# Nonlocality in Bell’s Theorem, in Bohm’s Theory, and in Many Interacting Worlds Theorising

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## Abstract

**:**

## 1. Introduction

- Is the universe local or nonlocal?
- What is the future of scientific explanation? Is scientific metaphysics, e.g., the notions of reality, causality, or physical influence, obsolete in mathematical accounts of the quantum world?

- What is David Bohm’s legacy for the future of quantum physics?

- Are nonlocal connections—e.g., “action-at-a-distance”—fundamental elements in a radically new conception of reality?

## 2. Is the Universe Local or Nonlocal?

In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote.

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.

It is simply not clear how to translate Bell’s words here (about locality) into a sharp mathematical statement in terms of which the EPR argument might be rigorously rehearsed. …[I]t must be admitted that Bell’s recapitulation of the EPR argument in this paragraph leaves something to be desired.

A consequence …of “local causality” [is] the outcomes [in the two labs] having no dependence on one another nor on the settings of the remote [measurement], but only on the local [measurement settings] and on the past causes.

Quantum mechanics…gives certain correlations which …cannot be [reproduced by] a locally causal theory.

## 3. David Bohm’s Legacy: Permission to Theorise Radically New Conceptions of Reality

[A] hidden variable interpretation of elementary quantum theory [4,5] has been explicitly constructed. That particular interpretation has …a grossly nonlocal structure. This is characteristic, according to the result to be proved here, of any such theory which reproduces exactly the quantum mechanical predictions.

## 4. Nonlocality in the Many Interacting Worlds Approach

#### 4.1. General Considerations

#### 4.2. Simulations

## 5. MIW Beyond the Toy Model

#### 5.1. Constructing Generalised Interworld Potentials

#### 5.2. Examples

## 6. Numerical Results

#### 6.1. Toy Model

#### 6.2. Higher Order Potential

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MIW | Many Interacting Worlds |

EPR | Einstein, Podolsky, and Rosen |

1D | One (spatial) dimension |

2D | Two (spatial) dimensions |

3D | Three (spatial) dimensions |

## Appendix A. A Equivariance Method

**Figure A1.**The first excited state of a harmonic oscillator is simulated using 5-world approximation in equivariance Mmethod. Five-thousand worlds are used and only the two world-particles next to the node are simulated. The rest are kept stationary, similar to Figure 4 for the rational smoothing case.

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**Figure 1.**(

**a**) 50 world-particles (that is, 50 different worlds describing a single particle) are distributed based on the probability density of the ground state of a harmonic oscillator. (

**b**) Trajectories of the 50 world-particles. As expected, the classical and quantum forces approximately cancel each other and the world-particles stay approximately stationary. The slight oscillations near the boundary are due to approximation of the Bohmian potential by the toy-model potential.

**Figure 2.**(

**a**) 40 world-particles are distributed based on the probability density of the first excited state of a harmonic oscillator. (

**b**) Trajectories for the first excited state of a harmonic oscillator using the potential in Equation (4). The world-particles do not stay stationary. Particularly those near the node move towards the middle and fill the gap.

**Figure 3.**(

**a**) 5000 world-particles are distributed based on the probability density of the first excited state of a harmonic oscillator. Only 20 world-particles around the node in the middle are shown. (

**b**) Trajectories of the world-particles for the initial distribution in (

**a**). To apply the correct boundary condition, we kept five world-particles, near each boundary, fixed and simulated the dynamics of the remaining 10 world-particles in the middle. time-steps are ${10}^{-8}$. Since the nearest neighbours of the node do not stay near the starting point and move to the middle of the node after approximately 0.1 of a period, we did not continue the simulation for a full period. (

**c**) The same test as (

**b**) with time-steps of ${10}^{-9}$. Thus, the failure of the simulation is not an artefact of large time-steps.

**Figure 4.**Simulation of only two worlds adjacent to the node for the first excited state of a harmonic oscillator. The initial positions were set by considering 5000 worlds in total to describe the excited state. To apply the boundary condition, the rest of the world-particles were kept fixed. For the evolution, rational smoothing with $L=4$ (equivalent to 5-world approximation) is used. Time step is ${10}^{-9}$. The two world-particles stay stationary as expected.

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**MDPI and ACS Style**

Ghadimi, M.; Hall, M.J.W.; Wiseman, H.M.
Nonlocality in Bell’s Theorem, in Bohm’s Theory, and in Many Interacting Worlds Theorising. *Entropy* **2018**, *20*, 567.
https://doi.org/10.3390/e20080567

**AMA Style**

Ghadimi M, Hall MJW, Wiseman HM.
Nonlocality in Bell’s Theorem, in Bohm’s Theory, and in Many Interacting Worlds Theorising. *Entropy*. 2018; 20(8):567.
https://doi.org/10.3390/e20080567

**Chicago/Turabian Style**

Ghadimi, Mojtaba, Michael J. W. Hall, and Howard M. Wiseman.
2018. "Nonlocality in Bell’s Theorem, in Bohm’s Theory, and in Many Interacting Worlds Theorising" *Entropy* 20, no. 8: 567.
https://doi.org/10.3390/e20080567