On the Dimensionality of the Real World

Abstract

The ``strength'' of systems of field equations is defined following a procedure due to Einstein. For an arbitrary dimension number, the strength of Einstein's field equations and of Maxwell's equations are calculated, and it is shown that only in four dimensions do the two systems of field equations determine their respective fields equally strongly. In addition, in four dimensions, Weyl's equation exhibits the same strength as the Einstein or Maxwell fields. The results are used to show that a necessary condition for a unified field-theoretic description of gravity and electromagnetism is that the world be four dimensional. If one also demands that the neutrino field be determined as strongly as the Einstein and Maxwell fields, one must have a two-component neutrino, rather than a four-component neutrino.