Blow-up in nonlinear models of extended particles with confined constituents.

Finding a satisfactory mechanism for quark confinement is no doubt one of the most relevant problems in fundamental physics. One off-the-beaten-track view on this question considers confinement as a prequantum effect because of the way in which the constituent cnumber fields are coupled. More precisely, with suitable interaction terms, each one of these fundamental fields can be a source to the rest of them in such a way that the set of solutions of the wave equations does not contain states in which only one of these constituent fields is excited. This parallels the nonexistence of isolated quarks. In the classical c-number models which realize this idea, confinement is a classical effect previous to any second quantization. They make use of six constituent fields, three of them being charge conjugate to the other three; this has the advantage that antiparticles can be treated in the same way as particles, but, as a necessary price to pay, the charge has an indefinite character. In this work we show that this indefinite character is a fundamental handicap for the validity of these models, even from a purely classical point of view, since it is the cause of the existence of solutions which blow-up in finite time. For simplicity, we shall consider the version of the model presented in the last paper of Ref. 1. It refers to (1+1)-dimensional spacetime and makes use of six spinors g„P„a= 1,2, 3, the field equations being y =iy y'. The current g (sky Pk sky 0k ) k