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.

Observable mean $1.09$

Correct uncertainty $0.110$

Frozen-covariance uncertainty $0.189$

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 bandTransformed prediction overlays the referenceWrong: 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

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

Read the course: gauge fields and effective theories

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 misconceptionThe narrowest-looking ellipse must contain the most information.

Better mental modelCoordinate 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 misconceptionTransforming only the individual standard deviations is enough.

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

Run the experiment

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.
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.
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.

Operator Bases & False Constraints

Follow the prediction through the change of basis