Subadditive average distances and quantum promptness

• Published in: Classical and quantum gravity 2023-08, Vol.40 (16), p.165013
• Main Authors: Piazza, Federico, Tolley, Andrew J
• Format: Article
• Language: English
• Subjects: average distance; causality; emergent geometry; General Relativity and Quantum Cosmology; High Energy Physics - Theory; Physics; quantum gravity
• ISSN: 0264-9381; 1361-6382
• DOI: 10.1088/1361-6382/ace583

A central property of a classical geometry is that the geodesic distance between two events is additive . When considering quantum fluctuations in the metric or a quantum or statistical superposition of different spacetimes, additivity is generically lost at the level of expectation values. In the presence of a superposition of metrics, distances can be made diffeomorphism invariant by considering the frame of a family of free-falling observers or a pressureless fluid, provided we work at sufficiently low energies. We propose to use the average squared distance between two events ⟨ d 2 ( x , y ) ⟩ as a proxy for understanding the effective quantum (or statistical) geometry and the emergent causal relations among such observers. At each point, the average squared distance ⟨ d 2 ( x , y ) ⟩ defines an average metric tensor. However, due to non-additivity, ⟨ d 2 ( x , y ) ⟩ is not the (squared) geodesic distance associated with it. We show that departures from additivity can be conveniently captured by a bi-local quantity C ( x , y ) . Violations of additivity build up with the mutual separation between x and y and can correspond to C   0 (superadditive). We show that average Euclidean distances are always subadditive: they satisfy the triangle inequality but generally fail to saturate it. In Lorentzian signature there is no definite result about the sign of C , most physical examples give C