Ph. D. Thesis

Collective Coordinates for Nonlinear Klein-Gordon Field Theories

Over the last 200 years a powerful set of mathematical tools has been developed to attach linear problems, the most powerful being Fourier analysis. In the second half of the twentieth century, a nonlinear analog -- Inverse Scattering or Inverse Spectral Transform -- was discovered for a special type of nonlinear PDE. While inverse scattering is a very beautiful method, it involves relatively sophisticated mathematics, at least for an average physicist. For the uninitiated, the elegant mathematics yielded little intuitive understanding of the system and hence, one searched for a more simple set of ``collective coordinates''. The often observed fact that the soliton solutions to the nonperturbed nonlinear PDEs acted very much like particles lead to ansatz that the entire solution be written as the sum of a soliton whose center moves according to the dynamical variable X(t) plus a radiation or phonon field y(x,t) For nonlinear Klein-Gordon field theories this lead to a very nice result, namely that the soliton center of mass variable X(t) obeyed Newtonian mechanics, the effective potential being determined by the perturbation.

While this is indeed a satisfying result, especially for a physicist, the approach suffers from the deficiency that for many perturbations the radiation field y(x,t) did not satisfy the following ``well-behaved'' boundary conditions:

My Thesis removes this deficiency by adding a term to the ansatz which reflects a constant deformation of the field due to the perturbation. This allows the radiation term to have asymptotic behavior which is normal to such a field.

The development of a perturbation theory for nonlinear Klein-Gordon field theories given this new ansatz forms the basis of my thesis and may be found in Chapter 3. The following chapters further develop this perturbation

Various long calculations are relegated to the nine Appendices.

Unfortunately I did not save postscript versions of the figures and hence all of the PDF files contain room for the figures and captions, but no graphics. While having the figures is of course preferable, the vast majority of the thesis is analytic this, at least in my opinion, is far more interesting.