Loop Quantum Gravity's Boundary Maps

TL;DR

This paper develops a mathematical framework in loop quantum gravity to relate bulk quantum states to boundary states, enabling boundary-to-bulk reconstruction and analysis of correlations in quantum geometry.

Contribution

It introduces a boundary map formalism in loop quantum gravity, establishing a method for bulk reconstruction from boundary states and analyzing boundary density matrices.

Findings

Established a boundary-to-bulk reconstruction procedure.

Defined boundary density matrices from bulk spin network states.

Analyzed how bulk correlations are reflected in boundary states.

Abstract

In canonical quantum gravity, the presence of spatial boundaries naturally leads to a boundary quantum states, representing quantum boundary conditions for the bulk fields. As a consequence, quantum states of the bulk geometry needs to be upgraded to wave-functions valued in the boundary Hilbert space: the bulk become quantum operator acting on boundary states. We apply this to loop quantum gravity and describe spin networks with 2d boundary as wave-functions mapping bulk holonomies to spin states on the boundary. This sets the bulk-boundary relation in a clear mathematical framework, which allows to define the boundary density matrix induced by a bulk spin network states after tracing out the bulk degrees of freedom. We ask the question of the bulk reconstruction and prove a boundary-to-bulk universal reconstruction procedure, to be understood as a purification of the mixed boundary state into a pure bulk state. We further perform a first investigation in the algebraic structure of induced boundary density matrices and show how correlations between bulk excitations, i.e. quanta of 3d geometry, get reflected into the boundary density matrix.