Lattice-to-Continuum Constructive Proof Obligations

A finite lattice gauge or matter theory is a well-defined statistical system. A constructive continuum theorem requires substantially more: estimates uniform in lattice spacing and volume, convergence of a determining family of renormalized observables, survival of positivity and locality, and reconstruction of a continuum theory. A smooth $a^2$ fit for several observables is an extrapolation method, not a substitute for those functional and probabilistic steps.

Required background. Continuum limits and universality supply the topology and uniqueness obligations. Nonperturbative gauge measures supply the measure target, and bare parameters and continuum targets supply the regulator sequence. Helpful background. Hamiltonian continuum cross-validation, continuum-limit validation for QEC, lattice-to-continuum entropy, and geometric discretization checks illustrate observable-specific extra requirements.

From a lattice law to a continuum hierarchy

For a finite lattice $\Lambda_{a,L}$ and compact gauge group $G$, the Wilson measure

$$d\mu_{a,L}(U)=Z_{a,L}^{-1}e^{-S_{a,L}(U)} \prod_{e\subset\Lambda_{a,L}}dU_e$$

is normalized, positive, and gauge invariant. At this stage all observables are finite-dimensional integrals. Choose a bare trajectory $\beta(a)$ and define renormalized smeared observables $\mathcal O_{a,i}(f)$. The continuum target may be their joint characteristic functional

$$\mathcal Z(f_1,\ldots,f_k) =\lim_{a\to0,\,L\to\infty} \left\langle \exp\!\left(i\sum_i\mathcal O_{a,i}(f_i)\right) \right\rangle_{a,L}.$$

Existence for every test-function tuple, continuity at the origin, and positive definiteness can define a limiting random distribution. A finite list of means and covariances does not. Composite local fields need their own subtractions and mixing matrix; Wilson loops require control as their contours are approximated; topological observables may require sector-dependent normalization.

The uniform estimates

A credible proof program separates the following steps.

Step	Typical mathematical input	Failure if omitted
Bare tuning	Critical surface or controlled RG trajectory	Limit becomes massive, divergent, or trivial for unintended reasons
Volume control	Cluster, correlation, or free-energy bounds uniform in $a$	Thermodynamic limit may not exist or may select uncontrolled phases
Tightness	Sobolev/Besov moment bounds or compactness of correlations	No subsequential continuum object
Observable convergence	Uniform renormalization and all-$n$ correlation bounds	A few fitted quantities do not determine a QFT
Positivity	Reflection positivity stable under the chosen limit	OS quotient can have negative norm
Euclidean structure	Covariance, symmetry, regularity, clustering	Reconstruction hypotheses remain incomplete
Uniqueness and universality	Subsequence independence and comparison maps	Only a family of possible limits is known
Reconstruction	OS theorem or an algebraic continuum construction	Euclidean data have not yet produced Lorentzian dynamics

For Wilson lattice gauge theory, Osterwalder and Seiler prove finite-cutoff reflection positivity and the transfer-matrix structure under the stated reflection setup 1978, §§2–4, pp. 440–458. This settles one row. It does not supply tightness as $a\to0$ or convergence of local gauge-invariant composite fields.

First application: a Wilson ensemble with step scaling

Return to reflection positivity and transfer-matrix criteria. Suppose a sequence of Wilson ensembles has a measured step-scaling function and continuum fits for two glueball channels. A constructive route would proceed as follows:

1. define the tuned sequence $\beta(a)$ and physical volume scaling;
2. prove volume-uniform moment or generating-functional bounds for a separating gauge-invariant observable class;
3. prove tightness in a continuum configuration or distribution space;
4. show all limiting Schwinger functions retain reflection positivity and Euclidean symmetry;
5. establish uniqueness, nontriviality, and cluster properties;
6. reconstruct the Hilbert space and then prove any claimed spectral gap.

The step-scaling data constrain item one and test universality. The glueball fits constrain selected two-point spectral information. Neither provides the uniform all-observable estimates in items two through five.

Hamiltonian cross-validation is valuable but equally typed. Agreement between transfer-matrix energies and a Hamiltonian truncation checks selected regulated spectral quantities. Strong-resolvent convergence of Hamiltonians, convergence of local algebras, and equality with the OS-reconstructed continuum operator remain separate theorems.

A narrow proved scaling limit

Chatterjee’s Yang–Mills–Higgs result shows what an actual lattice scaling theorem records. It fixes $G=SU(2)$, includes a fundamental Higgs field, chooses unitary gauge, sends $a\to0$ jointly with a rapidly vanishing gauge coupling and diverging Higgs length, specifies the stereographically projected gauge field, and proves distributional convergence to a massive Gaussian one-form 2026, Theorem 3.2 and §3.3, pp. 12–17. The output is exact because every one of those restrictions is visible. It does not imply a non-Gaussian or pure-gauge continuum limit.

Positivity must match the observable

Reflection positivity for the underlying lattice action does not automatically settle a derived observable with nonlocal subtraction or analytic continuation. One must check that the reflection operation, support condition, and renormalization preserve the positive quadratic form. Entanglement entropy obtained by replica continuation, for example, adds an analytic-continuation obligation not present for ordinary Schwinger functions.

Failure test: two smooth extrapolations

Fit $O_1(a)=O_1(0)+c_1a^2$ and $O_2(a)=O_2(0)+c_2a^2$ and infer existence of every continuum correlator. Infinitely many inequivalent limiting laws can agree on two observables. The strongest surviving claim is a controlled extrapolation of $O_1$ and $O_2$ under the fit model and error budget. It cannot close tightness, positivity, uniqueness, or reconstruction.

Exercises

What does a uniform bound $\mathbb E\|\Phi_a\|_{H^{-s}}^p\le C$ contribute, and what does it not prove?

Solution

With a compact embedding into a slightly weaker space, the bound can give tightness and hence subsequential continuum laws. It does not prove that different subsequences agree, that the limit is reflection positive or non-Gaussian, or that the chosen field family determines all observables.

References

• Chatterjee, Sourav. “A Scaling Limit of $SU(2)$ Lattice Yang–Mills–Higgs Theory.” Probability and Mathematical Physics 7 (2026): 339–381. DOI; Open PDF.
• Jaffe, Arthur, and Edward Witten. “Quantum Yang–Mills Theory.” In The Millennium Prize Problems, 129–152. Clay Mathematics Institute and American Mathematical Society, 2006; problem description released 2000. Official PDF.
• Osterwalder, Konrad, and Erhard Seiler. “Gauge Field Theories on a Lattice.” Annals of Physics 110 (1978): 440–471. DOI.