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The appearance of linear spaces, describing physical quantities by vectorsand tensors, is ubiquitous in all of physics, from classical mechanics to themodern notion of local Lorentz invariance. However, as natural as this seems tothe physicist, most computer scientists would argue that something like a"local linear tangent space" is not very typical and in fact a quite surprisingproperty of any conceivable world or algorithm. In this paper, we take theperspective of the computer scientist seriously, and ask whether there could beany inherently information-theoretic reason to expect this notion of linearityto appear in physics. We give a series of simple arguments, spanning quantuminformation theory, group representation theory, and renormalization in quantumgravity, that supports a surprising thesis: namely, that the local linearity ofspace-time might ultimately be a consequence of the linearity of probabilities.While our arguments involve a fair amount of speculation, they have the virtueof being independent of any detailed assumptions on quantum gravity, and theyare in harmony with several independent recent ideas on emergent space-time inhigh-energy physics.

Is the local linearity of space-time inherited from the linearity of probabilities?