Even for holographic theories that obey boundary causality, the full bulk Lorentzian path integral includes metrics that violate this condition. This leads to the following puzzle: the commutator of two field theory operators at space-like-separated points on the boundary must vanish. However, if these points are causally related in a bulk metric, then the bulk calculation of the commutator will be nonzero. It would appear that the integral over all metrics of this commutator must vanish exactly for holography to hold. This is puzzling since it must also be true if the commutator is multiplied by any other operator. Upon a careful treatment of boundary conditions in holography, we show how the bulk path integral leads to a natural resolution of this puzzle.

• Received 29 April 2021
• Revised 22 June 2021
• Accepted 20 July 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.081603

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Published by the American Physical Society