De Broglie-Bohm theory

The De Broglie Bohm theory or Bohmian mechanics is - depending on the definition of the terms - an interpretation or an alternative formalism of quantum mechanics . It reproduces all the predictions of ( non-relativistic ) quantum mechanics, but, understood as an interpretation, is based on an understanding of reality that radically deviates from the Copenhagen interpretation .

Bohm's mechanics is a deterministic theory and allows a simple solution to the measurement problem of quantum mechanics , i. H. the act of measurement or observation does not play an important role (for the measurement problem see also the article on Schrödinger's cat ).

As with most interpretations of quantum mechanics, there is no way of experimentally distinguishing between Bohmian mechanics and usual quantum mechanics; That is, Bohmian mechanics and quantum mechanics make the same predictions in all experimentally verifiable situations. The de Broglie-Bohm theory is only supported by a small minority of physicists.

history

Louis-Victor de Broglie (1929)

Bohmian mechanics was developed in the 1920s by the French physicist Louis de Broglie . De Broglie referred to it as the "theory of the leadership wave " ( theory de l'onde pilote , English pilot wave theory ). However, this received little attention and was forgotten. Without knowing de Broglie's work, the American physicist David Bohm developed an equivalent version of this theory in the 1950s . Bohm later referred to the theory as the ontological or causal interpretation of quantum mechanics.

Since the 1970s, the Irish physicist John Stewart Bell was one of the few prominent physicists who advocated Bohmian mechanics.

Since the 1990s, there has been increased research activity in this area, e. B. with a working group at LMU ( Detlef Dürr ), Rutgers University in New Jersey ( Sheldon Goldstein ) and the University of Genoa ( Nino Zanghì ). Through this working group, the name was Bohmian Mechanics (Engl. Bohmian mechanics ) coined. This designation can be criticized for suppressing de Broglie's role. In view of the history of its origins, the name “De Broglie Bohm Theory” seems more appropriate. In the following, both terms are used synonymously .

formalism

The basic idea of ​​the De Broglie-Bohm theory is to describe a system not by the wave function ( ) alone, but by the pair of wave function and the particle locations ( ) of the respective objects ( electrons , atoms , etc.). The trajectories of the particles are the so-called hidden parameters of the theory. Bohm's mechanics is thus defined by two basic equations: on the one hand by the usual time-dependent Schrödinger equation of quantum mechanics: ${\displaystyle \psi }$ ${\displaystyle Q_{i}}$ ${\displaystyle Q_{i}(t)}$

${\displaystyle (1)\qquad \ \mathrm {i} \hbar {\frac {\partial \psi }{\partial t}}=H\psi \,}$,

as well as by the equation of motion ("guide equation ") for the particle locations : ${\displaystyle Q_{i}}$

${\displaystyle (2a)\qquad \ {\frac {\mathrm {d} {\vec {Q}}_{i}}{\mathrm {d} t}}={\frac {\hbar }{m_{i}}}\operatorname {Im} {\frac {{\vec {\nabla }}_{i}\psi }{\psi }}\,}$.

${\displaystyle m_{i}}$denotes the mass of the i-th particle and the Nabla operator applied to the coordinates of the i-th particle. ${\displaystyle {\vec {\nabla }}_{i}}$

If one writes the wave function in polar representation ,

${\displaystyle \psi =R\exp \left({\frac {\mathrm {i} }{\hbar }}S\right),}$

then the control equation can also be equivalent using

${\displaystyle (2b)\qquad {\frac {\mathrm {d} {\vec {Q}}_{i}}{\mathrm {d} t}}={\frac {{\vec {\nabla }}_{i}S}{m_{i}}}}$

be formulated.

