---
title: "Changing resolution while keeping predictions"
description: "Integrate out alternate sites, match low momenta and test a prediction that was not fitted."
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last_modified: "2026-10-27"
---

[Fields, scales and critical phenomena](https://sandboxphysics.com/learn/field-theory/)

FR01 · L3 · Chapter 6

# Changing resolution while keeping predictions

Integrate out alternate sites, match low momenta and test a prediction that was not fitted.

Before you start: Chapter 5. Completing a square is extended from one variable to a block matrix.

You will be able to: Derive a Schur complement and distinguish exact decimation from a truncated effective action.

A coarser field can preserve a fine field's predictions if its parameters change with the resolution. We can show this exactly for a Gaussian field: integrate out the sites we stop displaying, then calculate the probability distribution of those that remain. Approximating that new distribution with a shorter formula is a second step, and its error can be tested.

Read [Euclidean lattices and errors](https://sandboxphysics.com/learn/field-theory/euclidean-lattice-and-errors/) first. We will use its Gaussian action, covariance matrix and Fourier modes. By the end, you should be able to eliminate a Gaussian variable, explain why a parameter changes, and choose an observable that actually tests a matched model.

## Keep the values at every other site

Imagine photographing a field on a ring, then erasing every other dot from the photograph. The surviving dots keep their values. We want a probability distribution that assigns those values exactly the same probabilities as before.

Write the field as a vector. Its action is $S=\frac12\phi^TQ\phi$, where the positive-definite matrix $Q$ specifies the coupling between sites. Split the vector into retained values $r$ and eliminated values $e$. These letters label groups of variables; they do not label spatial directions.

The action becomes

$$
S=\frac12r^TQ_{rr}r+r^TQ_{re}e+\frac12e^TQ_{ee}e.
$$

Every possible value of $e$ contributes to the probability of a specified $r$. Summing continuously over those possibilities is an integral. A Gaussian integral is manageable because its exponent is a square.

Define $e_0=-Q_{ee}^{-1}Q_{er}r$ and substitute $e=e_0+\eta$. The terms linear in $\eta$ cancel. Expanding the remaining terms gives

$$
S=\frac12r^T\left(Q_{rr}-Q_{re}Q_{ee}^{-1}Q_{er}\right)r+\frac12\eta^TQ_{ee}\eta.
$$

Integrating over $\eta$ contributes a factor independent of $r$. It changes the normalization of the distribution. The part that determines the relative probabilities of retained configurations is

$$
Q_{\mathrm{eff}}=Q_{rr}-Q_{re}Q_{ee}^{-1}Q_{er}.
$$

This expression is called a Schur complement. Its meaning is simple: eliminated sites used to transmit correlations between the surviving sites. The effective coupling includes that influence.

The covariance of the retained variables is the corresponding submatrix of the original covariance. Therefore

$$
\left(Q_{\mathrm{eff}}^{-1}\right)_{ij}=\left(Q^{-1}\right)_{bi,bj},
$$

when every $b$th site is kept. This identity is an independent prediction we can check with two different matrix calculations.

## A mass coefficient can change while the correlation stays

Take a nearest-neighbor ring with lattice spacing $a$, positive mass parameter $m$, and action

$$
S=\frac a2\sum_i\left[m^2\phi_i^2+\left(\frac{\phi_{i+1}-\phi_i}{a}\right)^2\right].
$$

Use a fixed reference length and a corresponding field unit so that the displayed action is dimensionless. The numerical example uses $a=1$ and $m=0.5$ in those reference units. There are $64$ periodic sites. The precision matrix has diagonal $d=am^2+2/a=2.25$ and nearest-neighbor entry $-k=-1/a=-1$.

