A note on Heisenberg's uncertainty principle and Cantorian space-time
Considering the Cantorian space-time as a nondifferentiable manifold with the Hausdorff dimension \(d_ c^{(\mu)}\cong 4\), which replaces the smooth classical space-time in the description of the quantum processes, the author derives the Heisenberg's uncertainty principle for a Cantorian-type free particle, called Cantorion: \[ \Delta p\cdot\Delta x=h(\eta/\lambda)^{2-{1\over\Omega}}\cdot[\Omega(1-\Omega)^{1/2}] \] where \(\eta\) is the measurement resolution, \(\lambda\) is the wavelength of the Cantorian construction and \(\Omega=2/d_ c^{(4)}\) is the intrinsic geodesic dependent probability ``felt'' by the Cantorion on its nondifferentiable Peano-like and space filling path. It can be easily observed that inserting the explicit value of \(\Omega\cong 1/2\) into the previous formula the author comes very close to exactly the same form of the uncertainty principle derived by \textit{R. P. Feynman} and \textit{A. R. Hibbs} for the two slits experiment [Quantum Mechanics and Path Integrals, McGraw-Hill, New York (1965; Zbl 0176.549)].
scientific article; zbMATH DE number 96225