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# Discrete Space-time and Lorentz Transformations

Published:2015-10-21
Printed: Mar 2016
• Gerd Jensen,
Sensburger Allee 22 a , D--14055 Berlin
• Christian Pommerenke,
Institut für Mathematik , Technische Universität, D--10623 Berlin
 Format: LaTeX MathJax PDF

## Abstract

Alfred Schild has established conditions that Lorentz transformations map world-vectors $(ct,x,y,z)$ with integer coordinates onto vectors of the same kind. The problem was dealt with in the context of tensor and spinor calculus. Due to Schild's number-theoretic arguments, the subject is also interesting when isolated from its physical background. The paper of Schild is not easy to understand. Therefore we first present a streamlined version of his proof which is based on the use of null vectors. Then we present a purely algebraic proof that is somewhat shorter. Both proofs rely on the properties of Gaussian integers.
 Keywords: Lorentz transformation, integer lattice, Gaussian integers
 MSC Classifications: 22E43 - Structure and representation of the Lorentz group 20H99 - None of the above, but in this section 83A05 - Special relativity

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