Loop quantum gravity's boundary maps

In canonical quantum gravity, the presence of spatial boundaries naturally leads to boundary quantum states, representing quantum boundary conditions for the bulk fields. As a consequence, quantum states of the bulk geometry need to be upgraded to wave-functions valued in the boundary Hilbert space: the bulk becomes 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.

Automated summary

This study explores how canonical quantum gravity establishes the relationship between volume and boundary by upgrading the quantum states of volume geometry to wave-functions valued in the boundary Hilbert space. Through tracing out volume degrees of freedom and performing a universal reconstruction procedure from boundary to bulk, correlations between bulk excitations are investigated in the induced boundary density matrices.