# Observables and Unobservables in Quantum Mechanics: How the No-Hidden-Variables Theorems Support the Bohmian Particle Ontology

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## Abstract

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## 1. Introduction

## 2. Quantum Orthodoxy and Bohmian Mechanics

(Q) A measurement does not, in general, reveal a preexisting value of the measured property.

- How do the quantum observables acquire definite values upon measurement?It is now generally acknowledged that measurements are not a new type of interaction—let alone a primitive metaphysical concept—that requires a special treatment, but come under the common types of physical interactions (electromagnetism, gravitation, etc.). Hence, our physical theories should be able, at least in principle, to describe them. This, in turn, entails that the notion of measurement must not be part of the axioms of a physical theory. Thus, if quantum theory implies that the observable values are not merely revealed but produced by the measurement process—that is, by the interaction between the measurement device and the measured system, the theory should tell us how they are produced.
- What characterizes a physical system prior to—or better: independent of—measurement?After all, there must be some sort of ontological underpinning to the measurement process and the empirical data that it yields. That is, there must be something in the world on which the measurement is actually performed—something with which the observer or measurement device interacts, and there must be something definite about the physical state of the observer or measurement device that does not, in turn, require a measurement of the measurement (and so on, ad infinitum).

- Particle configuration: There always is a configuration of N permanent point particles in the universe that are characterized only by their positions ${X}_{1},\cdots ,{X}_{N}$ in three-dimensional, physical space at any time t.
- Guiding equation: A wave function $\Psi $ is attributed to the particle configuration, being the central dynamical parameter for its evolution. On the fundamental level, $\Psi $ is the universal wave function attributed to all the particles in the universe together. The wave function has the task to determine a velocity field along which the particles move, given their positions. It accomplishes this task by figuring in the law of motion of the particles, which is known as the guiding equation:$$\frac{\mathrm{d}{X}_{k}}{\mathrm{d}t}=\frac{\hslash}{{m}_{k}}\mathrm{Im}\frac{{\nabla}_{k}\Psi}{\Psi}({X}_{1},\cdots ,{X}_{N}).$$This equation yields the evolution of the k-th particle at a time t as depending on, via the wave function, the position of all the other particles at that time.
- Schrödinger equation: The wave function always evolves according to the Schrödinger equation:$$i\hslash \frac{\partial \Psi}{\partial t}=-\sum _{k=1}^{N}\frac{{\hslash}^{2}}{2{m}_{k}}{\Delta}_{k}\Psi +V\Psi .$$
- Typicality measure: On the basis of the universal wave function $\Psi $, a unique stationary (more precisely: equivariant) typicality measure can be defined in terms of the ${|\Psi |}^{2}$–density (see Goldstein and Struyve [9] for a proof and precise statement of the uniqueness result). Given that typicality measure, it can then be shown that for nearly all initial conditions, the distribution of particle configurations in an ensemble of sub-systems of the universe that admit of a wave function $\psi $ of their own (known as effective wave function) is a ${|\psi |}^{2}$–distribution. A universe in which this distribution of the particles in sub-configurations obtains is considered to be in quantum equilibrium.

## 3. No-Hidden-Variables Theorems

There is no “good” map $\widehat{A}\mapsto {Z}_{A}$ from the set of self-adjoint operators on a Hilbert space $\mathcal{H}$ to random variables on a common probability space $\mathsf{\Omega}$ such that the possible values of ${Z}_{A}$ correspond to the eigenvalues of $\widehat{A}$ (that is, the possible measurement values).

#### 3.1. Von Neumann

#### 3.2. Kochen–Specker

(NC) Whenever the quantum mechanical joint distribution of a set of self-adjoint operators $({A}_{1},\dots ,{A}_{m})$ exists, that is, when they form a commuting family, the joint distribution of the corresponding set of random variables, that is, of $({Z}_{{A}_{1}},\dots ,{Z}_{{A}_{m}})$, must agree with the quantum mechanical joint distribution.

- (a)
- The observables in each of the three rows and each of the three columns are mutually commuting.
- (b)
- The product of the three observables in each of the three rows is 1.
- (c)
- The product of the three observables in first two columns is 1, while the product of the right column is $-1$.

