From: jos.bergervoet@xs4all.nl   
      
   I'm looking for a toy model to solve the classical field   
   equations in 1+1 dimensions and then from random initial   
   conditions after some time get a localized bound state.   
   Is this possible? I could try:   
      
   1) One real field, with phi^4 interaction:   
         g phi^4 - m^2 phi^2   
   This would probably give domain walls between the two stable   
   vacua, moving around and sometimes bumping into each other.   
   But they are not truly bound states of attracting partiuicles.   
      
   2) A matter field Psi and force field phi, interaction:   
         g phi |Psi|^2   
   I'm not sure however that once g is strong enough to get a   
   bound state, its spectrum will be bound from below (with the   
   3rd order interaction). Also, this is not a state of two   
   attracting particles but (if it comes out as a solution at   
   all) more like a concentration of self-gravitating matter.   
      
   3) Using two complex matter fields and the electrical force:   
         g Psi^* A_mu d^mu Psi   
   where in this interaction Psi^* denotes complex conjugate and   
   I'm coupling the current to the potential A_mu. I would use two   
   fields Psi1 and Psi2 in this way to conserve the two particle   
   species, and give them opposite charge in the initial conditions   
   (to keep the universe charge-neutral, which will allow simple   
   cyclic boundaries in numerical solving).   
      
   I'd expect approach 3) would work. Or do I have to change the   
   scalar matter fields to spinors and completely resort to QED   
   in 1+1 dimensions? That would hardly be a simplified toy model   
   any more.. So what is the simplest choice?   
      
   --   
   Jos   
      
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