The Constructive Program and Cutoff Removal

Constructive quantum field theory replaces the symbol $e^{-S(\phi)}D\phi$ by a sequence of honest probability measures and proves that selected observables converge when every regulator is removed. The central point is uniformity: existence at each cutoff, or convergence of a formal expansion, does not by itself produce a continuum relativistic model.

Required background. Euclidean random fields and Schwinger hierarchies supplies the measure-to-correlator dictionary; Osterwalder–Schrader axioms and reflection positivity supplies the Euclidean positivity condition; existence, construction, reconstruction, and continuum claims separates those logically different conclusions.

Helpful background. Clustering, vacuum uniqueness, and mass-gap implications explains the spectral consequences of decay; OS–Wightman comparison directions and failure modes records why reconstruction is not a consequence of two-point continuation alone.

A cutoff-removal theorem has several inputs

Fix a spacetime dimension $d$, a Gaussian reference measure $\mu_C$ on $\mathcal S'(\mathbb R^d)$, a bounded region $\Lambda$, and an ultraviolet smoothing scale $\varepsilon>0$. A typical regulated measure is

$$d\mu_{\Lambda,\varepsilon}(\phi)=Z_{\Lambda,\varepsilon}^{-1} \exp\{-V_{\Lambda,\varepsilon}(\phi)\}\,d\mu_C(\phi).$$

This formula becomes a theorem only after four questions are answered. First, is $V_{\Lambda,\varepsilon}$ measurable and is $0<Z_{\Lambda,\varepsilon}<\infty$? Second, which counterterms and Wick-ordering covariance define the family? Third, in which topology are the laws or Schwinger functions to converge? Fourth, which estimates remain independent of $\Lambda$ and $\varepsilon$?

A useful abstract statement is the following. Let $\varepsilon_j\downarrow0$ and $\Lambda_j\uparrow\mathbb R^d$. Suppose the regulated laws are tight in a specified distribution space $E\subset\mathcal S'$, their moments have uniform bounds strong enough to give tempered Schwinger distributions, and every subsequential limit has Euclidean invariance, symmetry, reflection positivity, and the required growth property. If the limiting moments are unique, then $\mu_{\Lambda_j,\varepsilon_j}$ converges weakly to a Euclidean field measure $\mu$, its moments satisfy those Osterwalder–Schrader hypotheses, and OS reconstruction yields a Wightman theory. Clustering is an additional hypothesis or conclusion of additional estimates; non-Gaussianity requires a nonzero connected correlation or another separating observable.

The conclusion is one-way. Tightness gives subsequences, not uniqueness. Moment convergence need not identify a law without a determinacy estimate. Reflection positivity can be lost if a regulator or counterterm is not compatible with time reflection. Finally, an OS measure gives a relativistic theory only after all reconstruction hypotheses, not merely covariance and positivity, have been checked. The original Euclidean construction strategy and this separation of obligations are described in Summers 2016, §3, pp. 9–11.

How uniform estimates do the work

The ultraviolet step controls local singularities. Wick ordering, power counting, multiscale decomposition, and counterterms produce estimates that do not deteriorate as $\varepsilon\to0$. The volume step controls the accumulation of interactions over space. Cluster expansions, correlation inequalities, or phase-specific contour estimates bound connected correlations and make the influence of a far boundary small. Compactness then converts bounds into a limit; identities satisfied uniformly by the approximants pass to that limit by continuity or dominated convergence.

For a polynomial $P$ bounded below in two dimensions, take the massive covariance $C=(-\Delta+m_0^2)^{-1}$ and

$$V_{\Lambda}(\phi)=\lambda\int_\Lambda :P(\phi(x)):_C\,d^2x, \qquad \lambda\ge0.$$

At fixed $\Lambda$, the interaction density belongs to every finite $L^p(\mu_C)$ needed for normalization. At weak coupling relative to the mass, a convergent cluster expansion supplies bounds uniform in growing rectangles. The finite-volume laws therefore have a weak infinite-volume limit, and the limiting Schwinger functions satisfy the OS axioms. This is the classic massive $P(\phi)_2$ application; Summers 2016, §3.1, pp. 11–12 states the regulated measure, weak convergence, OS conclusion, and exponential clustering result.

That worked construction is an existence result for a named two-dimensional model and regime. It is not a proof that arbitrary path integrals exist, nor a proof of a four-dimensional interacting continuum theory. The model-by-model boundary is carried into rigorous status, construction, and open problems.

Adversarial removal tests

Drop the lower-bound hypothesis by taking $P(u)=-u^4$. Along the constant-field direction, the supposed density grows like $e^{+\lambda |\Lambda|u^4}$; its normalizing integral diverges already at fixed cutoff. There is nothing to remove.

A subtler failure swaps limits without a uniform estimate. For numbers $a_{L,N}=L/(L+N)$, one has $\lim_{L\to\infty}\lim_{N\to\infty}a_{L,N}=0$ but $\lim_{N\to\infty}\lim_{L\to\infty}a_{L,N}=1$. A constructive proof must exclude the field-theoretic analogue by joint bounds, a cofinal-limit theorem, or a demonstrated order of limits. Perturbative coefficient convergence supplies none of these measure-theoretic facts.

An independent check is to test the free case $V=0$: the covariance and all Wick moments are regulator-independent after smearing, reflection positivity reduces to positivity of the free kernel, and connected functions above order two vanish. Any proposed general argument that fails this fixture has mishandled normalization or topology.

Exercises

1. Tightness versus uniqueness. Explain why a uniform bound $\sup_j\mathbb E\|\phi\|_E^p<\infty$ can yield a subsequential limit but not a unique continuum measure.

Solution

Markov’s inequality makes high-norm sets uniformly unlikely; when compact embeddings are available, this proves tightness and Prokhorov gives subsequences. Two different subsequences may still converge to different laws, as happens at phase coexistence. A uniqueness argument, boundary-condition control, or equality of a determining family of observables is separate.

2. A surviving identity. If each cutoff law is invariant under $\phi\mapsto-\phi$ and the smeared field moments converge, show that every limiting odd Schwinger function vanishes.

Solution

For every cutoff, changing variables gives $\mathbb E[\phi(f_1)\cdots\phi(f_{2k+1})]=-$ the same expectation, hence it is zero. Moment convergence passes this zero to the limit. This checks symmetry, not normalization, reflection positivity, clustering, or nontriviality.

References

• Glimm, James, Arthur Jaffe, and Thomas Spencer. “The Wightman Axioms and Particle Structure in the $P(\phi)_2$ Quantum Field Model.” Annals of Mathematics 100 (1974): 585–632. 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.