From Euclidean Sources to Lorentzian Spacetimes in Holographic Conformal Field Theories

TL;DR

This paper explores how Euclidean path-integral states in holographic conformal field theories can generate a wide range of bulk Lorentzian spacetimes, providing explicit formulas and analyzing their limitations.

Contribution

It extends previous work by deriving explicit first-order formulas for Lorentzian fields from Euclidean sources in arbitrary dimensions, and analyzes the conditions for generating localized initial data.

Findings

Arbitrary bulk initial data can be generated by Euclidean sources at linear order.

Localized initial data requires small amplitude sources to avoid divergence.

Perturbative approach is valid for small sources, but diverges for highly localized data.

Abstract

We consider states of holographic conformal field theories constructed by adding sources for local operators in the Euclidean path integral, with the aim of investigating the extent to which arbitrary bulk coherent states can be represented by such Euclidean path-integrals in the CFT. We construct the associated dual Lorentzian spacetimes perturbatively in the sources. Extending earlier work, we provide explicit formulae for the Lorentzian fields to first order in the sources for general scalar field and metric perturbations in arbitrary dimensions. We check the results by holographically computing the Lorentzian one-point functions for the sourced operators and comparing with a direct CFT calculation. We present evidence that at the linearized level, arbitrary bulk initial data profiles can be generated by an appropriate choice of Euclidean sources. However, in order to produce initial data that is very localized, the amplitude must be taken small at the same time otherwise the required sources diverge, invalidating the perturbative approach.