Mini-curso

The Pilot Wave Theory and Soliton-Like Solutions

Prof. Vieri Benci
Università di Pisa

11 e 13 de agosto de 1998

Artigos disponíveis

Solitons on Manifolds dvi ps
Solitons and Relativistic Dynamics dvi ps
Soliton like solutions of a Lorentz invariant equation in dimention 3 dvi ps
Solitons and the Electromagnetic Field dvi ps
Solitons and Particles dvi ps
Existence of String-like Solitons dvi ps
Solitons and the Pilot Wave Theory dvi ps
A New Variational Principle for the Fundamental Equations of Classical Physics dvi ps

Resumo

Sometimes the phenomena arising from nonlinear equations exhibit some feature which might appear strange and even paradoxical. Thus, it is possible that many strange facts of nature can be explained by nonlinear models without requiring new principles.

The subject of thse lectures is part of a program which has the aim to investigate which phenomena can be explained by nonlinear effects in Classical Mechanics and which ones require the new axioms of Quantum Mechanics.

The model which we consider is related to the De Broglie-Bohm pilot wave theory; the existence of such a pilot wave was considered impossible by many of his contemporaries both for Physical and Mathematical reasons.

For the discussion of the Physics we refer to the beautiful work of Bell ``On the impossible pilot wave'' which inspired the title of this conference. Here, we discuss some of the related mathematics.

In these lectures we present a classic model which exhibit some of the features of Quantum Mechanics. By ''classic model'', we mean a field described by a partial differential equation of the second order, defined in the classical space-time (i.e. $\bf {R}^{4}$ ) and invariant for the Poincaré group. We assume that this equation is nonlinear; this nonlinearity is responsable of the existence of soliton-like solutions.

Roughly speaking a soliton is a solution of a field equation whose energy travels as a localized packet and which preserves its form under perturbations. In this respect solitons have a particle-like behavior. A solution of our equation can be written as follows:

$\begin{displaymath}u(x,t)=\psi (x,t)+\delta _{\varepsilon }\left( x-\bf {q}(t)\right)\end{displaymath}$

Considering this decomposition the solutions to our equation can be thought as a combination of a wave and a particle.

A solution of our equation can be written as follows

$\begin{displaymath}u(x,t)=\psi (x,t)+\delta _{\varepsilon }\left( x-\bf {q}(t)\right)\end{displaymath}$

where $\psi (x,t)$ can be considered as a wave and $\delta_{\varepsilon}\left( x-\bf {q}(t)\right)$ is our soliton: a bump of energy concentrated around the point $\bf {q}(t)$ in a ball of radius of the order $\varepsilon .$ Considering this decomposition the solutions to our equation can be thought as a combination of a wave and a particle. $\varepsilon$ occurs in the equation as a parameter and it will be interpreted as the radius of the soliton; an other parameter occurring in our equation is c, the speed of light.

From the technical point of view, these lectures use the usual Variational and Toplogial methods of Nonlinear Analysis.

About this document ...