The quantum mechanical origin of the Klein–Gordon equation hides its capability to model many classical systems. We consider three examples of vibrating systems whose mathematical descriptions lead to the Klein–Gordon equation. These examples are adapted to applications such as the motion of suspended cables and Inca rope suspension bridges. We also discuss the correspondence between the classical and quantum settings of this equation as a way to provide an explanation of the concept of mass.

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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.