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arXiv:1309.6502

The Uncertainty Principle derived by the finite transmission of light and information

Piero Chiarelli

Published Sep 24, 2013• 0 citations
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Abstract

This work shows that in the frame of the stochastic generalization of the quantum hydrodynamic analogy (QHA) the uncertainty principle can be derived by the postulate of finite transmission speed of light and information . The theory shows that the measurement process performed in the large scale classical limit of stochastic QHA (SQHA), cannot have a duration smaller than the time need to the light to travel the distance up to which the quantum non-local interaction extend itself. The product of the minimum measuring time multiplied by the variance of energy fluctuation due to presence of stochastic noise shows to lead to the minimum uncertainty principle. The paper also shows that the uncertainty relations can be also derived if applied to the indetermination of position and momentum of a particle of mass m in a quantum fluctuating environment.

DOI
10.48550/arxiv.1309.6502

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This work shows that in the frame of the stochastic generalization of the quantum hydrodynamic analogy (QHA) the uncertainty principle can be derived by the postulate of finite transmission speed of light and information . The theory shows that the measurement process performed in the large scale classical limit of stochastic QHA (SQHA), cannot have a duration smaller than the time need to the light to travel the distance up to which the quantum non-local interaction extend itself. The product of the minimum measuring time multiplied by the variance of energy fluctuation due to presence of stochastic noise shows to lead to the minimum uncertainty principle. The paper also shows that the uncertainty relations can be also derived if applied to the indetermination of position and momentum of a particle of mass m in a quantum fluctuating environment.
This work shows that in the frame of the stochastic generalization of the quantum hydrodynamic analogy (QHA) the uncertainty principle is fully compatible with the postulate of finite transmission speed of light and information. The theory shows that the measurement process performed in the large scale classical limit in presence of background noise, cannot have a duration smaller than the time need to the light to travel the distance up to which the quantum non-local interaction extend itself. The product of the minimum measuring time multiplied by the variance of energy fluctuation due to presence of stochastic noise shows to lead to the minimum uncertainty principle. The paper also shows that the uncertainty relations can be also derived if applied to the indetermination of position and momentum of a particle of mass m in a quantum fluctuating environment.
The paper investigates the non-local property of quantum mechanics in the quantum hydrodynamic analogy (QHA) given by Madelung. The role of the quantum potential in generating the non-local dynamics of quantum mechanics is analyzed. The work shows how in presence of noise the non-local properties as well as the quantization of the action are perturbed. The resulting stochastic QHA dynamics much depend by the strength of the interaction: Strongly bounded systems (such as linear ones) lead to quantum entangled stochastic behavior, while weakly bounded ones may be not able to maintain the quantum superposition of states on large distances and may loose their macro-scale quantum coherence acquiring the classical stochastic evolution . The work shows that in the frame of the stochastic approach it is possible to have freedom between quantum weakly bounded systems. The stochastic QHA model shows that the wave-function collapse to an eigenstates (deriving by interaction of a quantum microscopic system with a classical (macroscopic) one) can be described by the model itself as a kinetic quantum (relaxation) process to a stationary state. Since the time of the wave function decay to the eigenstate represents the minimum duration time of a measurement, the minimum uncertainty principle is shown to be compatible with the relativistic postulate about the light speed as the maximum velocity of transmission of interaction. About this topic, the paper shows that the Lorenz invariance of the relativistic quantum potential (coming from the Dirac equation) enforces the hypothesis that the superluminal transmission of information are not present in measurements on quantum entangled state.
The paper investigates the non-local property of quantum mechanics by analyzing the role of the quantum potential in generating the non-local dynamics and how they are perturbed in presence of noise.The resulting open quantum dynamics much depend by the strength of the Hamiltonian interaction: Weakly bounded systems may not be able to maintain the quantum superposition of states on large distances and lead to the classical stochastic evolution.The stochastic hydrodynamic quantum approach shows that the wave-function collapse to an eigenstates can be described by the model itself and that the minimum uncertainty principle is compatible with the relativistic postulate about the light speed as the maximum velocity of transmission of interaction.The paper shows that the Lorenz invariance of the quantum potential does not allow superluminal transmission of information in measurements on quantum entangled states.
The work analyzes the compatibility between the classical freedom, the local relativistic causality and the non-local behavior of quantum mechanics in the frame of the stochastic approach of the quantum hydrodynamic analogy (SQHA). The work describes the role of the quantum potential in generating the quantum non-local dynamics in a fluctuating environment. The analysis shows that it is possible to maintain the concept of classical freedom between far away weakly bounded systems (moderate non-locality) as well as to make compatible the uncertainty principle with the relativistic postulate of invariance of light speed. The work shows that the paradox of instantaneous quantum non local behavior at infinite distances of the standard formalism is an artifact due to the non-relativistic non-stochastic ambit of such theory where the light speed is infinite and the non-local interaction owns an infinite range of action. The work envisages that the SQHA can possibly lead to a fully theoretically self-standing quantum mechanics where the wave function collapse, during a measurement process, can be described by the theory itself without empirical postulates. Under this light the paper discusses the need of searching for (both local and non-local) hidden variables quantum mechanics as well as the need of superluminal communications in quantum experiments. The analysis shows that all these hypotheses are attempts of interpreting the outputs of quantum measurements that cannot be fully explained by the semi-empirical formalism of quantum mechanics, based on the statistical postulates of the measuring process as well as the existence of a classical observer. A two photon experiment is discussed to the light of the SQHA approach. Key words: Quantum non-locality, superluminal transmission of quantum information, classical freedom, local relativistic causality, Einstein, Podolsky, and Rosen (EPR) paradox, macroscopic quantum decoherence, Bell’s inequalities, quantum hydrodynamic analogy.
This work shows that the stochastic generalization of the quantum hydrodynamic analogy (QHA) has its corresponding stochastic Schrödinger equation (SSE) as similarly happens for the deterministic limit. The SSE owns an imaginary random noise that has a finite correlation distance, so that when the physical length of the problem is much smaller than it, the SSE converges to the standard Schrödinger equation comprehending it. The model shows that in non-linear (weakly bounded) systems, the term responsible of the non-local interaction in the SSE may have a finite range of efficacy maintaining its non-local effect on a finite distance. A non-linear SSE that describes the related large-scale classical dynamics is derived. The work also shows that at the edge between the quantum and the classical regime the SSE can lead to the semi-empirical Gross-Pitaevskii equation.
The quantum hydrodynamic analogy (QHA) equivalent to the Schrodinger equation is generalized to its stochastic version by a systematic technique. On large scale, the quantum stochastic hydrodynamic analogy (QSHA) shows dynamics that under some circumstances may acquire the classical evolution. The QSHA puts in evidence that in presence of spatially distributed noise the quantum pseudo-potential restores the quantum behavior on a distance shorter than the correlation length of fluctuations (named here lc) of the quantum wave function modulus. The quantum mechanics is achieved in the deterministic limit when lc tends to infinity with respect to the scale of the problem. When the physical length of the problem is of order or larger than lc, the quantum potential may have a finite range of efficacy maintaining the non-local behavior on a distance lL (named here "quantum non-locality length") depending both by the noise amplitude and by the inter-particle strength of interaction. In the deterministic limit (quantum mechanics) the model shows that the "quantum non-locality length" lLalso becomes infinite. The QSHA unveils that in linear systems fluctuations are not sufficient to break the quantum non-locality showing that lL is infinite even if lc is finite.