Heisenberg’s and Hardy’s Uncertainty Principles for Special Relativistic Space-Time Fourier Transformation

Abstract

The special relativistic (space-time) Fourier transform (SFT) in Clifford algebra $$Cl_{(3,1)}$$ of space-time, first introduced from a mathematical point of view in Hitzer (Adv Appl Clifford Algebras 17:497–517, 2007), extends the quaternionic Fourier transform to functions, fields and signals in space-time. The purpose of this paper is to advance the study of the SFT and investigate important properties such as continuity, Plancherel identity, Riemann–Lebesgue lemma, and to establish the associated Hausdorff–Young inequality. Moreover, using the observer related space-time split several uncertainty inequalities are established, including Heisenberg’s uncertainty principle and Hardy’s theorem.