Spinor waves in a space-time lattice (II)

Abstract

In a previous note, an exceptional space-time lattice was found by a roundabout heuristic process. This process was far from convincing; here a more translucent characterization of the lattice is presented. A cornerstone is the consideration of pairs of reciprocal lattices, together with the basic symmetry (S 4) of the metric tensor. The basic requirement is that one member of a pair of reciprocal lattices contains the other as a sublattice. One preferred lattice is discussed in some detail; it contains three copies of its reciprocal lattice, and it is the simplest example satisfying the requirements. In the expression of the metric tensor in terms of the lattice generators a possible topology on the lattice is suggested. By means of this topology, propagation of spinor waves can be formulated. This proposed—the simplest—propagation mechanism is inhibited, though, by the fact that the three sublattices are required to carry the two types of spinors alternatively. This inhibition can be lifted by introducing a second type of elementary propagation, to next nearest neighbors. If this inhibition is only feebly lifted, this would result in particles with mass small as compared to the inverse of the lattice constant, presumably the Planck mass. Including the propagation to next nearest neighbors leads to spinor waves with six components, two components for each sublattice. In the long-wavelength limit four of them obey a massive Dirac equation, while the remaining two obey a Weyl equation. These considerations conceivably provide a root for the lack of parity invariance in nature, and for the joint occurrence of pairs of massive and massless spinor waves. The construction, furthermore, allows one to accommodate just three different families of spinor waves of this type. Extension of the above arguments outside the realm of the long-wavelength limit forcibly makes the lattice concept independent of the original continuous Minkowski spacetime: the latter is no longer a unique embedding space for the lattice, but appears as anapproximate interpolation, valid near the long-wavelength limit. This may be the minimal requirement to be imposed on a lattice theory in the light of the empirical evidence, if the scale of the lattice structure is, compared to the empirical scales, as small as the Planck scale.