Schwarzschild field inn dimensions and the dimensionality of space problem

See, for example, the very interesting monograph byR. Weitzenböck:Der Vierdimensionale Raum (Braunschweig, 1929).

A slightly different statement of this proposition is customarily attributed toJ. Überweg:System der Logik (various editions, Bonn, 1857–1882), although it was probably known toG. Green and other mathematicians who studied problem inn dimensions somewhat earlier

P. Ehrenfest:Proc. Amsterdam Acad.,20, 200 (1917);Ann. Physik,61, 440 (1920).

For example, no reference is made to Ehrenfest’s work in the recent historical treatment ofK. Jammer:Concepts of Space (Cambridge, 1954); nor in the discussion ofH. Weyl:Philosophy of Mathematics and Natural Science (Princeton, 1949), p. 136.

G. J. Whitrow:The Structure and Evolution of the Universe (New York, 1959) Appendix.

The modern topological theory of dimensionality begins with Poincaré’s essay (1912),Pourquoi l’Espace a Trois Dimensions, Dernières Pensées (Paris, 1926). For later developments seeK. Menger:Dimension Theorie (Leipzig, 1926) Chapter II.W. Hurewicz andH. Wallman:Dimension Theory (Princeton, 1941).

F. R. Tangherlini:Phys. Rev. Lett.,6, 147 (1961).

For the energy levels see Ehrenfest’s paper —to be found in his Collected Scientific Papers (Amsterdam, 1959), p. 400.

See,e.g.,A. Sommerfeld:Partial Differential Equations in Physics (New York, 1949), Appendix IV.

It may be of interest to supplement these investigations with a similar analysis for spinor wave equations inn dimensions as given byR. Brauer andH. Weyl:Am. Journ. Mat.,57, 425 (1935).

J. Callaway:Phys. Rev.,112, 290, (1958).

A. Peres:Phys. Rev.,120, 1044 (1960).

É. Cartan:Oeuvres Complètes (Paris, 1952), part I, vol. I, p. 112.

A. Einstein:Geometry and Experience, to be found, for example inIdeas and Opinions (New York, 1954).

At least to within the limits set by the proton stability experiment ofC. C. Giamiti andF. Reines:Phys. Rev.,126, 2178 (1962).

H. Lehmann, K. Symanzik andW. Zimmermann:Nuovo Cimento,1, 1425 (1955). For a recent discussion seeW. Thirring:Phys Rev.,126, 1209 (1962).