18 - Action-Angle Variables in the Statistical Mechanics of the Sine-Gordon Field

Summary

INTRODUCTION

The sine-Gordon field in one space dimension and covariant form has Hamiltonian

It is well known that the spectral transform which solves the sine- Gordon equation

in light cone co-ordinates is one example of the now famous Zakharov-Shabat- Ablowitz-Kaup-Newell-Segur (ZS-AKNS) or Dirac spectral problem (see, for example, Bullough and Caudrey (1980) and especially the references to ZS and AKNS there). Indeed the solution of (1.3) by the Dirac spectral problem was published first by AKNS (1973a) before the more general case was described (AKNS, 1973b).

It is also well known that the spectral transform is a canonical transform (Flaschka and Newell, 1974; Dodd and Bullough, 1979, Faddeev, 1980). In the case of (1.3) new canonical co-ordinates can be found so that the ‘total momentum1 and H, in light cone co-ordinates (£,n),can be expressed in the forms

By taking the Lorentz covariant combinations etc., defining a new mass m=2m, and a new coupling constant then dropping this primed notation one finds (Bullough, 1980) that one can obtain the following Hamiltonian in covariant form

Evidently (1.5) is an equivalent of (1.1) under canonical transformation. The number M is a rest mass: M = Smy₀^-1. The kink solution

while a similar one can be made for the breather solution

which has energy

One would conclude from the results (1.6) that (1.5) is the sum of independent kink, antikink, breather and ‘radiation1 parts — the radiation part being the integral in (1.5) which is associated with the continuous part of the eigenspectrum of the spectral problem. However, the co-ordinates in (1.5) are actually collective co-ordinates with no indication of any interaction between them via phase shifts. In contrast, the multisoliton solutions of (1.2), which can be derived from (1.5) by inversion, contain explicit phase shifts induced between any pair of solitons, kinks, antikinks or breathers (see e.g. Caudrey, Eilbeck and Gibbon, 1975): these solitons also phase shift the small amplitude harmonic modes which, for small enough amplitude, become solutions of (1.2) (Rubinstein, 1970; Currie, 1977).