What the continuum limit must preserve

Taking a continuum limit means comparing a sequence of regulated theories at the same physical scales. It requires more than making a grid visually finer.

Prerequisites: dimensional transmutation, the lattice definition, and finite volume.

By the end:

• Relate a physical mass to its dimensionless lattice value.
• Identify a needed estimate and the parameter in which it must be uniform.

Match physical questions across lattices

Let $a$ be the lattice spacing and $n$ a separation measured in lattice steps. The physical separation is $t=an$. If $a$ changes while $n$ stays fixed, the measurement moves toward zero physical separation. To compare the same positive $t$, one must use $n\simeq t/a$.

A simple example makes the issue explicit. Suppose

$$C_a(n)=e^{-ma n}$$

for a fixed physical energy $m$. Then $C_a(n)\to1$ at every fixed $n$ as $a\to0$. Nevertheless, at fixed physical time,

$$C_a(t/a)=e^{-mt}.$$

The first limit sampled a shrinking time interval. It did not reveal a gapless physical theory.

Lattice masses must shrink

Write the dimensionless lattice decay rate as $\widehat m(a)$. Its physical counterpart is

$$m_{\mathrm{phys}}(a)=\frac{\widehat m(a)}{a}.$$

For a finite nonzero continuum mass,

$$\widehat m(a)\to0, \qquad \frac{\widehat m(a)}a\to m>0.$$

Equivalently, the correlation length in lattice sites diverges while the physical correlation length remains finite:

$$\xi_{\mathrm{latt}}\sim\frac1{am}, \qquad \xi_{\mathrm{phys}}=a\xi_{\mathrm{latt}}\sim\frac1m.$$

This explains why a fixed positive lower bound on a lattice-unit mass is generally the wrong target along a continuum trajectory with finite physical scales.

In practice, choose a physical reference length or energy and compare dimensionless ratios. Lüscher’s Wilson-flow construction provides useful gauge-invariant reference quantities. A numerical value in GeV additionally requires an explicit calibration convention.

Renormalized observables are part of the limit

The continuum limit is a statement about observables, not only bare link configurations. A schematic renormalized composite can involve

$$O_R(a)=Z_O(a)O_a+\sum_j Z_{Oj}(a)O_{j,a},$$

including possible operator mixing. Subtractions and normalization conditions must be specified. For singular quantum fields, convergence is naturally tested after smearing rather than demanding pointwise limits.

A candidate construction might establish limits of the form

$$\lim_{a\to0} \langle O_{1,R}(f_1)\cdots O_{k,R}(f_k)\rangle_a =S_k(f_1,\ldots,f_k).$$

One must then control the properties of the resulting family of correlation functions: symmetry, regularity, positivity, locality after reconstruction, and the appropriate Yang–Mills identification. A subsequence of numbers converging is not enough by itself.

The original Osterwalder–Schrader paper explains the kind of connection that Euclidean data can have to a physical quantum theory when its hypotheses hold.

Numerical extrapolation and mathematical construction

Numerical work can estimate a dimensionless observable at several small spacings and fit a justified cutoff-dependence model, for example

$$R(a)=R(0)+c(a/\ell_{\mathrm{ref}})^2+\cdots$$

in an appropriate setting. The chosen action, observable, logarithmic corrections, fit range, and higher-order terms determine whether that model is adequate.

Such an extrapolation is valuable evidence. It carries uncertainties from measurements and from assumptions about the extrapolation. A constructive theorem must instead establish the relevant limiting objects and properties with mathematical control.

The distinction is about the type of conclusion. It does not diminish the quantitative importance of high-quality simulations.

Why weak bare coupling need not mean a free theory

Along an asymptotically free continuum trajectory, the bare coupling at scale $1/a$ tends to zero. The renormalized interaction at a fixed physical scale can remain nonzero. Sending the ultraviolet scale to infinity is different from turning off the physical dynamics at all scales.

Nontriviality must still be demonstrated in a rigorous construction. Correct ultraviolet behavior alone cannot be substituted for control of long-distance observables.

Use the continuum lab to hold a physical gap fixed while changing the spacing. Watch both the lattice-unit mass and the number of lattice sites per correlation length.

Exercises

1. Convert the numbers

At spacings $a=0.20,0.10,0.05$ in fixed length units, measured lattice masses are $0.40,0.20,0.10$. What physical mass does this idealized sequence suggest?

Hint. Divide each lattice mass by its spacing.

Solution. Every ratio equals $2$ in inverse length units. The vanishing lattice-unit masses are consistent with a nonzero physical limit.

2. A bound with the wrong scale

Suppose $\widehat m(a)\geq c>0$ for all small $a$. What happens to the lower bound on $m_{\mathrm{phys}}$?

Hint. Restore the factor of $a$.

Solution. It becomes $m_{\mathrm{phys}}\geq c/a$, which diverges. This does not establish the desired finite physical threshold with the chosen reference scale.

Continue at your own depth

Limits and uniform control develops this lesson without adding a prerequisite to the next lesson.

The reconstruction references must be read as a pair: Osterwalder–Schrader I (1973) and II (1975). The latter extends and corrects the earlier formulation. The spectral and Euclidean-data reasoning here assumes the required physical Hilbert space exists; it does not replace the corrected regularity, covariance and reconstruction hypotheses. See the reconstruction guide for the scope of that statement.

Read further

• Lüscher, Wilson flow and physical reference scales.
• Osterwalder and Schrader, Euclidean reconstruction.
• Athenodorou and Teper, numerical continuum extrapolations.

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