Frame independence of the fundamental length in relativistic quantum field theory

Some time ago we proposed a relativistic quantum field theory with a fundamental length in terms of tempered ultrahyperfunctions [Bruening, E. and Nagamachi, S., 'Relativistic quantum field theory with a fundamental length', J. Math. Phys. 45, 2199 (2004); e-print arXiv:math-ph/0804.1663; 'Solutions of a linearized model of Heisenberg's fundamental equation II', J. Math. Phys. 49, 052304 (2008)]. The definition of the fundamental length by Bruening and Nagamachi ['Relativistic quantum field theory with a fundamental length', J. Math. Phys. 45, 2199 (2004)] seems to depend on the Lorentz frame which is used to define it. In this article we show that this is actually not the case. In addition we discuss in some detail the geometric and analytic realization of the fundamental length and point out some important difference to standard relativistic quantum field theory in the sense of Gaarding and Wightman.