Description:
<jats:title>A<jats:sc>bstract</jats:sc>
</jats:title>
<jats:p>Rindler positivity is a property that holds in any relativistic Quantum Field Theory and implies an infinite set of inequalities involving the exponential of the Rényi mutual information <jats:italic>I</jats:italic>
<jats:sub>
<jats:italic>n</jats:italic>
</jats:sub> (<jats:italic>A</jats:italic>
<jats:sub>
<jats:italic>i</jats:italic>
</jats:sub>
<jats:italic>,</jats:italic>
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$ \overline{A} $$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mover>
<mml:mi>A</mml:mi>
<mml:mo>¯</mml:mo>
</mml:mover>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
<jats:sub>
<jats:italic>j</jats:italic>
</jats:sub>) between <jats:italic>A</jats:italic>
<jats:sub>
<jats:italic>i</jats:italic>
</jats:sub> and <jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$ \overline{A} $$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mover>
<mml:mi>A</mml:mi>
<mml:mo>¯</mml:mo>
</mml:mover>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
<jats:sub>
<jats:italic>j</jats:italic>
</jats:sub>, where <jats:italic>A</jats:italic>
<jats:sub>
<jats:italic>i</jats:italic>
</jats:sub> is a spacelike region in the right Rindler wedge and <jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$ \overline{A} $$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mover>
<mml:mi>A</mml:mi>
<mml:mo>¯</mml:mo>
</mml:mover>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
<jats:sub>
<jats:italic>j</jats:italic>
</jats:sub> is the wedge reflection of <jats:italic>A</jats:italic>
<jats:sub>
<jats:italic>j</jats:italic>
</jats:sub>. We explore these inequalities in order to get local inequalities for <jats:italic>I</jats:italic>
<jats:sub>
<jats:italic>n</jats:italic>
</jats:sub> (<jats:italic>A,</jats:italic>
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$ \overline{A} $$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mover>
<mml:mi>A</mml:mi>
<mml:mo>¯</mml:mo>
</mml:mover>
</mml:math>
</jats:alternatives>
</jats:inline-formula>) as a function of the distance between <jats:italic>A</jats:italic> and its mirror region <jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$ \overline{A} $$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mover>
<mml:mi>A</mml:mi>
<mml:mo>¯</mml:mo>
</mml:mover>
</mml:math>
</jats:alternatives>
</jats:inline-formula>. We show that the assumption, based on the cluster property of the vacuum, that <jats:italic>I</jats:italic>
<jats:sub>
<jats:italic>n</jats:italic>
</jats:sub> goes to zero when the distance goes to infinity, implies the more stringent and simple condition that <jats:italic>F</jats:italic>
<jats:sub>
<jats:italic>n</jats:italic>
</jats:sub>
<jats:italic>≡ e</jats:italic>
<jats:sup>(<jats:italic>n−</jats:italic>1)<jats:italic>I</jats:italic>
</jats:sup>
<jats:sub>
<jats:italic>n</jats:italic>
</jats:sub> should be a completely monotonic function of the distance, meaning that all the even (odd) derivatives are non-negative (non-positive). In the case of a CFT, we show that conformal invariance implies stronger conditions, including a sort of monotonicity of the Rényi mutual information for pairs of balls. An application of these inequalities to obtain constraints for the OPE coefficients of the 4-point function of certain twist operators is also discussed.</jats:p>