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Dear Dr B: Why is Lorentz-invariance in conflict with discreteness?

20 Apr 2016, 11:03 UTC
Dear Dr B: Why is Lorentz-invariance in conflict with discreteness?
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Can we build up space-time fromdiscrete entities?“Could you elaborate (even) more on […] the exact tension between Lorentz invariance and attempts for discretisation?Best,Noa”Dear Noa: Discretization is a common procedure to deal with infinities. Since quantum mechanics relates large energies to short (wave) lengths, introducing a shortest possible distance corresponds to cutting off momentum integrals. This can remove infinites that come in at large momenta (or, as the physicists say “in the UV”). Such hard cut-off procedures were quite common in the early days of quantum field theory. They have since been replaced with more sophisticated regulation procedures, but these don’t work for quantum gravity. Hence it lies at hand to use discretization to get rid of the infinities that plague quantum gravity.Lorentz-invariance is the symmetry of Special Relativity; it tells us how observables transform from one reference frame to another. Certain types of observables, called “scalars,” don’t change at all. In general, observables do change, but they do so under a well-defined procedure that is by the application of Lorentz-transformations.We call these “covariant.” (Or at least we should. Most often invariance is conflated with covariance in the literature.)(To be precise, Lorentz-covariance isn’t the full symmetry of Special Relativity because there ...

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