Figuratively speaking, the wave function “guides” or “guides” the movement of the particles. Within this theory, the quantum objects move on continuous (and deterministic) tracks. This movement is of course only clearly defined when initial conditions are specified. Note that the guide equation is a first order differential equation , i.e. the specification of the particle locations at a point in time already defines the movement. In contrast to this, in classical mechanics, only position and speed (or impulse) clearly determine the movement. All predictions of quantum mechanics can be reproduced by the De Broglie-Bohm theory if one stipulates that at the beginning the distribution of the locations fits the probability interpretation of the wave function . This distribution is called the "quantum equilibrium distribution". The initial condition therefore means the validity of the following “quantum equilibrium hypothesis”. ${\displaystyle Q_{i}}$ ${\displaystyle \psi }$

Quantum equilibrium hypothesis

The quantum equilibrium hypothesis reads: The spatial distribution of a system described by the wave function is . ${\displaystyle \rho }$ ${\displaystyle \psi }$ ${\displaystyle \rho =|\psi |^{2}}$

The distribution is the so-called quantum equilibrium distribution. ${\displaystyle |\psi |^{2}}$

Due to the quantum equilibrium hypothesis, the Heisenberg uncertainty principle is not violated in Bohm's mechanics either. In contrast to usual quantum mechanics, the probability statements of Bohm's mechanics are only due to our ignorance of the concrete initial conditions.

The quantum mechanical continuity equation

${\displaystyle {\frac {\partial \rho }{\partial t}}+{\vec {\nabla }}\cdot \left(\rho {\frac {{\vec {\nabla }}S}{m}}\right)=0\,}$

ensures that a once -distributed system retains this property. However, it remains to be seen why this distribution should be available at any point in time. There are various approaches to answering this question. Obviously, it is unsatisfactory to ascribe this circumstance to very special initial conditions (e.g. of the universe). It would be physically intuitive if one could specify a dynamic mechanism that explains how (as many as possible) initial conditions approach quantum equilibrium. Valentini, for example, pursues this approach, who argues how a larger class of initial conditions lead to an approximate quantum equilibrium distribution due to the Bohmian dynamics. ${\displaystyle |\psi |^{2}}$

The question of whether many or a few initial conditions are compatible with the quantum equilibrium hypothesis naturally presupposes a measure with which these quantities can be measured. Dürr et al. chose this freedom as a starting point. These authors choose a measure for which, due to a special weighting, almost all initial conditions are compatible with the quantum equilibrium hypothesis, and argue why this measure is natural . With this they explain why a hypothetical Bohmian universe is in quantum equilibrium. The main result of this work is to (i) define the concept of the wave function of a subsystem, and (ii) to show that these subsystems satisfy the quantum equilibrium hypothesis. In this sense, according to Dürr, the quantum equilibrium hypothesis is not a postulate , but a consequence of Bohm's mechanics.

Properties of the Bohmian Trajectories

Figure 1: Simulation of some Bohmian trajectories for a double slit. The particles are guided through the wave function that interferes at the double slit. In this way, the well-known interference pattern occurs, although a movement of particles is described.

As mentioned, an initial condition uniquely defines each Bohmian trajectory, since the guide equation (2) is a first-order differential equation. The consequence of this is that the particle trajectories cannot intersect in the configuration space . In the single-particle case , the movement takes place without overlapping in the spatial space. In this way one can get a qualitative picture of the particle movement in simple systems without numerical simulation.

Figure 1 shows the simulation of some trajectories for the double slit . The property that there is no overlap, together with the symmetry of the arrangement, ensures that the tracks cannot intersect the center plane. This figure also illustrates that the Bohmian trajectories are completely non-classical. They show changes in direction, although the area behind the gap is field-free in the classic sense. In this sense, there is neither conservation of energy nor of momentum on the level of the individual orbits .

In the case of real wave functions, the situation is even simpler. Since the phase of the wave function disappears here, the particle rests in distributed locations. This situation is e.g. B. in the ground state of the hydrogen atom or in the energy eigenstates of the harmonic oscillator . ${\displaystyle |\psi |^{2}}$

Spin in Bohmian mechanics

It is instructive to look at how the De Broglie-Bohm theory describes spin . Here are several approaches, but one obvious possibility is not attributable to the spin of the particles, but only as a property of the wave function (or the Pauli - spinor ) specific.