All eliminated odd sites are uncoupled from one another once the even values are fixed. Completing each square gives the coarse diagonal $D=d-2k^2/d$ and coarse neighbor coupling $-K=-k^2/d$. Numerically,

$$
D=1.36111,\qquad K=0.444444.
$$

The new spacing is $A=2a=2$. To write the result in the familiar form, allow both the mass coefficient $m_{\mathrm{eff}}^2$ and the gradient coefficient $Z_{\mathrm{eff}}$ to vary. Comparing the diagonal and off-diagonal entries gives

$$
m_{\mathrm{eff}}^2=\frac{D-2K}{A}=0.236111,\qquad Z_{\mathrm{eff}}=AK=0.888889.
$$

The mass coefficient was $0.25$. It has changed. Yet the variance at a retained site is still about $0.970143$, and the covariance between neighboring retained sites is about $0.360531$. Those are the fine ring's covariances at separations zero and two.

These numbers can also be checked with the periodic image sum of the infinite-chain Green function, independently of matrix elimination. On this ring the finite-volume correction is below the displayed precision. The coefficient called “matched mass squared” is a coefficient in a lattice action. Extracting an exponential decay length from correlations is a separate calculation.

## Exact elimination can produce a complicated action

The preceding nearest-neighbor model is unusually kind: in one dimension its Gaussian family closes under this decimation. Suppose the original action also contains a bending term,

$$
\frac{a\kappa}{2}\sum_i\left(\frac{\phi_{i+1}-2\phi_i+\phi_{i-1}}{a^2}\right)^2.
$$

Here $\kappa$ has squared-length units before lengths are expressed in the reference unit. This term penalizes sharp changes in slope. Integrating out sites now generally leaves couplings extending beyond the nearest surviving neighbor.

We could keep the entire effective matrix. Often we want a simpler description with only a few parameters. For a periodic coarse ring with $N_c$ sites, try

$$
A_{\mathrm{local}}(q)=c_0+4c_1\sin^2(q/2),\qquad q_n=\frac{2\pi n}{N_c}.
$$

$A(q)$ is an eigenvalue of the precision, so its reciprocal is the covariance of that normalized Fourier mode. Set $c_0=A_{\mathrm{exact}}(0)$. Then choose $c_1$ so that the first nonzero mode also agrees:

$$
c_1=\frac{A_{\mathrm{exact}}(q_1)-c_0}{4\sin^2(q_1/2)}.
$$

Two observations fixed two coefficients. Agreement at those two observations is guaranteed. Mode two gives a new test. Its relative covariance error is

$$
\delta_2=\left|\frac{A_{\mathrm{exact}}(q_2)}{A_{\mathrm{local}}(q_2)}-1\right|.
$$

The omitted terms often matter more at shorter wavelengths. This is why a short action can be useful for low-energy questions even when it predicts the entire spectrum poorly.

## Compare resolutions using the saved field

Open [coarse graining and renormalization flow](https://sandboxphysics.com/labs/renormalization-flow/). The default is $64$ sites, $a=0.5$, $m=0.4$, $\kappa=0.25$, seed $310031$, with every second site retained. The gold row copies values from the turquoise row. It is a gallery of independent Euclidean configurations; its progress is a sample index.

The default two-mode match gives $m_{\mathrm{eff}}^2\approx0.159681$ and $Z_{\mathrm{eff}}\approx1.00750$. Mode two has a covariance error of about $1.68\%$. These are deterministic model calculations. The displayed field samples vary with the seed; these matrix results do not.

The covariance panel takes the Fourier power of each of those same twelve retained configurations, then averages the powers. Its gold points carry one standard error, calculated from the spread between independent configurations. The turquoise curve is the exact covariance, and the dashed curve is the local prediction. The match uses exact modes zero and one. Changing the seed changes the sampled points and error bars; the held-out truncation error stays fixed.

Predict what happens to the held-out error when the bending term vanishes. Choose “Exact local closure”, then explain the result using the square completion above. Next choose “Short-scale structure” and compare it with “Too coarse for the prediction”. The latter retains every eighth site of the same specified Gaussian model. Its mode-two error is about $50.7\%$.

Save samples and trajectory. The exported file contains the fine configurations, the effective matrices and the fitted spectra at all four resolutions. The field values and the spectral predictions can be checked separately.

Real-space decimation removes site variables. A momentum-shell transformation removes Fourier variables in a high-momentum interval. They are different transformations, but both require us to account for the variables we integrate over. [Tong's discussion of the Wilsonian action](https://sandboxphysics.com/learn/field-theory/wilson-fisher-and-universality/) takes this idea into an interacting field.

