# Geometry of the Shannon mutual information in continuum QFT

## TL;DR

This paper investigates the geometric and scaling properties of Shannon mutual information in continuum quantum field theory, specifically for a free massless scalar field, and introduces related information measures like Fisher information.

## Contribution

It provides a novel analysis of the geometric structure of mutual information in continuum QFT and derives an expression for the reduced probability density in this context.

## Key findings

- Derived geometric terms and scaling laws for mutual information
- Expressed the reduced probability density for field configurations on a ball
- Computed Fisher information related to the location of observation regions

## Abstract

We analyze geometric terms and scaling properties of the Shannon mutual information in the continuum. This is done for a free massless scalar field theory in $d$-dimensions, in a coherent state reduced with respect to a general differentiable manifold. As a by-product, we find an expression for the reduced probability density of finding a certain field on a ball. We will also introduce and compute the Fisher information that this probability carries about the location of the observation region. This is an interesting information measure that refers to points in physical space, although in relativistic QFT they are labels and not fluctuating quantum observables.