Specifically, one moves from the Schrödinger equation to the Pauli equation . The wave function becomes a 2-component spinor . There is - analogous to the description of spinless particles - a current : ${\displaystyle (\psi _{1},\psi _{2})^{\mathrm {T} }}$

${\displaystyle {\vec {j}}=\sum _{a}\left({\frac {\hbar }{2m\mathrm {i} }}(\psi _{a}^{*}{\vec {\nabla }}\psi _{a}-\psi _{a}{\vec {\nabla }}\psi _{a}^{*})-{\frac {e}{mc}}{\vec {A}}\psi _{a}^{*}\psi _{a}\right)\,}$.

Here denotes the vector potential and the spinor index. The control equation is analogous to the spinless case: ${\displaystyle {\vec {A}}}$ ${\displaystyle a}$

${\displaystyle {\frac {\mathrm {d} {\vec {Q}}}{\mathrm {d} t}}={\frac {\vec {j}}{\rho }}\,}$.

Even without an overview of the mathematical details, the following point should become clear: The property spin is not assigned to the particle, i. H. the object on the Bohmian trajectory, and the configuration space remains the same as in the case of spinless objects. In particular, no “hidden variable” is introduced for the spin. The usual way of speaking is that the spin is "contextualized" (see below).

Important properties

Characteristic properties of the De Broglie-Bohm theory are the following:

Solution of the quantum mechanical measurement problem

By far the most important property of the De Broglie-Bohm theory is that it solves the measurement problem of quantum mechanics or that within this theory the measurement problem does not even arise. As a reminder: the measurement problem in quantum mechanics essentially consists in interpreting superpositions of macroscopically different states. These occur naturally in the quantum mechanical treatment of the measurement, although every measurement actually carried out always has a defined result (i.e. it is not described by an overlay).

To clear up this contradiction, John von Neumann postulated a special change of state during the act of measurement, the so-called collapse of the wave function . However, this is less a solution than an admission of the measurement problem. Ultimately, it remains unclear which interaction has the rank of a measurement and how this mechanism is to be understood physically.

In Bohm's mechanics, on the other hand, there is a simple mechanism that characterizes the component of the wave function that corresponds to the actual measurement result: It is the particle location that has reached a branch of the wave function in a continuous movement. In other words: different measurement results are differentiated in the De Broglie Bohm theory by different configurations.

Status of observables and contextuality

A radical innovation of the De Broglie-Bohm theory is its reinterpretation of the observable concept of quantum mechanics. The usual quantum mechanics identifies all observable quantities with Hermitian operators that act on the Hilbert space of states. Not interchanging these operators is interpreted as an expression of the radical novelty of quantum mechanics.

The de Broglie-Bohm theory takes a different approach here. It explicitly characterizes the location and describes it using real coordinates and the particle speed using a real vector field (on the configuration space). All other quantities (spin, energy, momentum, etc.) only have a derived status. The reason for this is simple: When performing an experiment to "measure" e.g. B. the spin component (as with any other event) the output is determined by the wave function and the starting point. So there is no measurement in the literal sense of the word, i.e. that is, no intrinsic property is determined that also exists independently of the measurement. The somewhat unfortunate way of speaking is that these quantities (e.g. the spin) are contextualized , i.e. This means that the measured value depends on the respective context of the measuring arrangement and the starting location. In concrete terms, for example, cases can be constructed in which different spin components are “measured” by different starting locations in systems that are described by the same wave function . Incidentally, this property is the key to why the Cooking-Specker theorem does not affect the consistency of the De Broglie-Bohm theory. The Hermitian operators of common quantum mechanics do not play a fundamental role in Bohmian mechanics, but appear as mathematical objects that encode probability distributions (cf.).

