Abstract
We offer further evidence that discreteness of the sort inherent in a causal set cannot, in and of itself, serve to break Poincaré invariance. In particular we prove that a Poisson sprinkling of Minkowski spacetime cannot endow spacetime with a distinguished spatial or temporal orientation, or with a distinguished lattice of spacetime points, or with a distinguished lattice of timelike directions (corresponding respectively to breakings of reflection-invariance, translation-invariance, and Lorentz invariance). Along the way we provide a proof from first principles of the zero-one law on which our new arguments are based.
Highlights
We offer further evidence that discreteness of the sort inherent in a causal set cannot, in and of itself, serve to break Poincaré invariance
In particular we prove that a Poisson sprinkling of Minkowski spacetime cannot endow spacetime with a distinguished spatial or temporal orientation, or with a distinguished lattice of spacetime points, or with a distinguished lattice of timelike directions
Will a discrete structure prove to be the kinematical basis of quantum gravity and if so should we expect it to preserve the known symmetries of Minkowski spacetime, at least quasi-locally? One strand of thought has tended to answer these questions with ‘yes’ followed by ‘no’, and has held out effects like modi ed dispersion relations for electromagnetic waves as promising candidates for a phenomenology of spatiotemporal discreteness
Summary
Will a discrete structure prove to be the kinematical basis of quantum gravity and if so should we expect it to preserve the known symmetries of Minkowski spacetime, at least quasi-locally? One strand of thought has tended to answer these questions with ‘yes’ followed by ‘no’, and has held out effects like modi ed dispersion relations for electromagnetic waves as promising candidates for a phenomenology of spatiotemporal discreteness. F Dowker and R D Sorkin and in [2] it was proved rigorously that a ‘sprinkling’ of Minkowski spacetime induced by a Poisson process can determine a rest frame only with zero probability. This theorem, left open the possibility that a sprinkling, even if it could not remove all the symmetry of at spacetime, could cut it down to a proper subgroup H of the Poincaré group G. For further background on these questions we refer the reader to [1, 2]
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