Discrete Space-time and Lorentz Transformations
Printed: Mar 2016
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.
Lorentz transformation, integer lattice, Gaussian integers
22E43 - Structure and representation of the Lorentz group
20H99 - None of the above, but in this section
83A05 - Special relativity