Non-locality

Since the wave function is defined on the configuration space (with N the number of particles), the guiding equation basically links the movement of individual particles with the position of all others at the same point in time. In this way, objects that are separated in space can also influence one another, i. H. this form of interaction happens faster than light , even instantaneously . This mechanism explains the Bohmian mechanics the EPR effect or the violation of Bell's inequality . However, due to the quantum equilibrium hypothesis, signal transmission using these correlations is not possible. This form of the “Einstein locality” is therefore very well respected. ${\displaystyle R^{3N}}$

However, if the many-particle state is not entangled, i. H. Factored into the proportions of the individual parts, the equations of motion of Bohm's mechanics decouple, and the corresponding subsystems develop independently of one another.

Results such as the already mentioned violation of Bell's inequalities or the “Free Will Theorem” by John Horton Conway and Simon Cooking show that there can be no completions or formulations of quantum mechanics that are local and deterministic.

determinism

The de Broglie-Bohm theory describes the quantum phenomena deterministically, i.e. that is, all changes of state are completely determined by the initial conditions (wave function and configuration). All probability statements are only due to ignorance of the special starting locations.

In contrast to this, the usual view asserts the randomness of quantum phenomena in principle, for example in the act of measurement.

However, it must be emphasized that due to the quantum equilibrium hypothesis, ignorance about the initial conditions is fundamental in the De Broglie-Bohm theory and thus the descriptive content of both theories is identical. In philosophical terminology, the ontological indeterminacy of quantum physics (simplified: there is no place) becomes an “epistemic” indeterminacy in the De Broglie Bohm theory (there is a place, but it cannot be recognized).

Complementarity superfluous

The concept of complementarity was introduced to justify the common use of descriptions in quantum mechanics that contradict each other in the strict sense. For example, it is common perception that wave and particle properties are complementary to one another. This means that they complement each other and that the respective area of ​​application must be taken into account when using them.

In Bohm's mechanics wave and particle character of z. For example, electrons are a simple consequence of the fact that both a particle property (namely the location) and a wave-like quantity (the wave function) are used to describe them. The Figure 1 presents the simulation of some trajectories in the double slit experiment and illustrates this point very clearly.

Schools of de Broglie-Bohm theory

The de Broglie-Bohm theory allows - like every other theory - various equivalent representations. Our previous presentation, for example, did not attach any importance to the so-called quantum potential and thus followed Bell's reading, which was suggested by Dürr et al. a. has been further developed. In many representations of the De Broglie-Bohm theory, however, the quantum potential is emphasized as the decisive characteristic. For this reason, it should also be mentioned here (other differences between different schools of De Broglie-Bohm theory concern the status of observables and the wave function and the derivation of quantum equilibrium).

The quantum potential

In Bohm's presentation of the theory in 1952 (as well as the representations of other authors in 1993) the novelty of the De Broglie-Bohm theory is seen in the appearance of an additional potential term. If you insert the polar representation in the Schrödinger equation and separate the real and imaginary parts, you are led to the following equations: ${\displaystyle \psi =R\cdot e^{{\frac {\mathrm {i} }{\hbar }}S},\rho =R^{2}}$

${\displaystyle (3)\qquad -{\frac {\partial \rho }{\partial t}}={\vec {\nabla }}\cdot \left(\rho {\frac {{\vec {\nabla }}S}{m}}\right)\,}$

${\displaystyle (4)\qquad -{\frac {\partial S}{\partial t}}=V+{\frac {1}{2m}}({\vec {\nabla }}S)^{2}+U_{\text{quant}}\,}$.

Expression (3) is precisely the continuity equation of quantum mechanics. Equation (4) corresponds to the classic Hamilton-Jacobi equation for the effect . In addition to the kinetic term and the potential energy, an additional term occurs here, the so-called quantum potential: ${\displaystyle S}$ ${\displaystyle U_{\text{quant}}}$

${\displaystyle U_{\text{quant}}=-{\frac {\hbar ^{2}}{2m}}{\frac {{\vec {\nabla }}^{2}R}{R}}\,}$.

