Thanks sir,yes its totally related to dispersion relation only.you means sir that for a single wave length we'll get two modes or sub-bands you can say,right as well as left side.we can also think physically that like normal modes.

regards

It is not obvious what you mean by ballistic case. But the number of dispersion relations per mode is equal to the order of the dispersion equation, i.e. the order of time derivatives in the field equation.

I.e, a field equation which is of second order in time will necessarily lead to a square root when computing the dispersion relation. I relativistic theories one usually do not discover that there are two dispersion relations, since the two square roots only imply a difference in signs; this property implies particle-antiparticle symmetry to linear order.