On my website
www.hurbl.com I've written up some new results regarding
Kaluza-Klein geometry. The first paper (2009) presents the theory in the
simplest case (classical charged particles). These results are not new,
see Klein (1927) and Lichnerowicz (1955), but serve as introduction for
the new paper (2011).
This paper considers the stationary five-dimensional space with torsion.
This solves two problems: 1) Klein's constraint (55-component of the
metric is constant) is no longer an obstacle to fully covariant
equations, and 2) the Dirac equation can be formulated in the
five-dimensional space.
The torsion for 2) was introduced by Schouten and van Dantzig in 1933,
and later Jordan dropped the Klein constraint, which led to the modern
version of Kaluza-Klein theory, in which a new field must be considered
which does not have an exact equivalent in four dimensions.
Andreas