The classical Hamilton-Jacobi theory is a reformulation of Newtonian (or Hamiltonian) mechanics. The Hamilton-Jacobi equation is first order (but non-linear). The speed (or the pulse) is determined by the condition . This corresponds precisely to the management equation of the De Broglie-Bohm theory. ${\displaystyle m{\vec {v}}={\vec {\nabla }}S}$

With the help of the quantum potential, Bohm's equation of motion can finally be given a Newtonian appearance:

${\displaystyle m{\frac {\mathrm {d} ^{2}{\vec {Q}}}{\mathrm {d} t^{2}}}=-{\vec {\nabla }}(V+U_{\text{quant}})\,}$.

However, since the particle movement is already completely determined by the guide equation, the derivation via the analogy to the Hamilton-Jacobi theory and the additional potential term can be dispensed with. The management equation can e.g. B. can also be motivated directly from considerations of symmetry. The quantum potential formulation also invites one to misunderstand that the De Broglie-Bohm theory is essentially classical mechanics with an additional potential term. Basically, however, the preference for one or the other formulation of the theory is a matter of taste. In addition, each formulation can have a meaningful area of ​​application. For example, the problem of the classic limit value of the De Broglie-Bohm theory can be formulated particularly intuitively in the quantum potential version.

criticism

The de Broglie-Bohm theory is only supported by a small minority of physicists. However, this is only partly due to explicit criticism of this theory, but also to the fact that the De Broglie-Bohm theory does not make any new experimentally verifiable predictions, but is of interest as a contribution to questions of interpretation in physics. Most scientists do not take part in these discussions.

The criticism made of the De Broglie-Bohm theory can be divided into different groups. For example, one can reproach the theory that it characterizes spatial space and that the wave function acts on the particle locations, but not the other way around. In addition, it can appear unsatisfactory that the De Broglie-Bohm theory populates the world with “empty” wave functions; H. those components that do not contain a particle trajectory and due to decoherence should no longer have any influence on the particle dynamics. Instead of seeing it as a criticism, these properties are sometimes seen as notable innovations in the description of nature.

The non-locality of the De Broglie-Bohm theory is often raised as an objection. This can be countered on different levels. On the one hand, in view of the violation of Bell's inequalities, there are numerous physicists who also claim the “non-locality” of conventional quantum mechanics. At least the EPR effect between spatially distant objects is an experimental fact. This discussion suffers from the fact that the term non-locality or non-separability can be given numerous meanings. For example, the "signal locality", i.e. H. no signal propagation at faster than light speed, respected by both quantum mechanics and the De Broglie-Bohm theory.

One can take the view that the charge of non-locality does not apply to a non-relativistic theory such as Bohm's mechanics. The accusation of non-locality is usually linked to the doubt that a satisfactory relativistic (and quantum field theoretical) generalization of the De Broglie-Bohm theory can be given. This objection questions the generalizability of the De Broglie-Bohm theory and seeks a more substantive examination of the theory, or demands its further development. In particular, the quantum equilibrium condition and the requirement of Lorentz covariance seem to contradict one another, i.e. H. in a “bohm-like” relativistic theory an excellent frame of reference must be introduced. However, there are “bohm-like” models of Dirac theory in which this excellent reference system has no experimental effect and all statistical predictions of relativistic quantum mechanics can be reproduced.

There are also various approaches to a “bohm-like” quantum field theory. While some maintain the "particle ontology" of the non-relativistic formulation, others introduce fields as hidden parameters . So far, however, this approach has only been successful for boson fields. The further development in this area is likely to play a major role in the reception of the De Broglie-Bohm theory. A more detailed discussion of the criticism of the De Broglie Bohm theory can be found in an article by Oliver Passon published in 2004. Ward Struyve gives an overview of the different approaches to the quantum field theoretical generalization of the De Broglie-Bohm theory.

