Boundary Causality Violating Metrics in Holography

TL;DR

This paper investigates how boundary causality is preserved in holography despite the inclusion of bulk metrics that violate causality, resolving a fundamental puzzle about boundary operator commutators.

Contribution

It demonstrates how the bulk path integral's boundary conditions naturally resolve causality violations in holographic theories.

Findings

Bulk path integral respects boundary causality despite including causality-violating metrics.

Boundary conditions in holography ensure vanishing commutators at spacelike separation.

The resolution maintains consistency of boundary operator algebra in holography.

Abstract

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 spacelike-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.