Space time quantification and generalisation of the Dirac equation: interpretation of an additional degree of freedom

The introduction of an elementary length (/b a/), defining the ultimate limit for the measurable distance, implies that Dirac's differential equation has to be replaced by a corresponding finite-difference equation. This generalized Dirac equation requires, however, the introduction of spinors with /b eight/ instead of the usual /b four/ components. It is shown that one has actually two four-component spinors, whose amplitudes are defined, respectively, for two possible sets of eigenvalues of the time variable, /b ct/ being an integer or a half-integer multiple of the quantum of length /b a/. Moreover, it appears that the two spinors can`vibrate' in phase or in opposite phase, so that the additional degree of freedom is analogous to the possible occurrence of `acoustical' and `optical' modes for the vibrations of a crystal lattice. The corresponding spinor-eigen-states are characterized, however, by the same energy-momentum relation, defining a new internal degree of freedom, which remains still degenerate in the presence of an electro-magnetic field. It is only in the case where the quantum of length is identically zero, that one gets a single four component spinor.