Relative Entropy and Proximity of Quantum Field Theories

TL;DR

This paper explores how the relative entropy between quantum field theories can quantify their distinguishability, relate to renormalization group flow, and assess fine-tuning and string landscape effective theories.

Contribution

It introduces a formalism using relative entropy to measure proximity of QFTs, connecting it to the Zamolodchikov metric and renormalization group flow.

Findings

Relative entropy quantifies distinguishability of QFTs.

The formalism relates to the Zamolodchikov metric for conformal theories.

Provides a criterion for differentiating low energy theories in string landscape.

Abstract

We study the question of how reliably one can distinguish two quantum field theories (QFTs). Each QFT defines a probability distribution on the space of fields. The relative entropy provides a notion of proximity between these distributions and quantifies the number of measurements required to distinguish between them. In the case of nearby conformal field theories, this reduces to the Zamolodchikov metric on the space of couplings. Our formulation quantifies the information lost under renormalization group flow from the UV to the IR and leads us to a quantification of fine-tuning. This formalism also leads us to a criterion for distinguishability of low energy effective field theories generated by the string theory landscape.