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Fractal and Fractional

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

Ivana Ilić, Jelena M. Višnjić, Vojislav V. Mitić, Branislav M. Randjelovic, Luka Ćirić

Published Feb 12, 2022• 0 citations
Publisher

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.

DOI
10.3390/fractalfract6020104

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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.
The famous EPR paradox shows that if we describe quantum particles in the usual way -- by their wave functions -- then we get the following seeming contradiction. If we entangle the states of the two particles, then move them far away from each other, and measure the state of the first particle, then the state of the second particle immediately changes -- which contradicts to special relativity, according to which such immediate-action-at-a-distance is not possible. It is known that, from the physical viewpoint, this is not a real paradox: if we measure any property of the second particle, the results will not change whether we perform the measurement on the first particle or not. What the above argument shows is that the usual wave function description of a quantum state does not always adequately describe the corresponding physics. In this paper, we propose a new, more physically adequate description of a quantum state, a description in which there is no EPR paradox: measurements performed at the first particle does not change the state of the remote second one.
A generalization of the 1935 Einstein-Podolsky-Rosen (EPR) argument for measurements with continuous variable outcomes is presented to establish criteria for the demonstration of the EPR paradox, for situations where the correlation between spatially separated subsystems is not perfect. Two types of criteria for EPR correlations are determined. The first type are based on measurements of the variances of conditional probability distributions and are necessary to reflect directly the situation of the original EPR paradox. The second weaker set of EPR criteria are based on the proven failure of (Bell-type) local realistic theories which could be consistent with a local quantum description for each subsystem. The relationship with criteria sufficient to prove entanglement is established, to show that any demonstration of EPR correlations will also signify entanglement. It is also shown how a demonstration of entanglement between two spatially separated subsystems, if not able to interpreted as a violation of a Bell-type inequality, may be interpreted as a demonstration of the EPR correlations. In particular it is explained how the experimental observation of two-mode squeezing using spatially separated detectors will signify not only entanglement but EPR correlations defined in a general sense.
The EPR paradox appears when measurement results of some properties of two distantly entangled particles are correlated in a way that cannot be explained classically, and apparently violate locality. The resolution of the paradox depends on one’s interpretation of quantum mechanics. Explanations from quantum mechanics remain commonplace today, but they fail to explain the EPR (Einstein, Podolsky and Rosen) paradox totally in a way than can be accepted by the whole community. Here, we present a simple resolution to this paradox in which the uncertainty in the energy of the two-particle system is reduced by its lack of interaction during the journey so that the uncertainty in time becomes greater than the time they have been separating. Consequently, the present and past become indistinguishable because when we measure an observable in the system its value is the same as if the two particle were still together or very close. It is also argued that the destruction of information as the present and past become identical should release heat by Landauer’s principle, and this might make this proposal testable.
Introduction The paradox of Einstein, Podolsky and Rosen was advanced as an argument that quantum mechanics could not be a complete theory but should be supplemented by additional variables. These additional variables were to restore to the theory causality and locality. In this note that idea will be formulated mathematically and shown to be incompatible with the statistical predictions of quantum mechanics. It is the requirement of locality, or more precisely that the result of a measurement on one system be unaffected by operations on a distant system with which it has interacted in the past, that creates the essential difficulty. There have been attempts to show that even without such a separability or locality requirement no ‘hidden variable’ interpretation of quantum mechanics is possible. These attempts have been examined elsewhere and found wanting. Moreover, a hidden variable interpretation of elementary quantum theory has been explicitly constructed. That particular interpretation has indeed a grossly non-local structure. This is characteristic, according to the result to be proved here, of any such theory which reproduces exactly the quantum mechanical predictions. Formulation With the example advocated by Bohm and Aharonov, the EPR argument is the following. Consider a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by Stern–Gerlach magnets, on selected components of the spins σ 1 and σ 2 .
Bohm version of the EPR paradox Einstein, Podolsky, and Rosen (EPR) were concerned with the following issue. Given two spatially separated quantum systems A and B and an appropriate initial entangled state, a measurement of a property on system A can be an indirect measurement of B in the sense that from the outcome of the A measurement one can infer with probability 1 a property of B , because the two systems are correlated. There are cases in which either of two properties of B represented by noncommuting projectors can be measured indirectly in this manner, and EPR argued that this implied that system B could possess two incompatible properties at the same time, contrary to the principles of quantum theory. In order to understand this argument, it is best to apply it to a specific model system, and we shall do so using Bohm's formulation of the EPR paradox in which the systems A and B are two spin-half particles a and b in two different regions of space, with their spin degrees of freedom initially in a spin singlet state (23.2). As an aid to later discussion, we write the argument in the form of a set of numbered assertions leading to a paradox: a result which seems plausible, but contradicts the basic principles of quantum theory. The assertions E1–E4 are not intended to be exact counterparts of statements in the original EPR paper, even when the latter are translated into the language of spin-half particles. However, the general idea is very similar, and the basic conundrum is the same.
In their original paper, Einstein, Podolsky, and Rosen (EPR) questioned the probabilistic nature of the quantum theory using an argument based on the quantum-mechanical correlations between two distant particles which had once interacted. Bell later proved that the quantum theory and any local theory differ significantly in their predictions for these experimentally measurable correlations. It is shown that the arguments of EPR can be extended to apply equally well to certain properties of a single particle or system and that, in analogy with Bell's theorem, the predictions of the quantum theory for certain single-particle experiments are in disagreement with the predictions of any local theory. For example, based on either the quantum theory or existing experimental observations, it is possible to arrange for a photon to have a negligible probability of not being absorbed by some system of finite dimensions, such as one of a large number of atoms, within a finite time interval. It is shown that any local theory which satisfies this requirement must also require that negligible interference effects occur in an optical interferometer at sufficiently low intensities from an incoherent source in combination with sufficiently large separation LLL between the two optical paths. Time-reversal invariance is used to estimate the requirement $L\ensuremath{\gg}\frac{3{\ensuremath{\lambda}}^{2}}{4{\ensuremath{\pi}}^{2}{r}_{c}}$, where $\ensuremath{\lambda}$ is the wavelength of the light and rc{r}_{c}rc​ is the classical radius of the electron; at optical wavelengths this distance is on the order of 10 m. The same separation should also be required between the photon detectors in any optical experiment based on Bell's theorem. A review of the literature indicates that no existing experimental data, either from optical interferometers or experiments based on Bell's theorem, can be used to distinguish between the local and quantum theories. An investigation of low-intensity optical interference at large path separations is proposed for this purpose.