This document contains comments about the article "De Broglie-Bohm theory" in Wikipedia
• The text in italics is copied from that url
• Immediate followed by some comments
In the last paragraph I explain my own opinion.

### Introduction

The article starts with the following sentence.
The theory is deterministic and explicitly nonlocal: the velocity of any one particle depends on the value of the guiding equation, which depends on the configuration of the system given by its wavefunction; the latter depends on the boundary conditions of the system, which in principle may be the entire universe.
The logic in this sentence is wrong. It should be reverse.
• The measurements come first. That means first you have to measure the velocities of the particles.
• Next come the calculations. That means based on the measurements to calculate the parameters of the wave functions.
Within these calculations the boundaries should be included.
• Finally are the predictions.

### 1.1 Double-slit experiment

The double-slit experiment is an illustration of wave-particle duality.
Okay
The system seems to exhibit the behaviour of both waves (interference patterns) and particles (dots on the screen).
Okay.
The problem is that the picture beside the text does not show the dots.
The Copenhagen interpretation states that the particles are not localised in space until they are detected, so that, if there is no detector on the slits, there is no information about which slit the particle has passed through. If one slit has a detector on it, then the wavefunction collapses due to that detection.
The Copenhagen interpretaion does not specify any law. In the physical reality there are reactions inwhich particles are changed, created, destroyed and or modified, all the time. In some cases reactions create photons which allow us to be aware that there are reactions. Otherwise we do not know. That is all.
When the particle hits the screen, the original particle reacts with the screen and dissolves into something else. There is no collapse of a wave function .

A whole different issue how do to explain interference patterns. A very important issue: a very clear description of the experiments performed.

The final position of the particle on the detector screen and the slit through which the particle passes is determined by the initial position of the particle.
Okay. See reference 1
Next:
Such initial position is not knowable or controllable by the experimenter, so there is an appearance of randomness in the pattern of detection.
The appearance of a random pattern implies that the initial conditions are not exactly the same.
In Bohm's 1952 papers he used the wavefunction to construct a quantum potential that, when included in Newton's equations, gave the trajectories of the particles streaming through the two slits.
What Bohm did he used a wave function to describe each individual particle and the interference pattern.
The problem is such a wave function does not explain the interference pattern.
To explain the behavior when the particle is detected to go through one slit, one needs to appreciate the role of the conditional wavefunction and how it results in the collapse of the wavefunction; this is explained below.
All of this does not explain the interference pattern. The interference pattern is a physical phenomena which requires a physical explanation.
The basic idea is that the environment registering the detection effectively separates the two wave packets in configuration space.
This sentence is "difficult" to understand, because it is not clear.

### 2.1 The ontology

The wavefunction itself, and not the particles, determines the dynamical evolution of the system: the particles do not act back onto the wave function.

### 2.4 Relation to the Born rule

For a given experiment, we can postulate this as being true and verify experimentally that it does indeed hold true, as it does.
Very tricky sentence.

### 3.1 Relativity

The relation between nonlocality and preferred foliation can be better understood as follows. In de Broglie–Bohm theory, nonlocality manifests as the fact that the velocity and acceleration of one particle depends on the instantaneous positions of all other particles.
This sentence has no contents. It has no physical meaning to claim that something (a parameter of a particle) is a function of all the other particles in the universe.
On the other hand, in the theory of relativity the concept of instantaneousness does not have an invariant meaning.
First of all you have to define what instantaneous means.
What has relativity to do with this?
Thus, to define particle trajectories, one needs an additional rule that defines which space-time points should be considered instantaneous.
What is the purpose of: defining a particles trajectory?
IMO it is impossible to define a particular path of a particle. Even if one does, how do you demonstrate that the particle actual took that path?
The simplest way to achieve this is to introduce a preferred foliation of space-time by hand, such that each hypersurface of the foliation defines a hypersurface of equal time.
How do you know that this is the simplest way? Very tricky sentence.

### 3.2 Spin

To incorporate spin, the wavefunction becomes complex-vector-valued.
That maybe true.

### 4.1 Measuring spin and polarization

According to ordinary quantum theory, it is not possible to measure the spin or polarization of a particle directly; instead, the component in one direction is measured; the outcome from a single particle may be 1, meaning that the particle is aligned with the measuring apparatus, or -1, meaning that it is aligned the opposite way.
This sentence is too simple. I expect that what they mean is that spin is identical as a vector which has three components in the x, y and z direction. Only one can be "measured" based on the alignment of the measuring apparatus.
For an ensemble of particles, if we expect the particles to be aligned, the results are all 1.
A better sentence is: "For an ensemble of particles, if the results are all 1 (in different directions), the particles are called: aligned.
In de Broglie–Bohm theory, the results of a spin experiment cannot be analyzed without some knowledge of the experimental setup.
This is the case for all different experiments. Why this sentence?
It is possible[43] to modify the setup so that the trajectory of the particle is unaffected, but that the particle with one setup registers as spin-up, while in the other setup it registers as spin-down.
In order to understand you must read reference [43].

