Manuscript VI@1 · VI.tex · proposition · b5b60afb

Proposition: Unrestricted switch dependence

Finite-renormalization spaces and certificate requirements › Derivative counting alone is insufficient · lines 2097–2129 · proof included or linked

VI@1#prop:switch-infinite, VI@1#eq:infinite-switch

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If arbitrary polynomial dependence on a dimensionless switch is permitted without a bound on switch degree, a derivative-order bound alone does not give a finite-dimensional local functional catalogue.

For example, at formal order $κ^{2}$ , the functionals

C_{n} [g, φ] = κ^{2} m^{4} \int g (x)^{n} φ (x)^{2} d^{4} x, n = 1, 2, \dots,

are linearly independent as functionals of the pair $(g, φ)$ .

Proof. Suppose a finite linear combination vanishes for all compact smooth $g, φ$ :

n = 1 \sum^{N} a_{n} C_{n} [g, φ] = 0.

Choose $φ \neq \equiv 0$ supported in a ball, and choose $g$ equal to an arbitrary constant $s$ on that ball. Then

m^{4} (n = 1 \sum^{N} a_{n} s^{n}) \int φ^{2} d^{4} x = 0.

Varying $s$ over an interval shows that the polynomial is zero. Therefore every $a_{n} = 0$ .

Displayed formulas in this passage

VI@1#eq:infinite-switch · line 2104
$C_{n} [g, φ] = κ^{2} m^{4} \int g (x)^{n} φ (x)^{2} d^{4} x, n = 1, 2, \dots,$
unlabeled · line 2118
$n = 1 \sum^{N} a_{n} C_{n} [g, φ] = 0.$
unlabeled · line 2123
$m^{4} (n = 1 \sum^{N} a_{n} s^{n}) \int φ^{2} d^{4} x = 0.$

Proposition: Unrestricted switch dependence

Displayed formulas in this passage

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