from uniqueness and decay.
- Constructing P(φ)₂ and φ⁴₃ Models — Explains the successful low-dimensional scalar constructions.
- Constructive Fermionic and Yukawa Models — Identifies the determinant and stability estimates needed for fermions.
- Constructive Scattering, Particle Structure, and Mass Gaps — Adds spectral and scattering conclusions to a constructed model.
- Stochastic Quantization: Invariant Measures and Convergence Theorems — Distinguishes invariant-measure construction from stochastic convergence.
- Three-Dimensional Yang–Mills–Higgs Scaling-Limit Theorems — States the precise gauge–Higgs scaling regime and its Gaussian output.
- From Euclidean Measures to Relativistic Models — Checks the complete reconstruction hypotheses after regulator removal.
- Constructive Existence by Model, Dimension, and Observable — Compares existence results by model, dimension, and observable.
How the ideas fit together
Section titled “How the ideas fit together”The proof has three stages: control the regulated measure, prove convergence with uniform estimates, and verify the structural properties of the limit. Tightness alone supplies subsequences; neither uniqueness nor interaction follows automatically. Low-dimensional scalar constructions illustrate the full sequence Glimm and Jaffe 1987, Chapters 8–12. Gauge–Higgs scaling is a distinct example: the stated coupled parameter limit produces a massive Gaussian field Chatterjee 2026, Theorems 3.1–3.2 (PDF), not a non-Gaussian pure Yang–Mills construction.
What each hypothesis licenses
Section titled “What each hypothesis licenses”| 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 |
Review the chapter
Section titled “Review the chapter”Why does a nonzero limiting covariance fail to prove that the limit is interacting?
Solution
A Gaussian field can have a nonzero covariance. Interaction requires a suitable non-Gaussian diagnostic, such as a nonvanishing higher truncated correlation function, together with control that it survives the limit. Normalization and two-point convergence do not supply that diagnostic.
Where to go next
Section titled “Where to go next”Continue to rigorous renormalization for an all-scale method of obtaining the uniform estimates needed by these constructions.