Markov property of the CFT vacuum and the a-theorem

TL;DR

The paper establishes an entropic proof of the $a$-theorem in $d=4$ by leveraging the Markov property of the CFT vacuum on the lightcone and the strong subadditivity of entanglement entropy. A master inequality $\,r\,\Delta S''(r) - (d-3)\,\Delta S'(r) \le 0$ is derived from a geometric arrangement of boosted spheres and, in conjunction with the Markov property, cancels UV wiggles to relate UV and IR universal terms. Applying this to the fixed-point entropy form $S_{\rho^0}(r)=\mu^0_2 r^2 - 4 A_{UV} \log(r/\epsilon)$ yields $A_{IR} \le A_{UV}$, i.e. the $a$-theorem, with the Markov property playing a crucial role. The framework unifies previous $c$-, $F$-, and $a$-theorems across dimensions and suggests extensions to higher dimensions and to relative entropy formulations, including potential defect generalizations of the $g$-theorem.

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.

Figures (1)

• Figure 1: Boosted and uniformly distributed circles lying on the null cone in $d=3$ spacetime dimensions. The vertical axis of the cone gives the time direction. A wiggly sphere, corresponding to one of the sets in (\ref{['ecu']}), is highlighted in black, and its corresponding limiting circle is highlighted in green.