Claude Gauthier

Classical applications of the Klein–Gordon equation

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.

A numerical model for the 3-D non-linear vibrations of an N-string

We present a non-linear numerical model describing the 3-D vibrations of a planar network of N sections of string which are connected together at one common mobile extremity. We call such a network N-string. For small-amplitude vibrations perpendicular to the N-string equilibrium plane, the numerical results coincide with the already known analytical solutions of the linear model. This non-linear model makes it possible to describe small- or large-amplitude 3-D vibrations of any kind of N-string subjected to an initial plucking. The equations of motion are also presented in a dimensionless form and a vast dimensionless physical parameter space is identified. The numerical model can be extended to more complex networks of strings.

Quantization of gauge fields through fibre bundle multiconnectivity

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 x 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 Aharonov-Bohm effect. Two examples show how this interpretation applies to the cases of electromagnetic and gravitational fields.

Conditions for periodic vibrations in a symmetric n-string

A symmetric N-string is a network of N ≥ 2 sections of string tied together at one common mobile extremity. In their equilibrium position, the sections of string form N angles of 2π/N at their junction point. Considering the initial and boundary value problem for small-amplitude oscillations perpendicular to the plane of the N-string at rest, we obtain conditions under which the solution will be periodic with an integral period.

On the periodicity of trigonometric functions generalized to quotient rings of R[x]

We apply a method of Euler to algebraic extensions of sets of numbers with compound additive inverse which can be seen as quotient rings of R[x]. This allows us to evaluate a generalization of Riemann’s zeta function in terms of the period of a function which generalizes the function sin z. It follows that the functions generalizing the trigonometric functions on these sets of numbers are not periodic.

MIRROR SYMMETRY AND CONFORMAL FLATNESS IN GENERAL RELATIVITY

Using symmetry arguments only, we show that every spacetime with mirror-symmetric spatial sections is necessarily conformally flat. The general form of the Ricci tensor of such spacetimes is also determined.