literature

• John S. Bell: Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy . 2nd Edition. Cambridge University Press, 2004, ISBN 0-521-81862-1 .
• David Bohm, BJ Hiley: The Undivided Universe: Ontological Interpretation of Quantum Theory . Routledge Chapman & Hall, New York 1993, ISBN 0-415-06588-7 .
• James T. Cushing: Quantum Mechanics: Historical Contingency and the Copenhagen Hegemony . University of Chicago Press, 1994, ISBN 0-226-13204-8 .
• Detlef Dürr, Sheldon Goldstein, Nino Zanghì: Quantum Physics Without Quantum Philosophy . 1st edition. Springer, 2012, ISBN 978-3-642-30690-7 .
• Detlef Dürr, Stefan Teufel: Bohmian Mechanics: The Physics and Mathematics of Quantum Theory . 1st edition. Springer, 2009, ISBN 978-3-540-89343-1 .
• Detlef Dürr: Bohm's mechanics as the basis of quantum mechanics . Springer, Berlin 2001, ISBN 3-540-41378-2 .
• Peter R. Holland: The Quantum Theory of Motion: An Account of the De Broglie-Bohm Causal Interpretation of Quantum Mechanics . Cambridge University Press, 1993, ISBN 0-521-48543-6 .
• Oliver Passon: Bohmsche Mechanik: An elementary introduction to the deterministic interpretation of quantum mechanics . Harri Deutsch, 2005, ISBN 3-8171-1742-6 .

Web links

Working groups

• Research network Bohmsche Mechanik (initiated by D. Dürr, S. Goldstein and N. Zanghì)
• Working group Bohmian Mechanics, University of Munich (D. Dürr)
• Bohmian Mechanics. (No longer available online.) University of Innsbruck (Gebhard Grübl), January 20, 2006, archived from the original on November 25, 2014 ; accessed on June 22, 2016 (English, Bohmsche Mechanik Uni Innsbruck). 
• Theoretical Physics at Birkbeck College, London (B. Hiley)
• Chris Dewdney's De Broglie Bohm Page ( Memento from November 13, 2010 in the Internet Archive )
• Homepage of Peter Holland

Introductions

• Brief, easily understandable introduction
• Videos with answers to frequently asked questions about Bohm's mechanics
• Lecture notes by Ilja Schmelzer
• Ilja Schmelzer on the underlying epistemology
• Understanding Bohmian mechanics: A dialogue, by R. Tumulka
• Entry in Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .
• Quanta Magazine: New Support for Alternative Quantum View

Individual evidence

1. ↑ L. de Broglie: La Mécanique ondulatoire et la structure atomique de la matière et du rayonnement . In: Journal de Physique, Series VI . tape VIII , no. 5 , 1927, pp. 225-241 . Reprinted in: L. de Broglie: La structure atomique de la matiere et du rayonnement et la mechanique ondulatoire . In: La Physique Quantique restera-t-elle Indeterministe . Gauthier Villars, Paris 1953.
   
