Limitations on Dimensional Regularization in Renyi Entropy

TL;DR

This paper investigates the limitations of using dimensional regularization for Renyi entropy in quantum field theory, demonstrating it cannot always be interpreted as a measure of entanglement in a Hilbert space.

Contribution

It provides a concrete proof that dimensionally regularized Renyi entropy lacks a Hilbert space interpretation, highlighting fundamental limitations of the regularization method.

Findings

Dimensional regularization of Renyi entropy does not correspond to a Hilbert space measure.

Regularized Renyi entropy cannot always be obtained as a limit of finite-dimensional quantum systems.

The study focuses on 4d unitary conformal field theories.

Abstract

Dimensional regularization is a common method used to regulate the UV divergence of field theoretic quantities. When it is used in the context of Renyi entropy, however, it is important to consider whether such a procedure eliminates the statistical interpretation thereof as a measure of entanglement of states living on a Hilbert space. We therefore examine the dimensionally regularized Renyi entropy of a 4d unitary CFT and show that it admits no underlying Hilbert space in the state-counting sense. This gives a concrete proof that dimensionally regularized Renyi entropy cannot always be obtained as a limit of the Renyi entropy of some finite-dimensional quantum system.