Quantum mechanics and the possibility of a Cantorian space-time
The author proposes a very interesting statement: the real physical model for space-time is actually represented by the 4-dimensional Peano-like Cantorian space (\(d_ c^{(4)}\)). The reasons can be summarized as follows: 1) Suspending the triadic Cantor set in \(n\)-dimensional spaces, \(n=d_ M^{(n)}\) is the Menger-Urysohn dimension, there is an obvious jump in the values of the Hausdorff capacity dimensions, \(d_ c^{(n)}\), and the Gibbs-Shannon entropies, \(S_ s^{(n)}\), which takes place exactly at \(n=4\). It results that for \(n>4\) the world set is totally unstable (chaotic) and for \(n<4\) despite the fact that the world set is stable it could not account for the physical reality as we know it. It is only the case \(n=4\) that leads to an almost metastable set and quasi ergodic behaviour. 2) In case that the twist movement obtained from the dynamics of the Peano-like curve in 3-dimensions is also preserved in four, then the quantum spin gets a natural geometrical meaning. 3) Because any 2-dimensional section in (\(d_ c^{(4)}\)) could be modelled by the Peano-Hilbert curve which is ``both a line and an area'', depending on the scale of observation, an unexpected connection to the measurement problem from quantum mechanics it arises. 4) The (\(d_ c^{(4)}\))-structure could explain the singularities from the quantum fields, the time-like irreversability and the initial singularity scenario. The universe possesses Peano-like structure in the small scale, Euclidean outlook on the normal one and is finally Riemannian on the large scale.
4-dimensional Peano-like Cantorian space
scientific article; zbMATH DE number 89972