Discrete Structure in Velocity Space

Abstract

The introduction of discreteness into the space-time continuum is perhaps one of the most interesting ideas in recent high energy particle physics. In fact, some authors1' have discussed this possibility, though the relation between their ideas and the experimental data has not been well treated; In a previous paper,2' we 'have attempted progress in this direction and proposed a theory in which we establish a relation between the observed enhancements in the distribution of the longitudinal Lorentz angles for the meson clusters emitted from cosmic ray jet interactions8'· and the simplest three-dimensional integral Lorentz It is remarkable in this theory that there exists a specific (minimum) velocity v = ( ..J8 I 3) c corresponding to the Lorentz angle O=cosh-1 3=1.7628 (=0,.) or the Lorentz factor r=3. It is the purpose of the present paper to call attention to the possible existence of another specific velocity ( v3/2)c. The velocity space is very important for the kinematical analysis of the multi-particle final states resulting from high energy particle collisions, since it is the very space where hodograph methods are available.5> Previously2' we used the longitudinal Lorentz angle 0=tanh-1 (v0/c), and found that the distribution of particles in this one-dimensional space is uniform with mesh size 0~1.7, as pointed out by Hasegawa,8> suggesting the presence.of discrete .:velocity levels. This mesh size was almost equal to the: minimum Lorentz angle o,., i.e., the Lorentz angle induced by the simplest three-dimensional. integral Lorentz transformation. However, from the theoretical viewpoint, there is no reason to exclude the four-dimensional case where the minimum velocity should be smaller than the three-dimensional case, i.e., v= (v3/2)c, O=cosh-1 2=1.31696 (=0.,.), r=2. · This indicates that there exists another kind of particle distribution with .mesh size 0.,;,. As Om is the minimum possible Lorentz angle, it is inferred that the reactions with mesh ·size Om occur most frequently. But the observation of the signals which come from discrete ·velocity levels are concentrated strongly at mesh size smaller than Om 'for many reasons, for example, a) the stronger secondary effects which mediate between observed mesons and primary objects, b) the stronger .9verlapping of clusters in velocity space. Though Hasegawa8> measured the mesh size directly for the reactions with mesh size 0/,, the same method may not be applicable to the events with mesh size Om.