Abstract
In this paper, we present a set of important properties of the special relativistic Fourier transformation (SFT) on the complex space–time algebra \({\mathcal {G}}{(3,1)}\), such as inversion property, the Plancherel theorem, and the Hausdorff–Young inequality. The main objective of this article is to introduce the concept of the vector derivative in geometric algebra and using it together with the notion of the space–time split to derive the Heisenberg–Pauli–Weyl inequality. Finally, we apply the SFT properties for proving the Donoho–Stark uncertainty principle for \({\mathcal {G}}{(3,1)}\) multi-vector functions.
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Christensen, J.G.: Uncertainty Principles, Master’s Thesis, Institute for Mathematical Sciences, University of Copenhagen (2003)
Dirac, P.A.M.: The quantum theory of the electron. Proc. R. Soc. A (1928). https://doi.org/10.1098/rspa.1928.0023
Donoho, D.L., Stark, P.B.: Uncertainty principles and signal recovery. SIAM J. Appl. Math. 49, 906–931 (1989)
Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003)
El Haoui, Y., Fahlaoui, S.: The Uncertainty principle for the two-sided quaternion Fourier transform. Mediterr. J. Math. 14, 221 (2017). https://doi.org/10.1007/s00009-017-1024-5
El Haoui, Y., Fahlaoui, S.: Donoho–Stark’s uncertainty principles in real Clifford algebras. Adv. Appl. Clifford Algebras 29, 94 (2019). https://doi.org/10.1007/s00006-019-1015-7
El Haoui, Y., Hitzer, E. & Fahlaoui, S. Heisenberg’s and Hardy’s Uncertainty Principles for Special Relativistic Space-Time Fourier Transformation. Adv. Appl. Clifford Algebras 30, 69 (2020). https://doi.org/10.1007/s00006-020-01093-5
Fu, Y., Li, L.: Uncertainty principle for multivector-valued functions. Int. J. Wavel. Multiresolution Inf. Process. 13(1), 1550005 (2015). https://doi.org/10.1142/S0219691315500058
Hitzer, E.: Relativistic physics as application of geometric algebra. In: Adhav, K. (ed.) Proc. of the Int. Conf. on Relativity 2005 (ICR2005), pp. 71–90. University of Amravati (2005). http://vixra.org/abs/1306.0121
Hitzer, E.: Quaternion Fourier transform on quaternion fields and generalizations. Adv. Appl. Clifford Algebras 17, 497–517 (2007). https://doi.org/10.1007/s00006-007-0037-8. (arXiv:1306.1023)
Hitzer, E.: Directional uncertainty principle for quaternion Fourier transform. Adv. Appl. Clifford Algebras 20, 271–284 (2010). https://doi.org/10.1007/s00006-009-0175-2
Hitzer, E.: Two-sided clifford Fourier transform with two square roots of \(-1\) in \(Cl(p, q)\). Adv. Appl. Clifford Algebras 24, 313–332 (2014). https://doi.org/10.1007/s00006-014-0441-9
Hitzer, E.: Special relativistic Fourier transformation and convolutions. Math. Methods Appl. Sci. 42, 2244–2255 (2019). https://doi.org/10.1002/mma.5502
Linares, F., Ponce, G.: Introduction to Nonlinear Dispersive Equations. Publicações Matemáticas, IMPA, Rio de Janeiro (2004)
Mawardi, B., Hitzer, E.: Clifford Fourier Transform and Uncertainty Principle for the Clifford Geometric Algebra \(Cl(3,0)\). Adv. Appl. Clifford Algebras 16(1), 41–61 (2006). https://doi.org/10.1007/s00006-006-0003-x
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The author is grateful to the editor and the two anonymous referees, whose insightful comments improved the paper immensely.
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El Haoui, Y. Uncertainty Principle for Space–Time Algebra-Valued Functions. Mediterr. J. Math. 18, 97 (2021). https://doi.org/10.1007/s00009-021-01718-4
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DOI: https://doi.org/10.1007/s00009-021-01718-4