Localized Propagating Tachyons in Extended Relativity Theories

Abstract

We examine the possibility of localized propagating tachyonic fields within a properly extended relativity. A possible extension is to include superluminal transformations and reference frames. This leads to complex 4D spacetime, or real 8D spacetime M_{4,4}. The mass shell constraint in M_{4,4} becomes, after first quantization, the ultrahyperbolic Klein-Gordon equation. The Cauchy problem for such equation is not well posed, because it is not possible to freely specify initial data on a 7D hypersurface of M_{4,4}. We explicitly demonstrate that it is possible to do it on a space-like 4-surface for bradyons, and on a time-like 4-surface for tachyons. But then the evolution of a bradyonic field into the four time-like directions, or the "evolution" of a tachyonic field into the four space-like directions, is not uniquely determined. We argue that this is perhaps no so bad, because in quantum field theory (after second quantization) the classical trajectories of fields are not determined anyway, and so it does not matter, if they are not completely determined already in the first quantized theory. A next possible extension of relativity is to consider 16D Clifford space, C, a space whose elements are oriented r-volumes, r=0,1,2,3,4 of real 4D spacetime. Then the evolution parameter can be associated with an extra light-cone coordinate, e.g., with the sum of the scalar and the pseudoscalar coordinate, and initial data can be given on a light-like hypersurface, in which case the Cauchy problem is well posed. This procedure brings us to the Stueckelberg theory which contains localized propagating tachyons in 4D spacetime.