Keep the physics. Remove the grid.

Prerequisites: the continuum-limit lesson and dimensional analysis. Learning outcomes: convert between physical and lattice units, distinguish refinement from increasing physical volume, and identify a scaling trajectory that changes the physics.

A positive physical mass does not mean a fixed positive mass in lattice units. The distinction is essential when interpreting lattice results.

THE CONTINUUM LIMIT

One mass. Two units.

Reset

Keep the physical mass fixed at m = 1. Make the lattice finer and see the lattice-unit mass shrink while the physical correlation length stays unchanged.

Lattice spacing a

0.20

Physical length in fixed reference units

Physical box length L

8.0

A separate limit from making a smaller

Lattice mass am0.20dimensionless

Correlation length ξ / a5.0lattice spacings

Physical correlation length ξ1.00reference length units

Spacings across the box L / a40.0continuous ratio

The mass in lattice units The marked point follows your selected spacing.

The continuum limit needs more sites, not a vanishing physical mass.

Units demonstration. This fixes m = 1 and assumes ξ = 1/m. It does not simulate Yang–Mills or calculate its gap. L/a is shown as a continuous ratio; an actual periodic lattice requires an integer number of sites and compatible parameters. The box indicator L/ξ is a scale comparison, not a certified finite-volume error bound.

Read the scaling

The physical mass in this model is fixed at one inverse unit of length. As the lattice spacing decreases, the mass in lattice units decreases and the number of lattice sites per physical correlation length increases. A dimensionless lattice mass tending to zero can therefore be exactly what a massive continuum limit requires.

Now hold the spacing fixed and increase the box length. That is a different limit. A useful numerical calculation needs both enough sites per correlation length and a box large compared with the relevant physical correlation lengths.

Exercise: Find two parameter choices with the same number of sites along one direction but different physical box lengths. Why does the site count alone not tell you whether the simulation controls the same physics?

This laboratory accounts for scales. It neither constructs a lattice gauge measure nor extrapolates measured data. Read the continuum-limit lesson for the additional mathematical requirements, or the benchmark protocol for comparisons between numerical studies.

Derive the scale conversion

Let the reference length be $\ell$. The lab fixes $m=1/\ell$, so the physical correlation length is $\xi=1/m=\ell$. A separation $x=na$ gives the same exponential in two descriptions:

$$e^{-mx}=e^{-(am)n},\qquad \xi/a=1/(am),\qquad N=L/a.$$

Here $a$ and $L$ carry length units, while $am$, $\xi/a$ and $N$ are dimensionless. The plotted $N$ is a continuous ratio for teaching; an actual periodic lattice must use an integer site count with compatible spacing and box size. The control never sets $a=0$: its limiting trend represents what must be controlled, not an existing continuum construction.

At $a=0.2\ell$ and $L=8\ell$, the lattice mass is $0.2$, the correlation length spans five spacings, and $N=40$. Halving $a$ while holding $L$ fixed gives $am=0.1$, ten spacings per correlation length and $N=80$. The physical mass and physical volume have not changed. Doubling $L$ at fixed $a$ also doubles $N$, but changes the box from eight to sixteen correlation lengths. Site count alone cannot distinguish these operations.

Solution to the exercise above: $(a,L)=(0.2\ell,8\ell)$ and $(0.1\ell,4\ell)$ both have $N=40$. The second box is physically half as wide and contains half as many physical correlation lengths. Their potential finite-volume effects are therefore different.

Exercise: Suppose instead that $am=0.2$ stays fixed while $a$ tends to zero. What happens to the physical mass?

Solution: $m=0.2/a$ grows without bound, so $\xi=a/0.2$ tends to zero in fixed physical units. This is a different trajectory from keeping a finite positive physical mass. A lower bound fixed in lattice units is not automatically the desired continuum mass statement.

Source and limitation

Lüscher’s Properties and uses of the Wilson flow in lattice QCD, arXiv:1006.4518v3, §3.4 and equation (3.3) provides an actual reference-scale construction and discusses lattice-spacing effects in dimensionless ratios. This lab’s elementary conversion is derived above; its numbers are synthetic scale accounting, not that paper’s data. It neither demonstrates convergence of observables nor bounds finite-volume errors. Continue to continuum scaling and scale setting for those distinctions.