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arXiv:1809.08653 · Physica Scripta

A natural cure for causality violations in Newton–Schrödinger equation

Filippo Maimone, Giovanni Scelza, Adele Naddeo

Published Jan 30, 2019• 3 citations
arXivHTMLPDF

Abstract

Abstract It is explicitly shown that a one-family parameter model reproducing the nonlinear Newton–Schrödinger equation as the parameter goes to infinity is free from any causality violation problem for any finite value of it. This circumstance arises from the intrinsic mechanism of spontaneous state reduction of the model, absent in the Newton–Schrödinger limit. A specific ideal EPR experiment involving a superposition of two distinct CM’s position states of a massive lump is analyzed, showing recovered compatibility with quantum mechanics. Besides, the new framework suggests a soft version of the many worlds interpretation, in which the typical indiscriminate proliferation of Everett branches, together with the bizarre inter-branches communications made possible by nonlinearity, are strongly suppressed in the macroscopic world by the same mechanism of state reduction.

DOI
10.1088/1402-4896/ab030d

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Abstract It is explicitly shown that a one-family parameter model reproducing the nonlinear Newton–Schrödinger equation as the parameter goes to infinity is free from any causality violation problem for any finite value of it. This circumstance arises from the intrinsic mechanism of spontaneous state reduction of the model, absent in the Newton–Schrödinger limit. A specific ideal EPR experiment involving a superposition of two distinct CM’s position states of a massive lump is analyzed, showing recovered compatibility with quantum mechanics. Besides, the new framework suggests a soft version of the many worlds interpretation, in which the typical indiscriminate proliferation of Everett branches, together with the bizarre inter-branches communications made possible by nonlinearity, are strongly suppressed in the macroscopic world by the same mechanism of state reduction.
Abstract We quote a definitive simple proof that neither classical stochastic dynamics nor quantum dynamics can be nonlinear if we stick to their standard statistical interpretations. A recently proposed optomechanical test of gravity’s classicality versus quantumness is based on the nonlinear Schrödinger—Newton equation (SNE) which is the nonrelativistic limit of standard semiclassical gravity. While in typical cosmological applications of semiclassical gravity the predicted violation of causality is ignored, it cannot be disregarded in applications of the SNE in high sensitive laboratory tests hoped for the coming years. We reveal that, in a recently designed experiment, quantum optical monitoring of massive probes predicts fake action-at-a-distance (acausality) on a single probe already. The proposed experiment might first include the direct test of this acausality.
The Schrödinger-Newton equation aims at describing the dynamics of massive quantum systems subject to the gravitational self-interaction. As a deterministic nonlinear quantum wave equation, it is generally believed to conflict with the relativistic no-signalling principle. Here we challenge this viewpoint and show that it is of key importance to study the quantitative and operational character of the superluminal effects. To this end we employ a rigorous formalism of probability measures on spacetime and quantify the probability of a successful superluminal bit transfer via the single-particle Schrödinger-Newton equation. We demonstrate that such a quantity decreases with the increasing size and mass of the system. Furthermore, we prove that the Einstein-Dirac system, which yields the Schrödinger-Newton equation in the non-relativistic limit, is perfectly compatible with the relativistic causal structure. Our study demonstrates that the Schrödinger-Newton equation, which is by construction non-relativistic, is in fact `more compatible' with the no-signalling principle than the ordinary free Schrödinger equation.
It has been suggested by Diósi and Penrose that the occurrence of quantum state reduction in macroscopic objects is related to a manifestation of gravitational effects in quantum mechanics. Although within Penrose's framework the dynamics of the quantum state reduction is not prescribed, it was suggested that the so-called Schrödinger–Newton equation can be used to at least identify the resulting classical end states. Here we analyse the extent to which the Schrödinger–Newton equation can be used as a model to generate a full, time-dependent description of the quantum state reduction process. We find that when supplied with an imaginary gravitational potential, the Schrödinger–Newton equation offers a rationalization for some of the hitherto unexplained characteristics of quantum state reduction. The description remains incomplete however, because it is unclear how to fully recover Born's rule.
A model is developed to describe state reduction in an EPR experiment as a continuous, relativistically invariant, dynamical process. The system under consideration consists of two entangled isospin particles each of which undergoes isospin measurements at spacelike separated locations. The equations of motion take the form of stochastic differential equations. These equations are solved explicitly in terms of random variables with a priori known probability distribution in the physical probability measure. In the course of solving these equations a correspondence is made between the state reduction process and the problem of classical nonlinear filtering. It is shown that the solution is covariant, violates Bell inequalities and does not permit superluminal signalling. It is demonstrated that the model is not governed by the free will theorem and it is argued that the claims of Conway and Kochen, that there can be no relativistic theory providing a mechanism for state reduction, are false.
This paper demonstrates the capability to simulate EPR experiments in a realistic way (i.e. without transmission of information between the source and the detector) and confirms even more the Bell’s theorem, by demonstrating that effectively non-local variables are required to violate the Bell’s inequality. Unlike all EPR experiments, this modeling instead of simply noting the inadequacy of a local modeling proposes a new way of non-local idealization. These simulations work on spreadsheet by idealizing objects with a non-linear notion, the extended object. This allows to set up a new principle of locality (or non-locality) and a new meaning on the loopholes of EPR experiments. It sheds new light on several concepts of Quantum Mechanics (QM): the quantum states (would represent an equivalence class of many different indistinguishable objects seen as equivalent in terms of measurement), the wave function (would represent a probability on this equivalence class seen as a single set despite different probabilities of happening for the individual objects composing this equivalence class), the measurement (in our simulation, the measured object is not modified), the superposition of states (in our simulation, Schrödinger's cat is never both dead and alive). These simulations lead to several kinds of EPR experiments with violations of the Bell’s inequalities with values different from the QM, allow to realize entanglements with more than 2 objects (whatever the number of Alice and Bob) or even with more than 2 results (beyond the only “+1” or “-1” outputs). Many questions remain to be explored in the physics domain on quantum phenomena (teleportation, encryption, quantum computing, etc.…), but this theoretical approach also reveals the necessity of developing new non-linear tools in mathematics domain and seems to show that QM could finally be a limit case of a more fundamental nonlinear physics theory founded on these extended objects. This observation is strangely reminiscent of developments such as string theory.
Nonlinear modifications of quantum mechanics generically lead to nonlocal effects which violate relativistic causality. We study these effects using the functional Schrödinger equation for quantum fields and identify a type of nonlocality which causes nearly instantaneous entanglement of spacelike separated systems. We describe a simple example involving widely separated wave packet (coherent) states, showing that nonlinearity in the Schrödinger evolution causes spacelike entanglement, even in free field theory.