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

Ivana D Ilić,Branislav Randjelović,Jelena M Visnjić,Vojislav V Mitić,Luka D Ćirić

doi:10.3390/fractalfract6020104

Abstract

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.

Highlights

In physics, the principle of locality implies that an object or an event at one position is directly influenced only by its proximate surroundings and it cannot cause an instantaneous action at another point
Let us observe a pair of electrons from the space X that we consider as one physical object
We introduced a special form of partial metric spaces, researched their properties and suggested some applications in certain domains of physics

Summary

Introduction

The principle of locality implies that an object or an event at one position is directly influenced only by its proximate surroundings and it cannot cause an instantaneous action at another point. In microscopic physics of quantum entities the result of a measurement on one particle can have a simultaneous effect on another entangled particle, regardless of their distance, which is in the contradiction with the principle of locality and with classical macroscopic physics. This fact is one of the main reasons for the appearance of the famous paper called the EPR paradox by Einstein, Podolsky, and Rosen [1], which implies that quantum physics may not be a complete theory and that maybe there exists some additional unrevealed variable witch integrates both microscopic and macroscopic physic laws. Hellman [4] reviewed certain arguments of EPR paradox and clarified some aspects of the “hidden variable”

Objectives

Results

Conclusion