Mathematical Approach to Distant Correlations of Physical Observables and Its Fractal Generalisation

In this paper, the new mathematical correlation of two quantum systems that were initially allowed to interact and then separated is being formulated and analyzed. These correlations are illustrated by many examples and are also connected with fractals at a certain level. The main idea of the paper arises from the EPR paradox, the paradox of Einstein, Podolsky, and Rosen that occurs when the measurement of a physical observable performed on one system has an immediate effect on the other separate system being entangled with it. That is a physical phenomenon, especially when the particles are separated by a large distance. In this paper, we define distant correlations as the advanced method for the exact interpretation of strong connection and influence among those particles even when they are widely separated. On the given topological space (X,τ), we define a notion of τ-metric such that the set X is a τ-metric space and we prove some properties of these spaces. By using this new proposed model, we nullify the contradiction that appears in the EPR paradox. An illustrative example involving fractals is given. This innovative mathematical approach to this physical phenomenon may be attractive for future research in the field of quantum physics.