# Physics Notes by Jakob Schwichtenberg

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quantum_field_theory:notions_solitons

# Solitons

## Definitions

Classical finite-energy solutions are interesting because they include “solitons” solutions that will be discussed later. This does not necessarily mean that classical infinite-energy solutions are irrelevant to physics. For example, classically, a plane wave has infinite energy, but leads to single particle states of finite energy in a quantum theory.

page 56 in Quarks, Leptons & Gauge Fields BY K. Huang

instanton is a saddle point of the action • soliton is a saddle point of the energy

A sphaleron is a maximum of the energy functional in the direction of changing winding number, and a minimum in the orthogonal direction. Thus it is a saddle point. (page 336 in An introduction to gauge theories and modern particle physics by E. Leader and E. Predazzi.)

[An instanton] is much like a topological soliton in field theory, except that it is localized in time rather than in space

Before suggesting why the paper has been so often cited, it is appropriate to explain what the term soliton means. As coined by Zabusky and Kruskal, 1 this term is generic for special solitary wave solutions of certain nonlinear wave equations. What then is a solitary wave? It is a pulse-like wave that travels with constant speed and shape; the effects of dispersion on the wave shape are just balanced by those of nonlinearity. There is just enough yin for the yang; it is a dynamically self-sufficient object, a ‘thing.’ “Solitons are solitary waves that preserve their speeds and shapes after mutual collision. They play a role in the construction of complete solutions for the nonlinear wave equation 2 that corresponds to the role played by Fourier components in the construction of solutions for linear wave equations

===== Conditions for Finite Energy Solutions =====

These conditions are where all the topological stuff and large gauge transformation stuff comes from!

The idea is to investigate Asymptotic Symmetries. These, are the set of all gauge transformation that respect some given boundary conditions modulo those gauge transformations that act trivially on all physical data. By looking for finite energy solutions, we derive boundary conditions. Then we investigate gauge transformations that respect these and ignore the boring ones (the “small” ones) that do not change anything. The remaining set of gauge transformation does indeed change something physical and thus are not really symmetries.

The conditions for finite energy solutions are given at page 94 in Quarks, Leptons & Gauge Fields by K. Huang:

\begin{align} F^{\mu \nu}(x) \rightarrow^{r\to \infty} O(r^{-2}), \quad r \equiv |x|, \notag \\ A^\mu(x) \rightarrow^{r\to \infty} -\frac{i}{g} U \partial^\mu U^{-1}+O(r^{-1}), \notag \\ D^\mu \phi(x) \rightarrow^{r\to \infty} O(r^{-2}), \notag \\ \phi(x) \rightarrow^{r\to \infty} \rho(x)+O(r^{-2}), \end{align} where $\rho(x)$ denotes minima of the Higgs potential.

See also page 55 in Quarks, Leptons & Gauge Fields by K. Huang!

<WRAP tip> The idea behind these boundary conditions is the following:

When we consider a large sphere, as we usually do, its surface grows as $r^2$. Thus, when we want that the energy flux is finite, we must demand that it decays as $1/r^2$ when we go to infinity. (Source: page 35 here) <WRAP>