Geometric Origin of Physical Constants in a Kaluza-Klein Tetrad Model

TL;DR

This paper demonstrates how to incorporate bispinor fields into a Kaluza-Klein model using a tetrad formulation, ensuring that physical constants derive from geometry, revealing a special relation in the Lagrangian related to Dirac's equation.

Contribution

It introduces a method to include bispinor fields in Kaluza-Klein theories while satisfying key geometric and dimensional conditions, linking physical constants to geometry.

Findings

Bispinor fields can be integrated into Kaluza-Klein models with tetrad formulation.

Physical constants in the model originate from geometric properties of higher dimensions.

A special relation among Lagrangian terms is revealed, connected to Dirac's bispinor equation.

Abstract

An important feature of Kaluza-Klein theories is their ability to relate fundamental physical constants to the radii of higher dimensions. In previous Kaluza-Klein theory, which unifies the electromagnetic field with gravity as dimensionless components of a Kaluza-Klein metric, i) all fields have the same physical dimensions, ii) the Lagrangian has no explicit dependence on any physical constants except mass, and hence iii) all physical constants in the field equations except for mass originate from geometry. While it seems natural in Kaluza-Klein theory to add fermion fields by defining higher dimensional bispinor fields on the Kaluza-Klein manifold, these Kaluza-Klein theories do not satisfy conditions (i), (ii), and (iii). In this paper, we show how conditions (i), (ii), and (iii) can be satisfied by including bispinor fields in a tetrad formulation of the Kaluza-Klein model, as well as in an equivalent teleparallel model. This demonstrates an unexpected feature of Dirac's bispinor equation, since conditions (i), (ii), (iii) imply a special relation among the terms in the Kaluza-Klein or teleparallel Lagrangian that would not be satisfied in general.