Local products require a closed basis and matrix evolution

Field renormalization makes separated elementary correlators finite, but it does not automatically define a product such as φ squared at one point. Loop momentum can become arbitrarily large around the insertion itself. Introduce a source for each relevant local operator and include the source-dependent counterterms required by that insertion. A finite operator basis at a chosen EFT order must be closed under the allowed ultraviolet mixing. Symmetries, dimensions, derivatives, and the chosen regulator restrict which partners are possible.

Operators related by equations of motion or total derivatives may be redundant for some on-shell amplitudes but still appear in off-shell renormalization. Gauge theories can require BRST-exact or gauge-variant representatives in intermediate bases, and dimensional continuation can introduce evanescent operators that vanish only in exactly four dimensions. Lower-dimensional mixing can involve masses or cutoff powers when allowed. One must therefore specify the purpose of the basis and the subtraction prescription before removing operators that look redundant in one particular matrix element.

Write an effective interaction or amplitude contribution as the row-column contraction C transposed times O. Differentiating it shows that the coefficient column obeys dC/dlnμ=γ transposed times C. The transpose is forced by which index is summed, while the opposite operator sign guarantees cancellation in the contraction. Using the same matrix with the same sign on both columns generally creates a false scale dependence. For matrix elements, any other running parameters and external normalizations must also be treated consistently.

If the anomalous-dimension matrices at different scales commute, their evolution can be expressed with an ordinary matrix exponential. Otherwise path ordering along the scale trajectory is required. Diagonalizing one fixed leading-order matrix can simplify the leading-log solution, but it is not a universal diagonalization at all orders. Jordan blocks can introduce logarithmic prefactors even when a matrix is constant. Exact physical predictions do not depend on the chosen operator basis, although individual operator matrix elements and Wilson coefficients do.

Under a constant invertible finite basis change O′=RO, coefficient covariance requires C′=R to the power minus transpose times C. The anomalous-dimension matrix transforms by similarity for constant R; a coupling-dependent R adds a derivative term. This basis covariance provides a simple check before any loop calculation. It also separates arbitrary mixing of coordinate representatives from dynamical evolution generated by an actual anomalous dimension. The presence of an off-diagonal entry alone does not specify a measurable transition probability between particles.

The matrix lab illustrates a constant shear basis change. Reset and set both controls to diagonal scale a=1 and shear b=0, predicting determinant one and singular values one. Reset and set a=1 and b=1, predicting determinant one and singular values approximately 1.618 and 0.618 for R=[[1,1],[0,1]]. Geometry is identity at Reset and interpolates under Play or Step, while readouts always describe the selected final matrix. The animation is not RG time or an anomalous-dimension solver; the unchanged determinant also does not mean every vector length is preserved.