Limits, tightness, and uniform control

Prerequisites: uniform estimates and weak convergence and tightness.

You will learn to specify a limiting trajectory, read the dependence of constants, and distinguish a subsequential limit from an identified quantum theory.

Name the objects before taking a limit

Suppose $\mu_{a,L}$ is a regulated measure, $a$ a lattice spacing and $L=N_sa$ the physical spatial size. The number of sites $N_s$ is not a physical length. Renormalized observables $O^R_{a,L}(f)$ may depend on a test function $f$ held fixed in physical coordinates.

A proposed continuum construction needs a common space in which these objects can be compared. For example, interpolated fields might be distributions and convergence might be tested against smooth functions. Merely placing all finite-lattice partition functions in one list does not define their limit as a field theory.

One possible strategy is to take $L\to\infty$ at each $a$, then $a\to0$. Another is a controlled joint sequence $(a_n,L_n)$. Neither is privileged by notation; each needs estimates appropriate to its path and an identification of the result.

A worked scaling trajectory

Fix a physical reference mass $M$ and choose

$$a_n=\frac1{nM},\qquad L_n=\frac nM,\qquad N_{s,n}=n^2.$$

Then $a_nM=1/n\to0$ and $L_nM=n\to\infty$. If a physical excitation has mass $m=M$, its lattice-unit mass is $am=1/n$, while its correlation length in sites is $n$. The number of correlation lengths fitting across the box is $L_nm=n$.

Compare $a_n=1/(nM)$ with $N_s=n$. That gives $L_n=1/M$, a fixed physical box. Refining resolution alone has not reached unbounded space. The continuum lab varies spacing and box size independently for precisely this reason.

Read the constant as part of the theorem

Consider a connected physical correlator with a bound

$$|C_{a,L}(t)|\leq K\,e^{-\Delta t},$$

where $t$ is physical Euclidean time. If $K<\infty$ and $\Delta>0$ are independent of $a,L$ along the chosen sequence and the correlators converge at fixed $t$, the inequality passes to that limit.

If instead $\Delta=\Delta_{a,L}$ tends to zero, the same conclusion does not follow. For the explicit sequence $C_L(t)=e^{-t/L}$, every finite $L$ has exponential decay, but $C_L(t)\to1$ at fixed $t$. A diverging prefactor can also make an otherwise attractive bound useless.

Even a retained decay estimate is not the entire mass-gap argument. The limit must have the required positive spectral representation and a sufficiently rich family of physical observables. Gap and correlation decay makes this last qualification precise.

Tightness gives candidates, not identification

Tightness controls escape of probability mass in a specified topology. Under the relevant compactness hypotheses it allows extraction of a weakly convergent subsequence. It does not choose the limit uniquely, identify it with Yang–Mills, or prove convergence of arbitrary unbounded observables.

For an elementary warning, let $X_n$ equal $n$ with probability $1/n$ and zero otherwise. Its distributions converge weakly to a point mass at zero, but $\mathbb E X_n=1$ for every $n$. Weak convergence alone did not pass the first moment to the limit. A suitable uniform-integrability estimate would address the missing tail control.

For field observables, the analogous issue is why a convergence theorem for bounded test functions cannot silently become a theorem for every polynomial composite. The probability bridge and renormalized-observables page separate these steps.

What must survive

Keep a ledger of the observable class, topology, normalization, boundary conditions, dimensions and groups. Record which constants are uniform in each parameter. Check positivity and covariance on the limiting observables, then the continuity and regularity needed for reconstruction. Finally establish the ultraviolet identification, nontriviality and physical spectral statement.

These are construction tasks, not a newly claimed theorem. The target and discussion of uniform finite-volume gaps are in Jaffe–Witten, §§4–5; the corrected Euclidean reconstruction framework is Osterwalder–Schrader II (1975).

Exercise: spot a noncommuting limit

Let $r>0$ be a fixed reference length. For the dimensionless toy function $f(a,L)=(aL/r^2)/(1+aL/r^2)$, compute $\lim_{a\to0}\lim_{L\to\infty}f$ and $\lim_{L\to\infty}\lim_{a\to0}f$. What happens on the dimensionally consistent path $L=r^2/a$?

Solution. At fixed $a>0$ and $r$, the inner large-volume limit is one, so the first iterated limit is one. At fixed finite $L$ and $r$, the inner cutoff limit is zero, so the second is zero. Along $L=r^2/a$, the dimensionless product $aL/r^2$ equals one and the answer is $1/2$. This toy function does not model Yang–Mills; it proves that writing “remove both cutoffs” without a controlling argument is insufficient.

Return to the construction map or examine ultraviolet stability with the question “uniform in which parameter?” in view.