Wetterich's Equation and its Boundary Conditions for Radon Measures on Locally Convex Spaces

TL;DR

This paper explores Wetterich's equation for Radon measures on locally convex spaces, linking it to effective actions and convex conjugates, with implications for quantum field theory and measure analysis.

Contribution

It extends Wetterich's equation to Radon measures on locally convex spaces and characterizes the effective average action's domain as a Lusin affine kernel.

Findings

Wetterich's equation applies to Radon measures under certain conditions.

The flow connects the cumulant-generating function's convex conjugate and the Onsager-Machlup function.

The metric is induced by measurable bilinear functionals.

Abstract

Wetterich's equation and corresponding flows of effective average actions are used frequently in theoretical physics to study the properties of quantum field theories. Under appropriate conditions, Wetterich's equation also holds for Radon measures on locally convex spaces and the domain of the effective average action is the Lusin affine kernel of the measure. The resulting flow interpolates between the convex conjugate of the cumulant-generating function of the measure in question and its (generalised) Onsager-Machlup function. The underlying metric of the latter is induced by a family of measurable bilinear functionals that can be understood as bilinear versions of Lusin measurable linear functionals.