Discreteness without symmetry breaking: a theorem. (English) Zbl 1179.83012

Summary: This paper concerns random sprinklings of points into Minkowski spacetime (Poisson processes). It proves that there exists no equivariant measurable map from sprinklings to spacetime directions (even locally). Therefore, if a discrete structure is associated to a sprinkling in an intrinsic manner, then the structure will not pick out a preferred frame, locally or globally. This implies that the discreteness of a sprinkled causal set will not give rise to “Lorentz breaking” effects like modified dispersion relations. Another consequence is that there is no way to associate a finite-valency graph to a sprinkling consistently with Lorentz invariance.


83A05 Special relativity
83C45 Quantization of the gravitational field
Full Text: DOI arXiv


[1] DOI: 10.1103/PhysRevLett.59.521
[2] Sorkin R. D., Lectures on Quantum Gravity (2003)
[3] Henson J., Approaches to Quantum Gravity: Toward a New Understanding of Space, Time and Matter (2009)
[4] DOI: 10.1002/andp.200410144 · Zbl 1080.83008
[5] DOI: 10.1142/S0217732304015026
[6] DOI: 10.1103/PhysRevD.67.064019 · Zbl 1222.83073
[7] Livine E. R., JHEP 06 pp 050–
[8] DOI: 10.1016/0550-3213(82)90222-X
[9] Stoyan D., Stochastic Geometry and Its Applications (1995) · Zbl 0838.60002
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