Charge fluctuations in fractionally charged excitations

The Dirac equation in the presence of the simplest kink-antiking scalar potential -x> is studied. The existence of the degenerate soliton and antisoliton excitations at zero energy is shown. The charge operator is defined and the charge of an antisoliton is shown to be +-,. In order to find out whether this charge is a sharp quantum observable, the mean square charge fluctuation about this value is calculated in the presence and in the absence of an antisoliton. Although both values are found to be logarithmically divergent, their difference quickly approaches zero as the two antisolitons get further apart. The implications of this result are discussed. ACKNOWLEDGEMENT I sincerely thank Professor Arthur K. Kerman for suggesting this problem and for his kind guidance throughout its progress. I enjoyed the opportunity to learn about physics from someone who makes it so challenging and interesting. INTRODUCTION For a number of years theoretical physicists have exhibited interest in fractionally charged excitations. It may be that further progress in elementary particle physics depends on the proper explanation of the origin and properties of quarks, the components of hadrons presumably carrying fractional charge. Recently it has been shown how a theory containing particles of charge e can lead to excitations of charge q equal to a fraction of e. Jackiw and Rebbil developed a relativistic field theoretical model in one dimension, in which a spinless Dirac field is coupled to a scalar field solving the classical Bose equation. They found that in the limit of widely separated kinks in the scalar field there exists a doubly degenerate bound state at zero energy. This state gives rise to doubly degenerate excitations of fermion number +1. The particle of fermion number 1 1 + is called a soliton, the particle of fermion number -2 is called an antisoliton. Independently J.R. Schrieffer2 studied a model for onedimensional conductors. He showed the existence of stable fractionally charged objects which are, in fact, kinks in the order parameter i describing the lattice distortion. Questions have been raised,4 however, as to whether the fractionally charged excitations can be called particles; more precisely, whether they are the eigenstates of the quantum charge operator. It may happen that the expectation of the charge of such an object is indeed fractional while every single measurement leads to an integer result. In this case the fractional charge is fictitious, and the object carrying it is not a particle but a quantum average of several particles. A way to resolve the issue is to calculate the fluctuation of the soliton charge. If the fluctuation is, indeed, zero, then the soliton can be rightly called a particle. Such a calculation was carried out by Kivelson and Schrieffer3 after the work on this thesis was actually started. They calculated the charge fluctuation of a soliton discovered by Schrieffer in dimerized and trimerized chains (non-relativistic models of one-dimensional conductors). They showed that, in the limit of widely separated solitons, the difference between the charge fluctuation of the system in the presence of a soliton and its absence approaches zero exponentially. Thus, when the solitons are far enough apart that their wave functions do not overlap, they do not contribute to the charge fluctuations. The conclusion reached by the authors was that each soliton, if looked at separately, is a particle. The whole system (all solitons existing in space) must possess an integer charge. There was a purpose, however, in continuation of work on this thesis. Right from the start we considered a relativistic field theoretical model outlined by Jackiw and Rebbi, rather than a non-relativistic one used by Schrieffer. It could happen that the relativistic effects would change the value of the difference in the charge fluctuations and make it non-zero. Even if the difference of the charge fluctuations was found to be zero, it would make sense to look more carefully into the issue of interpretation of this result. Namely, can we, from the difference of the charge fluctuations in the presence and in the absence of an excitation being zero, deduce the fact that this excitation is a particle. BOUND STATES IN THE DIRAC EQUATION AND SOLITONS Let us proceed now to the description of the model used in our calculation. In their original paper Jackiw and Rebbi suggested that if a scalar potential of the form for large L, is introduced into the one-dimensional Dirac equation, there exist soliton and antisoliton excitations. It turns out, however, that it is enough to consider the simplest scalar Bose field having two degenerate mean field ground states; namely, the kink-antikink pair of the form where a is large. Let us consider the one-dimensional Dirac equation with this scalar potential. at -'