#### 3.3. Bell

My own first paper on this subject (Physics 1, 195 (1965)) starts with a summary of the EPR argument from locality to deterministic hidden variables. However the commentators have almost universally reported that it begins with deterministic hidden variables.(Bell [2] (p. 157))

## 4. The Message of the Quantum

#### 4.1. Completeness of Quantum Mechanics

I suspect that you will run into problems at exactly that part of your theory that we just talked about ... You pretend that you could leave everything as it is on the side of observations, that is, that you could just talk in the former language about what physicists observe.(Quoted after Heisenberg [16] (p. 89); translation by the authors.)

#### 4.2. Metaphysical Indeterminacy

Sparse view: when the quantum state of A is not in an eigenstate of $\widehat{O}$, it lacks both the determinate and determinable properties associated with $\widehat{O}$.

#### 4.3. Quantum Logic

the truth-value true. However, according to the doctrine Q, neither$q\vee \neg q$: The particle has z-spin up or z-spin down

norq: The particle has z-spin up

can be considered true prior to a measurement or unless the particle happens to be in a z-spin eigenstate.$\neg q$: The particle has z-spin down

## 5. Measurements in Bohmian Mechanics: Spin

#### 5.1. The Bohmian Treatment of the Measurement Process

- There never are superpositions of anything in physical space. All there is in physical space are particle configurations with always definite positions. Thus, Schrödinger’s cat always is in a configuration of either a live cat or a dead cat. Superpositions concern only the wave function in physical space in its role to determine the trajectories on which the particles move.
- Consequently, quantum logic is irrelevant when it comes to an account of measurement: the particle configuration belongs unambiguously to one of the possible supports of the wave function, which in turn correspond to macroscopically different components of the experimental device, determining in this way the final outcome of the observation at hand.
- Nevertheless, there is entanglement in physical space: the motion of any particle depends on, strictly speaking, the positions of all the other particles in the universe via the wave function. Thus, for instance, in the double slit experiment, the motion of any particle after having passed one slit depends on the position of all the particles making up the experimental set-up, in particular on whether or not the other slit is open. This is the way in which Bohmian mechanics implements the quantum nonlocality proven by Bell’s theorem. The consequence is that the trajectories of the particles often are highly non-classical.
- A measurement is an interaction that will in general change the wave function of the measured system. “Incompatible measurements”—corresponding to non-commuting observables—are simply experiments in which the first measurement interaction changes the wave function in a way that influences the statistics of the second, etc.
- The fact that we cannot go beyond Born’s rule in making predictions is explained not by any indeterminacy of the properties of the particles, or any indeterminism of the dynamics, but by the fact that we cannot have more precise knowledge of the initial particle configuration. As mentioned in Section 2, in Bohmian mechanics, Born’s rule is derived from the laws of motion plus a probability (more precisely: typicality) measure linked with these laws.

#### 5.2. What Is Measured in a Spin Measurement?

#### 5.3. Is Bohmian Mechanics “Contextual”?

#### 5.4. Why Measurements?

## 6. Are Observables Observable?

- The physical properties are not observable.
- The physical properties are a small subset of the observables (small enough to avoid the no-hidden-variables results).

in physics the only observations we must consider are position observations, if only the positions of instrument pointers. It is a great merit of the de Broglie–Bohm picture to force us to consider this fact. If you make axioms, rather than definitions and theorems, about the “measurement” of anything else, then you commit redundancy and risk inconsistency.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Pattern created by a ray of silver atoms in the original Stern–Gerlach experiment:

**left**: without,

**right**: with magnetic field.

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Lazarovici, D.; Oldofredi, A.; Esfeld, M.
Observables and Unobservables in Quantum Mechanics: How the No-Hidden-Variables Theorems Support the Bohmian Particle Ontology. *Entropy* **2018**, *20*, 381.
https://doi.org/10.3390/e20050381

**AMA Style**

Lazarovici D, Oldofredi A, Esfeld M.
Observables and Unobservables in Quantum Mechanics: How the No-Hidden-Variables Theorems Support the Bohmian Particle Ontology. *Entropy*. 2018; 20(5):381.
https://doi.org/10.3390/e20050381

**Chicago/Turabian Style**

Lazarovici, Dustin, Andrea Oldofredi, and Michael Esfeld.
2018. "Observables and Unobservables in Quantum Mechanics: How the No-Hidden-Variables Theorems Support the Bohmian Particle Ontology" *Entropy* 20, no. 5: 381.
https://doi.org/10.3390/e20050381