On the nature of ensembles from gravitational path integrals

Abstract Spacetime wormholes in gravitational path integrals have long been known to lead to interpretations in terms of an ensemble of theories. Here we probe the question of precisely what sort of theories such ensembles might contain. Careful consideration of a simple d = 2 topological model indicates that the Hilbert space structure of a general ensemble element fails to factorize over disconnected Cauchy-surface boundaries, and in particular that its Hilbert space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">H</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>S</mml:mi> <mml:mi>∂</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>S</mml:mi> <mml:mi>∂</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> Cauchy-surface boundaries fails to be positive definite when the number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>S</mml:mi> <mml:mi>∂</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> of disconnected such boundaries is large. This observation then suggests a generalization of the AdS/CFT correspondence in which a bulk gravitational theory is dual to an ensemble of theories that deviate from standard CFTs by violating both locality and positivity (at least under certain circumstances). Since violations of positivity are undesirable, we propose that positivity-violating elements of the ensemble be removed when studying physics in asymptotically AdS spacetimes (or in other contexts in which Cauchy surfaces have asymptotic boundaries), perhaps reducing the ensemble to a single standard CFT. Nevertheless, properties of any remaining CFTs that are uncorrelated with positivity of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">H</mml:mi> </mml:mrow> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>S</mml:mi> <mml:mi>∂</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> at large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>S</mml:mi> <mml:mi>∂</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> will agree with those of typical elements of the full ensemble and may be computed using the ensemble average. On the other hand, elements that violate positivity at large <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>S</mml:mi> <mml:mi>∂</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> </mml:math> can still have a positive-definite cosmological sector with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow> <mml:mi>C</mml:mi> <mml:mi>S</mml:mi> <mml:mi>∂</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . The collection of such elements then defines a basis for a Hilbert space describing such cosmologies. In contrast to the cases in which Cauchy-surfaces are allowed to have boundaries, we argue that the resulting Hilbert space need not decohere into single-state theories. As a result, familiar physics might be more easily recovered from this new scenario.