# Why Bohmian Mechanics? One- and Two-Time Position Measurements, Bell Inequalities, Philosophy, and Physics

## Abstract

**:**

## 1. Introduction

## 2. Bohmian Positions

- Assumption
**H**:Position measurements merely reveal in which (spatially separated and non-overlapping) mode the Bohmian particle actually is.

## 3. Two-Time Position Correlation in a Bell Test

**H**, one does not need to actually measure the positions of the particles; it suffices to know that each is in one specific mode. Hence, one can undo Alice’s measurement as illustrated in Figure 3. After the phase shift $-x$, the quantum state is precisely back to the initial state ${\psi}_{0}$, see Equation (1). Alice can thus perform a second measurement with a freshly chosen phase x′ and a third beam splitter, see Figure 3. Moreover, as Bohmian trajectories cannot cross each other (in configuration space), if ${r}_{A}$ is in Mode 1 before the first BS, then ${r}_{A}$ is also in Mode 1 before the last BS.

**H**, if ${r}_{A}\in \phantom{\rule{0.166667em}{0ex}}$“1”, then any position measurement performed by Alice between the first and second beam splitter necessarily results in $a=1$. Similarly, ${r}_{A}\in \phantom{\rule{0.166667em}{0ex}}$“2” implies $a=2$. Thus, Alice’s position measurement after the third beam splitter is determined by r′${}_{A}$, and Bob’s measurement is determined by ${r}_{B}$. Hence, it seems that one obtains a joint probability distribution for Alice’s measurements results and Bob’s: $P(a,a$′$,b|x,x$′$,y)$. However, such a joint probability distribution implies that Alice does not have to make any choice (she merely makes both choices, one after the other), and in such a situation there cannot be any Bell inequality violation. Hence, as claimed in [2], it seems that the existence of two-time position correlations in Bohmian mechanics prevents the possibility of a CHSH-Bell inequality violation, in contradiction with quantum theory predictions and experimental demonstrations [3].

## 4. What Is Going on? Let’s Add a Position Measurement

**H**, so Assumption

**H**is wrong. Every introduction to Bohmian mechanics should emphasize this. Indeed, Assumption

**H**is very intuitive and appealing, but wrong and confusing.

## 5. What about Large Systems?

## 6. Assumption **H** Revisited

**H**is wrong. How should one reformulate it? Clearly, a position measurement does not merely reveal the Bohmian particle because of the following:

- A position measurement necessarily involves the coupling to a large system, some sort of pointer, and this coupling implies some perturbation. Hence the “merely” in assumption
**H**is wrong [7]. - Whether a position measurement reveals information about the Bohmian particle or not depends on how the coupling to a large system is done and on how that large system (the pointer) evolves. Hence, not all measurements that, according to quantum theory, are position measurements, are also Bohmian-position measurements: some quantum-position measurements do not reveal where the Bohmian particle is.

## 7. Why Bohmian Mechanics

## 8. Conclusions

**H**is wrong. Still, Bohmian mechanics is deeply consistent. Position measurements perturb the system, even in Bohmian mechanics. Hence, the existence of two-time position correlations is not in contradiction with possible violations of Bell inequalities.

## Acknowledgments

## Conflicts of Interest

## Appendix A. Slow Position-Measurements in Bohmian Mechanics

#### Appendix A.1. Bohmian Trajectories in a Semi-Interferometer

**Figure A1.**A Bohmian particle and its pilot wave arrive on a beam splitter (BS) from the left in Mode “in”. The pilot wave emerges both in Modes 1 and 2, as the quantum state in standard quantum theory. Modes 1 and 2 meet again, but there is no beam splitter at this meeting point. Nevertheless, the Bohmian trajectories bounce at this point as indicated by the red arrows. Intuitively, this can be understood because the evolution equation of the Bohmian position is a first order differential equation, so Bohmian trajectories never cross each other. This intuition is confirmed by numerical simulations.

#### Appendix A.2. Position Measurements in Modes 1 and 2

**Figure A2.**Semi-interferometer with two macroscopic pointers locally coupled to Modes 1 and 2. The pointers are initially at rest, $|{p}_{j}=0\rangle $, but when detecting a particle they get a kick and end in a quantum state with momentum k: $|{p}_{j}=k\rangle $.

#### Appendix A.3. Conclusions

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**Figure 1.**A Bohmian particle and its pilot wave arrive on a beam splitter (BS) from the left in Mode “in”. The pilot wave emerges both in Modes 1 and 2, as per the quantum state in standard quantum theory. However, the Bohmian particle emerges either in Mode 1 or in Mode 2, depending on its precise initial position. As Bohmian trajectories cannot cross each other (in configuration space), if the initial position is in the lower half of Mode “in”, then the Bohmian particle has the BS in Mode 1 or, if not, in Mode 2.

**Figure 2.**Two Bohmian particles spread over four modes. The quantum state is entangled, see Equation (1), so the two particle are either in Modes 1 and 4 or in Modes 2 and 3. Alice applies a phase x on Mode 1 and Bob a phase y on Mode 4. Accordingly, after the two beam splitters, the correlations between the detectors allow Alice and Bob to violate Bell inequality. The convention regarding mode numbering is that modes do not cross, i.e., the nth mode before the beam splitter goes to detector n.

**Figure 3.**Alice’s first “measurement”, with phase x, can be undone because in Bohmian mechanics there is no collapse of the wavefunction. Hence, after having applied the phase $-x$ after her second beam splitter, Alice can perform a second “measurement” with phase x′. Mode number convention implies, e.g., that Mode 1 is always the upper mode, i.e., the mode on which all phases x, $-x$ and x′, are applied.

**Figure 4.**We add a pointer that measures through which path Alice’s particle propagates between her first and second beam splitter. The pointer moves up if Alice’s particle goes through the upper path, i.e., ${r}_{A}\in \phantom{\rule{0.166667em}{0ex}}$“1”, and down if it goes through the lower path, i.e., ${r}_{A}\in \phantom{\rule{0.166667em}{0ex}}$“2”. Hence, by finding out the pointer’s position, one learns through which path Alice’s particle goes, i.e., one finds out Alice’s first measurement result, though it all depends how fast the pointer moves. See text for explanation.

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Gisin, N. Why Bohmian Mechanics? One- and Two-Time Position Measurements, Bell Inequalities, Philosophy, and Physics. *Entropy* **2018**, *20*, 105.
https://doi.org/10.3390/e20020105

**AMA Style**

Gisin N. Why Bohmian Mechanics? One- and Two-Time Position Measurements, Bell Inequalities, Philosophy, and Physics. *Entropy*. 2018; 20(2):105.
https://doi.org/10.3390/e20020105

**Chicago/Turabian Style**

Gisin, Nicolas. 2018. "Why Bohmian Mechanics? One- and Two-Time Position Measurements, Bell Inequalities, Philosophy, and Physics" *Entropy* 20, no. 2: 105.
https://doi.org/10.3390/e20020105