Markov Property of the Conformal Field Theory Vacuum and the a Theorem.

We use strong subadditivity of entanglement entropy, Lorentz invariance, and the Markov property of the vacuum state of a conformal field theory to give 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.