# Markov property of the CFT vacuum and the a-theorem

## TL;DR

This paper provides a new proof of the a-theorem in four-dimensional conformal field theories by leveraging entanglement entropy properties, extending previous proofs of c and F theorems in lower dimensions, and unifying irreversibility results in quantum field theory.

## Contribution

It introduces a novel proof of the a-theorem using entanglement entropy and the Markov property of the CFT vacuum, unifying various irreversibility theorems across dimensions.

## Key findings

- Proof of the a-theorem in d=4 using entanglement entropy.
- Extension of vacuum entanglement entropy methods to higher dimensions.
- Unified framework for irreversibility theorems in relativistic QFT.

## Abstract

We use strong sub-additivity of entanglement entropy, Lorentz invariance, and the Markov property of the vacuum state of a conformal field theory to give a new proof of the irreversibility of the renormalization group in d=4 space-time dimensions -- the a-theorem. This extends the proofs of the c and F theorems in dimensions d=2 and d=3 based on vacuum entanglement entropy, and gives a unified picture of all known irreversibility theorems in relativistic quantum field theory.