Loop quantum gravity’s boundary maps. (English) Zbl 1482.83079
Summary: 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.
MSC:
| 83C57 | Black holes |
| 83C60 | Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism |
| 57R15 | Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) |
| 81T35 | Correspondence, duality, holography (AdS/CFT, gauge/gravity, etc.) |
| 58J32 | Boundary value problems on manifolds |
| 81P16 | Quantum state spaces, operational and probabilistic concepts |
| 68Q12 | Quantum algorithms and complexity in the theory of computing |
| 81P68 | Quantum computation |
Keywords:
loop quantum gravity; spin network; holography; bulk-boundary; quantum boundary condition; quantum circuit; quantum informationReferences:
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