### 4.2 Measurements, the quantum formalism, and observer independence

De Broglie–Bohm theory gives the same results as quantum mechanics.
Than why both?
It treats the wavefunction as a fundamental object in the theory, as the wavefunction describes how the particles move.
This last sentence is 100% correct. The wave function is a description. It does not explain anything. The biggest problem is how to calculate this function in a real environment IMO this is impossible.
This means that no experiment can distinguish between the two theories.
This seems a logical conclusion from the first sentence.

### 4.2.1 Collapse of the wavefunction

De Broglie–Bohm theory is a theory that applies primarily to the whole universe.
Each theory applies to the whole universe. I think they mean involves instead of imply.
That is, there is a single wavefunction governing the motion of all of the particles in the universe according to the guiding equation.
But what does this physical mean? IMO nothing.
The following makes more sense: Each particle has a wave function. Not all the wave functions are identical. This difference is embedded in the parameters of the wave function. The wave function i.e. the parameters of identical particles are identical.
Theoretically, the motion of one particle depends on the positions of all of the other particles in the universe.
You can easily write but exactly what is this dependence?
I think that this looks like a self fullfilling prophecy.
In some situations, such as in experimental systems, we can represent the system itself in terms of a de Broglie–Bohm theory in which the wavefunction of the system is obtained by conditioning on the environment of the system.
This sentence is so vaque..
It requires a special setup for the conditional wavefunction of a system to obey a quantum evolution. When a system interacts with its environment, such as through a measurement, the conditional wavefunction of the system evolves in a different way.
That is true by definition. Each interaction involves a change implying that the wave function also changes, but what does that mean?
The evolution of the universal wavefunction can become such that the wavefunction of the system appears to be in a superposition of distinct states.
What mean superposition in this context?
But if the environment has recorded the results of the experiment, then using the actual Bohmian configuration of the environment to condition on, the conditional wavefunction collapses to just one alternative, the one corresponding with the measurement results.
Again here: I think that this looks like a self fullfilling prophecy.
In reality the sentence explains nothing.

### 4.2.2 Operators as observables

In the standard quantum formalism, measuring observables is generally thought of as measuring operators on the Hilbert space.
I think the article becomes overly complex.
A major point of the analysis is that many of the measurements of the observables do not correspond to properties of the particles; they are (as in the case of spin discussed above) measurements of the wavefunction.
Which are descriptions or properties of the particles.
If one believes that spin measurements are indeed measuring the spin of a particle that existed prior to the measurement, then one does reach contradictions.
This is not a belief. If one performs a measurement one tries to quantify a property. If one repeats such a measurements one hopes to get the same result. If that is the case than one assumes that this condition was also true before the measurement was done.

### 4.3 Heisenberg's uncertainty principle

The Heisenberg's uncertainty principle states that when two complementary measurements are made, there is a limit to the product of their accuracy.
I think this inaccuracy is even stronger.
As an example, if one measures the position with an accuracy of Delta (x) and the momentum with an accuracy of Delta (p), then Delta (x) * Delta (p) are greater or equal h.
I think the logic is different. If the accuracy of x is Delta(x) then the accuracy of p Delta(p) can only be such that Delta (x) * Delta (p) is equal to h or larger. That means how smaller Delta(x) how larger Delta(p).
In de Broglie–Bohm theory, there is always a matter of fact about the position and momentum of a particle.
From a physical point of view each particle has always a position and a momentum.
Each particle has a well-defined trajectory, as well as a wavefunction.
From a physical point of view this is true. The problem is we do not know what it is.
Observers have limited knowledge as to what this trajectory is (and thus of the position and momentum).
The problem is in practice if you want to know what the trajectory is, you have to disturb the trajectory.

### 6.2 Bohmian mechanics

This term is used to describe the same theory, but with an emphasis on the notion of current flow, which is determined on the basis of the quantum equilibrium hypothesis that the probability follows the Born rule.

### Reflection 1: Single particle interference patterns.

If you want to understand single particle interference patterns than it is important to perform certain experiments. A detailed description of each is required.
1. In the first experiment you have a source which creates a stream of single particles (at regular intervals). This stream is quided through a barrier with one slit and captured on one screen as invidual dots.
The result should be a normal distribution.
2. In the second experiment the first slit is closed and a second slit is opened. Nothing else is changed. The duration of this (second) experiment should be the same as the first.
The result should again be a normal distribution (of the same number of dots), but not exactly at the same location.
3. In the third experiment both slits should be open. The duration should be the same as the first two.
The result should be an interference pattern.
4. In the fourth experiment, at the same position of the barrier, the barrier is replaced by the screen. The duration is the same as from all the other experiments.
The result should be a rather circular pattern of dots. The total number of dots detected should be the same?
What the results of the experiment tells us (if as predicted) that you have a source, which creates a stream of particles, which the condition that each particle atleast can go through either slit. This conclusion is already remarkable because the particle can also interfer with the barrier.
IMO I would expect that in experiment 4 more particles are detected than in the other 3 experiments implying that not all the particles actual go through either one slit.

### Reflection 2: Importance of wave function.

The problem with the concept of a wave function that you can not realy explain something. You can not explain the outcome of any physical experiment. As such the concept of collapse of the wave function explains nothing.

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Created: 18 January 2017

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