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Finite-regulator derivative expansion and truncation

Physical scope

A finite-range derivative-counting convention for the positive Gaussian heavy-sector kernel; it asserts no generic continuum remainder or convergence.

Definition

For the finite undirected graph $Γ = (X, E_{Γ})$ , define $d_{Γ} (x, y)$ as the least number of edges in a path from $x$ to $y$ , and set $d_{Γ} (x, y) = + \infty$ if no path exists. Concatenating paths gives the triangle inequality for this extended distance. Identify $R^{X}$ with real coordinate vectors indexed by $X$ ; for a matrix $B$ on this space, $B_{x y}$ denotes its row- $x$ , column- $y$ entry in the standard coordinate basis. For an integer $r \geq 0$ , say that $B$ has range at most $r$ when $B_{x y} = 0$ whenever $d_{Γ} (x, y) > r$ . The positive heavy-sector matrix $A$ from the model data has range at most one. A derivative expansion of its inverse kernel is the formal operator series

(M^{2} I + A)^{- 1} \sim M^{- 2} \sum_{r = 0}^{\infty} (- A / M^{2})^{r} .

Its order- $N$ truncation, for an integer $N \geq 0$ , is the finite operator

S_{N} : = M^{- 2} \sum_{r = 0}^{N} (- A / M^{2})^{r} .

It has range at most $N$ in this regulator. For a source $J$ supported in the spectral interval $[0, E^{2}]$ of $A$ , the dimensionless power-counting parameter is $E / M$ . This definition is an organized finite-order approximation; it does not assert convergence for unrestricted sources, a generic error bound, or a continuum local-operator expansion. Under the Axiom of Choice and the positive heavy-mode Gaussian matching-data hypotheses, the low-momentum matching remainder theorem gives, for $J = P_{E} J$ and $K = M^{2} I + A$ , the conditional bound $∥ K^{- 1} J - S_{N} J ∥ \leq M^{- 2} (E^{2} / M^{2})^{N + 1} ∥ J ∥$ . In this finite real symmetric positive-semidefinite setting the theorem applies on the complexification with its coordinate Hermitian inner product and restricts to real sources. This bound is useful when $E < M$ ; it asserts no generic interacting-action remainder.

Definition references

Finite-regulator EFT model and scale data

Finite-regulator Wilsonian action

Low-momentum matching has a controlled remainder

Facts & Assumptions

Given: The finite graph $Γ = (X, E_{Γ})$ and range-one matrix $A$ from the model-data definition, and the finite-regulator Wilsonian action convention.

[F1]

Finite-regulator EFT model and scale data: $X$ is finite, $d_{Γ}$ is graph distance, and the real symmetric matrix $A$ vanishes when $d_{Γ} (x, y) > 1$ . The extended shortest-path convention and standard coordinate entries used here are fixed in this Definition.

[F2]

Finite-regulator Wilsonian action: the Wilsonian action is defined by the finite high-coordinate integral while the retained coordinates are held fixed.

Proof

technique · direct matrix-range induction

Given: The finite graph and matrix hypotheses in the Definition.

1.1givenF1algebra

The identity $A^{0} = I$ has range zero. If $A^{r}$ has range at most $r$ , then $(A^{r + 1})_{x y} = \sum_{z \in X} (A^{r})_{x z} A_{z y}$ . A nonzero summand requires $d_{Γ} (x, z) \leq r$ and $d_{Γ} (z, y) \leq 1$ . When both distances are finite, concatenating the corresponding paths gives $d_{Γ} (x, y) \leq r + 1$ ; if either is infinite, the summand is zero by the range hypotheses. Induction yields range at most $r$ for every $A^{r}$ .

2.1givenF1F2step 1.1algebra∎

Each term of $S_{N}$ is a scalar multiple of $A^{r}$ with $r \leq N$ . Therefore every matrix entry of $S_{N}$ vanishes when $d_{Γ} (x, y) > N$ . Thus $S_{N}$ is the finite-range truncation of the formal inverse series defined above; no convergence or generic remainder follows from this range calculation. The scalar Wilsonian action in [F2] does not by itself identify $(M^{2} I + A)^{- 1}$ with an action kernel for an arbitrary $S_{U V}$ .

Finite-regulator derivative expansion and truncation

Definition

Definition references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources