Spinor Soliton as an Elementary Particle

Abstract

Classical and quantized spinor soliton are investigated from the viewpoint that the ele­ mentary particles are regarded as a kind of soliton. The non-linear spinor equation proposed by Heisenberg is studied in two-dimensional case which is usually called the Thirring modeL The results obtained are as follows: The principle of superposition is realized in a certain sense. A wave packet and a soliton are the same content in this special model. The collision of two solitons are illustrated in detail. 2> Almost all of the known solitons are classical, spinless and two-dimensional (one­ dimensional space and time). Many particle physicists want to regard the solitons as the elementary particles. To do this the equation should be solved and quan­ tized in four-dimensional space-time. However a simple dimensional analysis tells us that the stability of soliton is usually lost in larger dimensions. An equation with higher derivatives might give a stable soliton in four dimensions, but it would cause other problems. On the quantization of soliton there are some inter­ esting approaches, 3> which, however, seem not to be successful completely, be­ cause of the complication of non-linearity. In this connection we are interested in whether the exact solution of a quantized non-linear field equation may allow a soliton. When we consider that the solitons represent the elementary particles, we encounter various problems. For example, the soliton should describe free waves as well as scattering waves. As to the free waves the soliton solution should have some freedom to choose its wave form such as plane wave or wave packet arbitrarily. In addition not only one-particle state but also many-particle state should be described by a soliton solution of an equation. If the particles are separated enough to each other, the wave forms of the particles should be chosen arbitrarily, because they are independent of each other. In other words the prin­ ciple of superposition should be satisfied by free waves. It is, however, generally believed that because of the non-linearity of equation the wave forms of solitons are not free but restricted or determined by the equation and the principle of superposition should never be realized in non-linear equations. In this note we want to show that it is not true for a special non-linear equations.