[Paper Review] Constructing Dualities from Quantum State Manifolds

Van Zyl|arXiv (Cornell University)|Sep 2, 2015

Black Holes and Theoretical Physics83 references3 citations

TL;DR

This paper presents a systematic method to construct semi-classical gravitational duals from quantum state manifolds in simple quantum systems, recovering key concepts of gauge/gravity duality naturally. It establishes explicit dictionaries for both low- and higher-dimensional cases, with the latter requiring many bulk coordinates and suggesting directions for future alignment with existing literature.

ABSTRACT

The thesis develops a systematic procedure to construct semi-classical gravitational duals from quantum state manifolds. Though the systems investigated are simple quantum mechanical systems without gauge symmetry many familiar concepts from the conventional gauge/gravity duality come about in a very natural way. The investigation of the low-dimensional manifolds link existing results in the $AdS_2/CFT_1$ literature. We are able to extend these in various ways and provide an explicit dictionary. The higher dimensional investigation is also concluded with a simple dictionary, but this dictionary requires the inclusion of many bulk coordinates. Consequently further work is needed to relate these results to existing literature. Possible ways to achieve this are discussed.

Motivation & Objective

• To develop a systematic procedure for deriving semi-classical gravitational duals from quantum state manifolds.
• To explore how familiar gauge/gravity duality concepts arise naturally in simple quantum systems without gauge symmetry.
• To extend existing results in the AdS2/CFT1 literature through explicit constructions of dual dictionaries.
• To investigate higher-dimensional quantum state manifolds and derive a corresponding dual dictionary, despite the need for many bulk coordinates.
• To identify pathways for connecting the derived duals to established literature in the broader gauge/gravity duality framework.

Proposed method

• The study constructs gravitational duals by analyzing the geometric structure of quantum state manifolds in simple quantum mechanical systems.
• It employs differential geometry to extract the metric and curvature of the state manifold, linking them to bulk gravitational dynamics.
• The procedure systematically maps quantum state parameters to bulk fields, forming a dictionary between boundary quantum data and bulk geometry.
• For low-dimensional cases, the method reproduces known AdS2/CFT1 results, validating the approach.
• In higher dimensions, the method generates a dual dictionary but requires an increased number of bulk coordinates, complicating direct comparison with existing literature.
• The paper discusses potential extensions to reduce bulk coordinate proliferation and align results with conventional gauge/gravity duality frameworks.

Experimental results

Research questions

• RQ1 How can gravitational duals be systematically derived from quantum state manifolds in the absence of gauge symmetry?
• RQ2 To what extent do standard gauge/gravity duality concepts emerge naturally from the geometry of quantum state manifolds?
• RQ3 How can the derived dual dictionary for higher-dimensional state manifolds be related to existing literature in the gauge/gravity duality program?
• RQ4 What role do bulk coordinate counts play in the feasibility and interpretation of the constructed duals?
• RQ5 What modifications or assumptions are needed to align the higher-dimensional duals with established gravitational dual descriptions?

Key findings

• The method successfully constructs semi-classical gravitational duals from quantum state manifolds in simple systems, recovering core features of gauge/gravity duality without requiring gauge symmetry.
• In low-dimensional cases, the construction reproduces known AdS2/CFT1 results, validating the approach through consistency with existing literature.
• An explicit dictionary is established for the AdS2/CFT1 case, linking quantum state manifold geometry to bulk gravitational fields.
• For higher-dimensional manifolds, a dual dictionary is derived, but it requires a large number of bulk coordinates, complicating direct comparison with standard formulations.
• The paper identifies potential strategies to reduce bulk coordinate dependence and align the results with conventional gauge/gravity duality frameworks.