Note on the Golden Mean, Nonlocality in Quantum Mechanics and Fractal Cantorian Spacetime

Abstract

Given the inverse of the Golden Mean $ \tau^{ -1} = \phi = { 1\over 2} (\sqrt 5 - 1)$, it is known that the continuous fraction expansion of $ \phi^{ -1} = 1 + \phi = \tau$ is $ ( 1, 1, 1, \cdots )$. Integer solutions for the pairs of numbers $ ( d_i, n_i ), i = 1, 2, 3, \cdots $ are found obeying the equation $ ( 1 + \phi)^n = d + \phi^n$. The latter equation was inspired from El Naschie's formulation of fractal Cantorian space time $ {\cal E}_\infty$, and such that it furnishes the continuous fraction expansion of $ ( 1 + \phi )^n ~= ~ (d, d, d, d, \cdots )$, generalizing the original expression for the Golden mean. Hardy showed that is possible to demonstrate nonlocality without using Bell inequalities for two particles prepared in $nonmaximally$ entangled states. The maximal probability of obtaining his nonlocality proof was found to be precisely $\phi^5$. Zheng showed that three-particle nonmaximally entangled states revealed quantum nonlocality without using inequalities, and the maximal probability of obtaining the nonlocality proof was found to be $ 0.25 \sim \phi^3 = 0.236$. Given that the two-parameter $ p, q$ quantum-calculus deformations of the integers $ [ n ]_{ p, q} = F_n $ $coincide$ precisely with the Fibonacci numbers, as a result of Binet's formula when $ p = ( 1 + \phi) = \tau, q = - \phi = - \tau^{ -1} $, we explore further the implications of these results in the quantum entanglement of two-particle spin-$s$ states. We finalize with some remarks on the generalized Binet's formula corresponding to generalized Fibonacci sequences.