Quantization of gauge fields through fibre bundle multiconnectivity

Abstract

A geometric model for the quantum nature of interaction fields is proposed. We utilize a trivial fibre bundle whose typical fibre has a multiconnectivity characterized by a discrete group Γ. By seeing Γ as a gauge group with global action on each fibre, we show that the corresponding field strength is non-zero only on the future part of the light cone whose vertex is at the interaction point. When the interaction is submitted to the symmetries of a Lie group G, we consider the gauge group G × Γ. The field strength of the gauge having this group includes a term expressing the quantization of the interaction field described by G. This geometric interpretation of quantization makes use of topological arguments similar to those applied to explain the AharonovBohm effect. Two examples show how this interpretation applies to the cases of electromagnetic and gravitational fields.

Discretization and Interaction of Fields via the Classical Kaluza-Klein Theory

“…According to [5], the second term on the right hand side of Equation (5) corresponds to a point of space-time where a change of horizontal section occurs in the hyperspace-time fibre bundle. This change thus takes place optimally in the sense that it is carried out in the direction of a gradient.…”

“…A more complete geometric modeling of electromagnetism must also allow expressing the discontinuity of certain functions associated with an electrically charged particle, such as its energy when it is involved in an interaction. To model this discontinuous variation, we shall postulate that the geometric structure of the fifth dimension has a global symmetry making it multiconnected [3][4][5]. The fifth dimension then decomposes into equal length intervals, each topologically equivalent to the circle S…”