## Exercises

- A model is matched to the variance and the first Fourier-mode covariance. Which would be a stronger check: plotting those two quantities again, or predicting the second Fourier mode? Explain.
- For the numerical nearest-neighbor example, compute the coarse variance from $1/\sqrt{D^2-4K^2}$. Why is this an independent check of the changing mass coefficient?
- Suppose a field on an open chain is decimated. Would you expect one common diagonal and one common neighbor coupling to describe every retained site? Think about the endpoints.
- A colleague keeps the fine mass and gradient coefficients unchanged after doubling the spacing, then obtains a different covariance. Does this show that coarse descriptions cannot work? Propose a better comparison.

Explained answers

The second Fourier mode is stronger because it did not determine the parameters. Replotting the two matching quantities checks implementation and arithmetic, but provides no new predictive test.

Using the displayed $D$ and $K$ gives approximately $0.970143$. This calculation uses the Green function of a translation-invariant chain instead of the eliminated-variable integral. The same covariance survives although the action coefficients have changed.

The endpoints have different neighborhoods. Exact elimination remains possible, but its effective matrix usually needs boundary-specific coefficients. Translation invariance was an assumption in the ring's two-parameter spectral fit.

Keeping the coefficients unchanged creates a new discretization whose covariance can differ. Integrate out the eliminated variables first, or match specified observables, then test an unused observable. The comparison should also keep the physical volume and boundary conditions explicit.

The Gaussian elimination follows directly from completing a positive quadratic form. The broader scale-changing construction is developed in [Tong, Statistical Field Theory, chapter 3, sections 3.1–3.3](https://davidtong.org/pdfs/teaching/statistical-field-theory/sft3.pdf). The numerical examples and saved samples here are generated from the defined lattice action.

## Try the physics

[Coarse Graining and Renormalization Flow →Keep every second field value and integrate out the sites between them. Match two long-wavelength modes, then test a third that was left out of the match.](https://sandboxphysics.com/labs/renormalization-flow/)

## Sources and reading scope

1. [David Tong, Statistical Field Theory, Chapter 2](https://davidtong.org/teaching/statistical-field-theory/sfthtml/S2)

   Gaussian integrals, inverse-kernel covariance, correlation length and the Euclidean quantum-mechanical interpretation. Finite-chain normalizations are derived for this course.

2. [David Tong, Statistical Field Theory, Chapter 3, §3.5, Figure 30](https://davidtong.org/pdfs/teaching/statistical-field-theory/sft3.pdf)

   One-loop Wilson–Fisher flow near four dimensions, pp. 90–91, Figure 30. The Lab reconstructs an equation-derived phase portrait, with its parameters and convergence checks saved separately.

[← Putting a Euclidean field on a lattice](https://sandboxphysics.com/learn/field-theory/euclidean-lattice-and-errors/) [Testing a convincing data collapse →](https://sandboxphysics.com/learn/field-theory/finite-size-scaling/)

Reading order

[1. How a chain of oscillators becomes a field](https://sandboxphysics.com/learn/field-theory/from-oscillators-to-fields/) [2. Symmetry tells us what flows](https://sandboxphysics.com/learn/field-theory/symmetry-and-causality/) [3. Quantizing the normal modes](https://sandboxphysics.com/learn/field-theory/quantizing-a-field/) [4. What a correlation function predicts](https://sandboxphysics.com/learn/field-theory/correlators-to-scattering/) [5. Putting a Euclidean field on a lattice](https://sandboxphysics.com/learn/field-theory/euclidean-lattice-and-errors/) [6. Changing resolution while keeping predictions](https://sandboxphysics.com/learn/field-theory/keeping-low-energy-physics/) [7. Testing a convincing data collapse](https://sandboxphysics.com/learn/field-theory/finite-size-scaling/) [8. Reading a Wilson–Fisher flow diagram](https://sandboxphysics.com/learn/field-theory/wilson-fisher-and-universality/)