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On the Structure of the Schild Group in Relativity Theory

Published online by Cambridge University Press:  20 November 2018

Gerd Jensen
Affiliation:
Sensburger Allee 22 a, D–14055 Berlin e-mail: [email protected]
Christian Pommerenke
Affiliation:
Institut für Mathematik, Technische Universität, D–10623 Berlin e-mail: [email protected]
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Abstract

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Alfred Schild has established conditions that Lorentz transformationsmap world-vectors $(ct,x,y,z)$ with integer coordinates onto vectors of the same kind. These transformations are called integral Lorentz transformations.

This paper contains supplements to our earlier work with a new focus on group theory. To relate the results to the familiar matrix group nomenclature, we associate Lorentz transformations with matrices in $\text{SL}(2,\mathbb{C})$. We consider the lattice of subgroups of the group originated in Schild's paper and obtain generating sets for the full group and its subgroups.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Baker, A., Matrix groups. An introduction to Lie group theory. Springer-Verlag, London, 2002. http://dx.doi.org/10.1007/978-1-4471-0183-3 Google Scholar
[2] Fine, B. and M. Newman, The normal subgroup structure of the Picard group. Trans. Amer. Math. Soc. 302(1987), no. 2, 769786. http://dx.doi.org/10.1090/S0002-9947-1987-0891646-3 Google Scholar
[3] Goldstein, H., Classical mechanics. 2nd edition. Addison-Wesley, Reading, MA, 1980. Google Scholar
[4] Jensen, G. and C. Pommerenke, Discrete space-time and Lorentz transformations. Canad Math. Bull. 59(2016), no. 2, 123135. http://dx.doi.org/10.41 53/CMB-2O1 5-066-4 Google Scholar
[5] LorenteandP, M.. Kramer, Representations of the discrete inhomogeneous Lorentz group and Dirac wave equation on the lattice. J. Phys. A 32(1999), 24812497. http://dx.doi.org/1 0.1088/0305-4470/32/12/01 9 Google Scholar
[6] Louck, J. D., A new parametrization and all integral realizations of the Lorentz group. J. Math. Phys. 43(2002), no. 10, 51085134. http://dx.doi.org/10.1063/1.1505124 Google Scholar
[7] Penrose, R. and W. Rindler, Spinors and space-time. Volume I. Cambridge University Press, Cambridge, 1984. http://dx.doi.org/10.1 01 7/CBO9780511 564048 Google Scholar
[8] Schild, A., Discrete space-time and integral Lorentz transformations. Canad. J. Math. 1(1949), 2947. http://dx.doi.org/! 0.41 53/CJM-1 949-003-4 Google Scholar