---
title: "Operator Bases & False Constraints Interactive Lab | Sandbox Physics"
description: "Can changing an operator basis make a physical prediction more precise?"
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FR02 · A new coordinate system for the same theory

# Operator Bases & False Constraints

**Can changing an operator basis make a physical prediction more precise?** A change of operator basis preserves predictions when the coefficients and their covariance are transformed together. Keeping the old covariance in the new coordinates changes the assumed uncertainty and can make a constraint appear stronger without adding any information.

Pull a coordinate grid through the same Gaussian distribution. Its fixed-weight quantiles and probability contours deform together, but the observable keeps the same uncertainty. Freeze the old covariance to see a false improvement.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The model has two independent polynomial responses and a stated Gaussian coefficient distribution. The innermost contour has unit Mahalanobis radius and contains 39.3 percent of a two-dimensional Gaussian. Outer contours have radii 1.5, 2, 2.5 and 3. Coordinates show offsets from each transformed mean. Prediction bands show one standard deviation. A basis change is invertible; dropping operators or changing the prior describes a different inference problem.

Gold coordinates carry the same 432 fixed-weight Gaussian quantiles; teal contours mark the original coordinates. The camera follows their spread, and the teal direction follows the same observable.

The observable keeps its uncertainty

## Follow the prediction through the change of basis

Correct mean and one-standard-deviation band Transformed prediction overlays the reference Wrong: keep the old covariance

The distribution is specified in the original coordinates. The transformed operators, coefficients and covariance describe exactly that same distribution.

Model, assumptions and sources

The model has two independent polynomial responses and a stated Gaussian coefficient distribution. The innermost contour has unit Mahalanobis radius and contains 39.3 percent of a two-dimensional Gaussian. Outer contours have radii 1.5, 2, 2.5 and 3. Coordinates show offsets from each transformed mean. Prediction bands show one standard deviation. A basis change is invertible; dropping operators or changing the prior describes a different inference problem.

$$
O'=BO,\quad c'=B^{-T}c,\quad\Sigma'=B^{-T}\Sigma B^{-1}
$$

The responses are a constant and a linear power of the dimensionless variable. The reference coefficients are 0.8 and 0.45. Their standard deviations are 0.12 and 0.15, with covariance minus 0.009. The innermost contour has unit Mahalanobis radius and contains 39.3 percent of the Gaussian. Additional radii 1.5, 2, 2.5 and 3 contain 67.5, 86.5, 95.6 and 98.9 percent. The external bands show one standard deviation of each scalar prediction.

Every dot is a deterministic Gaussian quantile with the same fixed probability weight. Colors mark Mahalanobis radius, not a numerical density value. The quantiles and the coordinate mesh undergo the same invertible coefficient transformation, centered on the transformed mean. The viewport follows the vertical spread and always includes every displayed contour and node; its camera scale is a drawing choice. These nodes are separate from the saved noisy-amplitude observations in the mediator Lab.

The coral band intentionally keeps the original covariance in the new coordinates. Its changed width comes from a changed distribution. The transformation itself adds no data. Eliminating a redundant operator with field equations is a further EFT operation and is outside this two-independent-operator example.

[Manohar · EFT lectures, sections 3–6](https://arxiv.org/pdf/1804.05863)

[Read the course: gauge fields and effective theories](https://sandboxphysics.com/stories/gauge-effective-theory/)

Physics tutorial

## Operator Bases & False Constraints

**Background** Pull a coordinate grid through the same Gaussian distribution. Its fixed-weight quantiles and probability contours deform together, but the observable keeps the same uncertainty. Freeze the old covariance to see a false improvement.

**Why it matters** Two authors can quote very different bounds on coefficients and still predict exactly the same experiment.

Start with the essentials

- **Focus question**: Can changing an operator basis make a physical prediction more precise?
- **One-sentence intuition**: A coefficient bound refers to a coordinate system; a prediction requires the operators and their correlations too.

### Core mathematical model

#### Operators and coefficients move oppositely

$$
O=\begin{pmatrix}1\\x\end{pmatrix},\quad O'=BO,\quad B=\begin{pmatrix}1&k\\0&a\end{pmatrix},\quad c'=B^{-T}c,\quad c'^TO'=c^TO
$$

The two response functions are a constant and a term linear in the dimensionless energy variable. The rescaling stays positive, so the transformation can always be undone.

#### Uncertainty follows the same change of coordinates

$$
\Sigma'=B^{-T}\Sigma B^{-1},\qquad \sigma_y^2=O^T\Sigma O=O'^T\Sigma'O'
$$

The narrow and wide directions of the ellipse move together. Keeping the old covariance while changing the operators describes a new distribution and changes the prediction band.

### Common difficulties

#### A narrow coefficient is not a precise observable

**Typical misconception** The narrowest-looking ellipse must contain the most information.

**Better mental model** Coordinate rescaling can make any coefficient range look small. Propagate the covariance through the operator response before comparing information.

#### An ellipse is not two separate error bars

**Typical misconception** Transforming only the individual standard deviations is enough.

**Better mental model** The off-diagonal covariance carries the tilt of the ellipse. Omitting it changes the uncertainty of a combined prediction.

### Run the experiment

1. 01

   #### Shear the basis

   Move the gold grid handle up and down, then drag the blue handle. Compare the changing coordinate fabric with the observable below it.

   **What to observe:** The contours and fixed-weight quantiles move together, but the correctly transformed prediction band stays fixed.

2. 02

   #### Inspect a false constraint

   Choose “A deceptively narrow band”. Compare the coral band with the teal band at several prediction points.

   **What to observe:** The coral calculation uses the old covariance in the new basis. Its apparent precision comes from changing the assumptions.

3. 03

   #### Restore the original coordinates

   Choose “Original basis”, then restore another preset and export the coefficient distribution.

   **What to observe:** The same physical mean and variance are recovered in every invertible basis.

Physical conclusion

Transform the whole distribution when changing a basis.

## Related experiments and articles

Gauge fields and effective theories · Reading path

### Gauge symmetry, effective theories and anomalies

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why can a new operator basis appear to tighten a constraint?**](https://sandboxphysics.com/stories/gauge-effective-theory/basis-and-inference/)
2. [Article **Does more precise data make a low-energy theory more reliable?**](https://sandboxphysics.com/stories/gauge-effective-theory/truncation-and-positivity/)
3. [Lab **Heavy Mediator & Effective Field Theory** Separate fit bias from statistical noise](https://sandboxphysics.com/labs/heavy-mediator-eft/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)