Path integral optimization from Hartle-Hawking wave function

Jan Boruch,Tadashi Takayanagi,Pawel Caputa

doi:10.1103/physrevd.103.046017

Abstract

We propose a gravity dual description of the path-integral optimization in conformal field theories arXiv:1703.00456, using Hartle-Hawking wave functions in anti-de Sitter spacetimes. We show that the maximization of the Hartle-Hawking wave function is equivalent to the path-integral optimization procedure. Namely, the variation of the wave function leads to a constraint, equivalent to the Neumann boundary condition on a bulk slice, whose classical solutions reproduce metrics from the path-integral optimization in conformal field theories. After taking the boundary limit of the semi-classical Hartle-Hawking wave function, we reproduce the path-integral complexity action in two dimensions as well as its higher and lower dimensional generalizations. We also discuss an emergence of holographic time from conformal field theory path-integrals.

Highlights

The anti–de Sitter (AdS)/conformal field theories (CFTs) correspondence [1] provides us with a surprising relation between gravity and quantum manybody systems
We show that the maximization of the Hartle-Hawking wave function is equivalent to the path integral optimization procedure
Note that the above construction assumes that a gravity solution which calculates the Hartle-Hawking wave function is given by a subregion in a Poincare AdS geometry

Summary

INTRODUCTION

The AdS/CFT correspondence [1] provides us with a surprising relation between gravity and quantum manybody systems. The idea of path integral optimization is to coarse grain the discretization as much as possible, which makes computational costs minimal, while keeping the correct answer to the final wave functional. The minimization procedure picks up the most efficient discretization of path integral which leads to the correct vacuum state This method was generalized to various CFT setups in [32,33,34]. A direct connection between the path integral optimization and AdS/CFT has remained an open problem Another subtle issue is that, in the solution (7), we find ð∂iφÞ2 and e2φ are of the same order, which does not satisfy the criterion (6).

HARTLE-HAWKING WAVE FUNCTION WITH BOUNDARY

SOLUTIONS

EXCITED STATES

CONCLUSIONS AND DISCUSSION