# Is the vanishing commutator of observables outside the light cone only a necessary or also a sufficient condition for causality?

The equal-time commutator of observables in QFT has to vanish outside the light cone in order to ensure causality. Mathematically spoken, $[ \bar{\psi}(x)\Gamma_1\psi(x),\bar{\psi}(y)\Gamma_2\psi(y)]|_{x^0=y^0}=0$ for space-like separations $(x-y)^2<0$, where $\Gamma_{1,2}$ are e.g. the gamma matrices $\gamma^\mu$.

Is the vanishing commutator only a necessary or also a sufficient condition for causality? In other words, are there any theories in which the commutator vanishes outside the light cone, but causality is still violated?

The reverse question has already been asked before (but the given answer is not sufficient in my eyes): are there theories in which the commutator is non-vanishing outside the light cone but the theory is still causal?

• How do you define causality here, and also do you mean in flat space or not – Slereah Dec 12 '16 at 13:24