Heisenberg's and Hardy's uncertainty principles in real Clifford algebras

Recently, many surveys are devoted to study the Clifford-Fourier transform (CFT). Dealing with the real Clifford-Fourier transform introduced by Hitzer [The Clifford Fourier transform in real Clifford algebras. In: Hitzer EE, Tachibana K, editors. Session on Geometric Algebra and Applications, IKM 2012. Special Issue of Clifford Analysis, Clifford Algebras and their Applications. Vol. 2, No. 3. First published in Guerlebeck K, Lahmer T, Werner F, editors. Electronic Proc. of 19th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering, IKM 2012; 4–6 July 2012; Weimar, Germany; 2013. p. 223–235. Available from: http://vixra.org/abs/1306.0130], we establish analogue of the classical Heisenberg's inequality in the real Clifford algebra Cl(p,q). We derive a variation of the Heisenberg's inequality. Furthermore, we provide Hardy's theorem for the real CFT which allows to obtain Hardy's theorem in quaternion algebra.