# Information geometry of Euclidean quantum fields

**Authors:** Stefan Floerchinger  
**arXiv:** [2303.04081](https://arxiv.org/abs/2303.04081)  
**Date:** 2023-03-07  
**Summary page:** https://www.summarizepaper.com/en/paper/2303.04081v1/

## Key points

- Information geometry provides differential geometric concepts for spaces of probability distributions
- Stefan Floerchinger applies these concepts to quantum field theories in the Euclidean domain
- The dual affine structure corresponds to sources and field expectation values as coordinates, allowing for a deeper understanding of the theory's relationships
- A new generating functional serves as a functional generalization of the Kullback-Leibler divergence, from which connected correlation functions and one-particle irreducible correlation functions can be obtained
- This generating functional encodes the Fisher information metric and two dual connections, determining asymptotic probabilities for field configurations through Sanov's theorem.
- Information geometry offers valuable insights into quantum field theories in the Euclidean domain and provides new tools for exploring these complex systems.

## Summary

The field of information geometry provides a range of differential geometric concepts such as Riemannian metrics, connections, and covariant derivatives on spaces of probability distributions. In this paper, Stefan Floerchinger explores how these concepts can be applied to quantum field theories in the Euclidean domain, which can also be seen as statistical field theories. One key aspect of this approach is the dual affine structure that corresponds to sources and field expectation values seen as coordinates. This structure allows for a deeper understanding of the relationships between different aspects of the theory. Another important concept introduced in this paper is a new generating functional that serves as a functional generalization of the Kullback-Leibler divergence. From its functional derivatives, one can obtain connected correlation functions as well as one-particle irreducible correlation functions. Additionally, this generating functional encodes directly the geometric structure of the theory, including both the Fisher information metric and two dual connections. It also determines asymptotic probabilities for field configurations through Sanov's theorem. Overall, Floerchinger's work highlights how information geometry can provide valuable insights into quantum field theories in the Euclidean domain and offers new tools for exploring these complex systems.