Abstract :
An idealized two-slit experiment is envisaged in which the hypothetical experimental set-up is constructed in such a way as to resemble a toy model giving information about the structure of quantum space–time itself. Thus starting from a very simple equation which may be interpreted as a physical realization of Gödel’s undecidability theorem, we proceed to show that space–time is very likely to be akin to a fuzzy Kähler-like manifold on the quantum level. This remarkable manifold transforms gradually into a classical space–time as we decrease the resolution in a way reversibly analogous to the processes of recovering classical space–time from the Riemannian space of general relativity.
The paper’s main philosophy is to emphasize that the quintessence of the two-slit experiment as well as Feynman’s path integral could be given a different interpretation by altering our classical concept of space–time geometry and topology. In turn this would be in keeping with the development in theoretical physics since special and subsequently general relativity. In the final analysis it would seem that we have two different yet, from a positivistic philosophy viewpoint, completely equivalent alternatives to view quantum physics. Either we insist on what we see in our daily experiences, namely, a smooth four-dimensional space–time, and then accept, whether we like it or not, things such as probability waves and complex probabilities. Alternatively, we could see behind the façade of classical space–time a far more elaborate and highly complex fuzzy space–time with infinite hierarchical dimensions such as the so-called Fuzzy K3 or E–Infinity space–time and as a reward for this imaginative picture we can return to real probabilities without a phase and an almost classical picture with the concept of a particle’s path restored. We say almost classical because non-linear dynamics and deterministic chaos have long shown the central role of randomness in classical mechanics and this is reinforced once more in our model which is directly related not to Newtonian motion, but rather to a diffusion-like random walk similar to that used with great skill by Einstein and later on by Nagasawa and particularly the English-Canadian physicist Garnet Ord.