2. ↑ ^{a b} David Bohm: A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. I . In: Physical Review . tape 85 , no. 2 , January 15, 1952, p. 166-179 , doi : 10.1103 / PhysRev.85.166 . David Bohm: A Suggested Interpretation of the Quantum Theory in Terms of "Hidden" Variables. II . In: Physical Review . tape
    85 , no. 2 , January 15, 1952, p. 180-193 , doi : 10.1103 / PhysRev.85.180 .
3. ↑ See numerous articles in: JS Bell, John S. Bell: Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy . 2nd Edition. Cambridge University Press, 2004, ISBN 0-521-81862-1 .
4. ^ A. Valentini: Signal-locality, uncertainty, and the subquantum H-theorem. I . In: Physics Letters A . tape 156 , no. 1-2 , 1991, doi : 10.1016 / 0375-9601 (91) 90116-P . A. Valentini: Signal-locality, uncertainty, and the subquantum H-theorem. II . In: Physics Letters A . tape
    158 , no. 1-2 , 1991, doi : 10.1016 / 0375-9601 (91) 90330-B .
5. ↑ ^{a b c d} Detlef Dürr, Sheldon Goldstein, Nino Zanghí: Quantum equilibrium and the origin of absolute uncertainty . In: Journal of Statistical Physics . tape 67 , no. 5 , 1992, pp. 843-907 , doi : 10.1007 / BF01049004 , arxiv : quant-ph / 0308039 .
6. ↑ Detlef Dürr, Sheldon Goldstein, Nino Zanghì: Quantum Equilibrium and the Role of Operators as Observables in Quantum Theory . In: Journal of Statistical Physics . tape 116 , no. 1 , 2004, p. 959-1055 , doi : 10.1023 / B: JOSS.0000037234.80916.d0 , arxiv : quant-ph / 0308038 .
7. ↑ John Conway, Simon Cooking: The Free Will Theorem . In: Foundations of Physics . tape 36 , no. 10 , 2006, p. 1441–1473 , doi : 10.1007 / s10701-006-9068-6 , arxiv : quant-ph / 0604079 .
8. ↑ John Conway, Simon Cooking: The Strong Free Will Theorem . In: Notices of the AMS . tape 56 , no. 2 , 2009, p. 226–232 , arxiv : 0807.3286v1 ( ams.org (PDF) [accessed June 14, 2009]).
9. ^ ^{A b} Peter R. Holland: The Quantum Theory of Motion: An Account of the De Broglie-Bohm Causal Interpretation of Quantum Mechanics . Cambridge University Press, 1993, ISBN 0-521-48543-6 .
10. ↑ ^{a b c} David Bohm, BJ Hiley: The Undivided Universe: Ontological Interpretation of Quantum Theory . Routledge Chapman & Hall, New York 1993, ISBN 0-415-06588-7 .
11. ↑ K. Berndl, D. Dürr, S. Goldstein, N. Zanghe: Nonlocality, Lorentz invariance, and Bohmian quantum theory . In: Physical Review A . tape 53 , no. 4 , 1996, pp. 2062-2073 , doi : 10.1103 / PhysRevA.53.2062 , arxiv : quant-ph / 9510027 .
12. ↑ JS Bell: Quantum field theory of without observers . In: Physics Reports . tape 137 , no. 1 , 1986, doi : 10.1016 / 0370-1573 (86) 90070-0 .
13. ↑ Detlef Dürr, Sheldon Goldstein, Roderich Tumulka, Nino Zanghì: Bohmian Mechanics and Quantum Field Theory . In: Physical Review Letters . tape 93 , no. 9 , 2004, p. 090402 , doi : 10.1103 / PhysRevLett.93.090402 , arxiv : quant-ph / 0303156 .
14. ↑ Detlef Dürr, Sheldon Goldstein, Roderich Tumulka, Nino Zanghì: Bell-type quantum field theories . In: Journal of Physics A: Mathematical and General . tape 38 , no. 4 , 2005, p. R1-R43 , doi : 10.1088 / 0305-4470 / 38/4 / R01 , arxiv : quant-ph / 0407116 .
15. ↑ S. Colin: Beables for Quantum Electrodynamics . In: Annales de la Fondation Louis de Broglie . tape 29 , 2004, pp. 273-295 , arxiv : quant-ph / 0310056 .
16. ↑ Ward Struyve, Hans Westman, Angelo Bassi, Detlef Dürr, Tullio Weber, Nino Zanghì: A new pilot-wave model for quantum field theory . In: Quantum mechanics: Are There Quantum Jumps? - and On the Present Status of Quantum Mechanics . tape 844 . AIP, Trieste (Italy) / Losinj (Croatia) May 27, 2006, p. 321-339 , doi : 10.1063 / 1.2219372 , arxiv : quant-ph / 0602229 .
17. ^ O. Passon: Why isn't every physicist a Bohmian? 2004, arxiv : quant-ph / 0412119 .
18. ↑ W. Struyve: The de Broglie-Bohm pilot-wave interpretation of quantum theory . 2005, arxiv : quant-ph / 0506243 .