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Discrete Space-time and Integral LorentzTransformations

Published online by Cambridge University Press:  20 November 2018

Alfred Schild
Alfred Schild*
Affiliation:
University of Toronto
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Modern physical theory, both classical and quantal, faces serious difficulties which arise from the divergence of certain integrals. Perhaps the best known of these “infinities” is the self-energy of the point electron. Most of the simpler devices used to eliminate the infinities, such as the introduction of a finite electron radius, are non-relativistic and must therefore be rejected. Relativistic theories1 which do avoid some or all of the infinities are very complicated and often suffer from difficulty in physical interpretation.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 1 , 01 February 1949 , pp. 29 - 47
Copyright
Copyright © Canadian Mathematical Society 1949

References

1 Wentzel, G., Rev. Mod. Phys., vol. 19 (1947), 1-18.Google Scholar

Ambarzumian, V. and Iwanenko, D., Z. f. Phys., vol. 64 (1930), 563-567;Google Scholar L. Silberstein, “Discrete Space-Time,” University of Toronto Studies, Physics Series (1936). For a short outline of the present paper, see Phys. Rev.,vol. 73 (1948), 414- 415.

3 “Hypercubic” would be the appropriate adjective—but we shall retain the shorter form.

4 Heisenberg, Cf. W., Ann. Phys., vol. 32 (1938), 20–33.Google Scholar

5 Schrödinger, E., Sitz. Ber. Preuss. Akad. Wiss., vol. 24 (1930), 418-428.Google Scholar

6 Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers (Oxford, 1938), 184, Theorem 215.Google Scholar

7 Hardy, and Wright, , Theory of Numbers, 219, Theorem 252.Google Scholar

8 Laporte, O. and Uhlenbeck, G. E., Phys. Rev., vol. 37 (1931), 1381; Infeld, L., Phys. Zeitsckrift, vol. 33 (1932), 475. For an early use of a similar technique see also Goursat, E., Ann. École Norm. (3), vol. 6 (1889), 20, § 5.Google Scholar

9 O. Veblen and J.|von Neumann, “Geometry of Complex Domains,” Institute for Advanced Study mimeographed notes (Princeton, 1936).

10 I2 + m2 + n2 is not necessarily 1.

11 By saying that a set of vectors is spatially dense, we mean, more precisely, that the directions of the spatial projections of the vectors in the set are dense. This remark applies also to Sec. 8.