14th May, 2011
E-infinity Communication No. 86
Heisenberg uncertainty, quantum entanglement and von Neumann’s entropy from the view point of fractal spacetime and E-infinity theory
There has recently been a flurry of papers reassessing the validity of Heisenberg’s uncertainty in view of quantum nonlocality and maximal quantum entanglement. Two of the many recent publications which are worth mentioning are first an article in Science, 19 November (2010), Vol. 330 by Oppenhein and Wehner entitled “The uncertainty principle….”. The second is published in Nature–Physics-Letter (2010), 6, pp. 659-662 by M. Berta et al entitled “The uncertainty principle in the presence of quantum memory”.
Readers of E-infinity communications may be interested and slightly surprised to know that the new relations indicated in the mentioned recent research are a straight forward consequence of the fractal nature of quantum spacetime. For E-infinity experts and seasoned researchers the connection between entropy, quantum entanglement and Heisenberg’s uncertainty are natural and obvious facts. For the benefit of those not that familiar with fractal spacetime and E-infinity theory we offer the following few remarks, comments and elucidation.
Let us start by recalling two well known facts. First that one of the eigenvalue of Arnold’s mixing cat map or the two degrees of freedom unit oscillation is the golden mean. Consequently the topological entropy of these two fundamental models is the natural logarithm of 1.61803398 which is equal to 0.4812118252. The second well known fact is that the Hausdorff dimension of the path of a quantum point particle is not one, but rather unexpectedly 2. This fact was used by Abbott and Wisse as well as Ord, Nottale, El Naschie and Parisi. To find the entropic content of the quantum path we divide 2 by the natural logarithm of the golden mean and find 4.1561738141. This is a good approximation to 4 plus the golden mean power 3 namely the exact value of the Hausdorff dimension of the core of E-infinity manifold, the well known value 4.23606799. To obtain this exact value from the preceding entropic consideration we just expand the natural logarithmic term and retain only the linear terms. That way we find the familiar formula of E-infinity theory (1 + ϕ) divided by ( 1 ̶ ϕ) which gives (1/ϕ) to the power 3 or in decimal form 4.236067977… From this simple derivation we see how a quantum path is linked to the Hausdorff dimension of a peano curve which is D(H) = 2. Since a point on the peano curve is fundamentally resolution dependent and fuzzy it is clear that it can explain nonlocality. Also the dimension D(H) = 2 is essential for the derivation of the Heisenberg uncertainty principle as shown by Ord, Nottale and El Naschie (see for instance M.S. El Naschie – Quantum mechanics, Cantorian spacetime and Heisenberg’s uncertainty principle. VISTAS in Astronomy, Vol. 37, pp. 249-252 (1993). It should be noted that the ultimate explanation of nonlocality is the Lebesgue measure zero of the building blocks of spacetime which according to E-infinity are elementary random Cantor sets with a Hausdorff dimension, namely the golden mean as per a well known theorem due to Mauldin and Williams. This is the ultimate geometrical justification for the appearance of the golden mean in all quantum probabilities of entanglement as explained long ago by M.S. El Naschie. See for instance:
1 Chaos, Solitons & Fractals:
i. Vol. 1, No. 5, pp. 485-487 (1992).
iii. Vol. 9, No. 6, pp. 975-978 (1998).
iii. Vol. 26, No. 1, pp. 1-6 (2005).
iv. Vol. 9, No. 3, pp. 517-529 (1998).
2. Journal of the Franklin Institute:
i. Vol. 330, No. 1, pp. 183-198 (1993).
ii. Vol. 330, No. 1, pp. 199-211 (1993).
3. Il Nuovo Cimento, Vol. 107B, No. 5 (1992).