Proper time and conformal problem in Kaluza-Klein theory
E. Minguzzi
TL;DR
This paper addresses the conformal ambiguity in Kaluza-Klein theory by demonstrating how to remove extra terms in charged particle equations through conformal coupling, and fixes the conformal factor using the Klein-Gordon equation.
Contribution
It proves that charged particles can be coupled conformally to neutral particles to eliminate extra forces and resolves the conformal ambiguity by linking it to the Klein-Gordon equation's classical limit.
Findings
Extra terms in charged particle equations can be removed via conformal coupling.
The conformal factor is uniquely determined by the Klein-Gordon equation.
Each Fourier mode satisfies the Klein-Gordon equation even with a variable scalar field.
Abstract
In the traditional Kaluza-Klein theory, the cylinder condition and the constancy of the extra-dimensional radius (scalar field) imply that timelike geodesics on the 5-dimensional bundle project to solutions of the Lorentz force equation on spacetime. This property is lost for non constant scalar fields, in fact there appear new terms that have been interpreted mainly as new forces or as due to a variable inertial mass and/or charge. Here we prove that the additional terms can be removed if we assume that charged particles are coupled with the same spacetime conformal structure of neutral particles but through a different conformal factor. As a consequence, in Kaluza-Klein theory the proper time of the charged particle might depend on the charge-to-mass ratio and the scalar field. Then we show that the compatibility between the equation of the projected geodesic and the classical limit…
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