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Time-space duality in 2D quantum gravity. (English) Zbl 1484.83025

Summary: An important task faced by all approaches of quantum gravity is to incorporate superpositions and quantify quantum uncertainties of spacetime causal relations. We address this task in 2D. By identifying a global \(Z_2\) symmetry of 1 + 1D quantum gravity, we show that gravitational path integral configurations come in equal amplitude pairs with timelike and spacelike relations exchanged. As a consequence, any two points are equally probable to be timelike and spacelike separated in a Universe without boundary conditions. In the context of simplicial quantum gravity we identify a local symmetry of the action which shows that even with boundary conditions causal uncertainties are generically present. Depending on the boundary conditions, causal uncertainties can still be large and even maximal.

MSC:

83C45 Quantization of the gravitational field
83C40 Gravitational energy and conservation laws; groups of motions
62D20 Causal inference from observational studies
55U10 Simplicial sets and complexes in algebraic topology
81S40 Path integrals in quantum mechanics
83C27 Lattice gravity, Regge calculus and other discrete methods in general relativity and gravitational theory
81S07 Uncertainty relations, also entropic
58J32 Boundary value problems on manifolds
83C80 Analogues of general relativity in lower dimensions

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