The Broglie Theory - Bohm

The de Broglie-Bohm theory , also known as the wave-pilot theory , Bohm mechanics, Bohm interpretation and causal interpretation , is an interpretation of quantum theory . In addition to the wave function in the space of all possible configurations, it postulates a real configuration that exists, even without being measurable . The evolution of the configuration in time (i.e., the position of all particles or the configuration of all fields) is determined by the wave function using the control equation . The evolution of the wave function in time is determined by the Schrödinger equation . The theory is named after Louis de Broglie (1892-1987) and David Bohm (1917-1992).

The theory is deterministic ^[1] and clearly nonlocal : the velocity of any particle depends on the value of the control equation, which depends on the configuration of the system given by its wave function; the latter depends on the boundary conditions of the system, which in principle can be the whole Universe.

From the theory, a formalism for measurements arises, similar to thermodynamics for classical mechanics, which gives a standard quantum formalism, usually associated with the Copenhagen interpretation . The apparent nonlocality of the theory eliminates the "measurement problem", which usually refers to the interpretation of quantum mechanics in the Copenhagen interpretation. The Bourne rule in de Broglie-Bohm theory is not a basic law. It would be more correct to say that in this theory the connection between the probability density and the wave function has the status of a hypothesis, called the quantum equilibrium hypothesis, which complements the basic laws governing the wave function.

The theory was developed by de Broglie in the 1920s, but in 1927 he was forced to abandon it in favor of the then prevailing Copenhagen interpretation. David Bom, dissatisfied with the prevailing orthodox theory, reopened the de Broglie wave-pilot theory in 1952 . Bohm’s proposals were not widely accepted then, in part because Bohm was a communist in his youth ^[2] . The de Broglie-Bohm theory was considered unacceptable by the main theorists, mainly because of its apparent non-locality. Bell's theorem (1964) was inspired by the work of David Bohm discovered by Bell and the subsequent search for a way to eliminate the obvious nonlocality of the theory. Since the 1990s, interest in developing extensions of the de Broglie – Bohm theory has been reviving in attempts to reconcile it with special relativity and quantum field theory , among other features, such as spin or curved spatial geometry ^[3] .

In the Stanford Philosophical Encyclopedia , in an article on quantum decoherence ( Guido Bacciagaluppi, 2012 ), “ approaches to quantum mechanics ” are collected in five groups, one of which is the “wave-pilot theory” (the rest are the Copenhagen interpretation, objective reduction , many-world interpretation and modal interpretation ).

There are several equivalent mathematical formulations of the theory and several of its names are known. The de Broglie wave has a macroscopic analog, known by the term Faraday wave . ^[four]

Overview

The theory of de Broglie - Bohm is based on the following postulates:

There is a configuration $q$ ${\ displaystyle q}$ Universe described by coordinates $q^{k}$ ${\ displaystyle q ^ {k}}$ , which is an element of the configuration space $Q$ ${\ displaystyle Q}$ . The configuration spaces differ for different versions of the wave-pilot theory. For example, it could be a coordinate space $\mathbf {Q} _{k}$ ${\ displaystyle \ mathbf {Q} _ {k}}$ for $N$ ${\ displaystyle N}$ particles, or, in the case of field theory, the space of field configurations $\phi (x)$ ${\ displaystyle \ phi (x)}$ . The configuration evolves (for spin 0) in accordance with the control equation

m_{k}{\frac {dq^{k}}{dt}}(t)=\hbar \nabla _{k}\operatorname {Im} \ln \psi (q,t)=\hbar \operatorname {Im} \left({\frac {\nabla _{k}\psi }{\psi }}\right)(q,t)={\frac {m_{k}\mathbf {j} _{k}}{\psi ^{*}\psi }}=\mathrm {Re} \left({\frac {\mathbf {\hat {P}} _{k}\Psi }{\Psi }}\right)

{\ displaystyle m_ {k} {\ frac {dq ^ {k}} {dt}} (t) = \ hbar \ nabla _ {k} \ operatorname {Im} \ ln \ psi (q, t) = \ hbar \ operatorname {Im} \ left ({\ frac {\ nabla _ {k} \ psi} {\ psi}} \ right) (q, t) = {\ frac {m_ {k} \ mathbf {j} _ { k}} {\ psi ^ {*} \ psi}} = \ mathrm {Re} \ left ({\ frac {\ mathbf {\ hat {P}} _ {k} \ Psi} {\ Psi}} \ right )}

,

Where $\mathbf {j}$ ${\ displaystyle \ mathbf {j}}$ Is the current of probability , or the stream of probability, and $\mathbf {\hat {P}}$ ${\ displaystyle \ mathbf {\ hat {P}}}$ Is the momentum operator . Here $\psi (q,t)$ ${\ displaystyle \ psi (q, t)}$ Is a standard complex-valued wave function known from quantum theory that evolves according to the Schrödinger equation

i\hbar {\frac {\partial }{\partial t}}\psi (q,t)=-\sum _{i=1}^{N}{\frac {\hbar ^{2}}{2m_{i}}}\nabla _{i}^{2}\psi (q,t)+V(q)\psi (q,t)

{\ displaystyle i \ hbar {\ frac {\ partial} {\ partial t}} \ psi (q, t) = - \ sum _ {i = 1} ^ {N} {\ frac {\ hbar ^ {2} } {2m_ {i}}} \ nabla _ {i} ^ {2} \ psi (q, t) + V (q) \ psi (q, t)}

These postulates complete the formulation of the theory for any quantum theory with a Hamiltonian of type $H=\sum {\frac {1}{2m_{i}}}{\hat {p}}_{i}^{2}+V({\hat {q}})$ ${\ displaystyle H = \ sum {\ frac {1} {2m_ {i}}} {\ hat {p}} _ {i} ^ {2} + V ({\ hat {q}})}$ .

Configuration is distributed according to $|\psi (q,t)|^{2}$ ${\ displaystyle | \ psi (q, t) | ^ {2}}$ at time $t$ ${\ displaystyle t}$ , and therefore, this is true for all times. This state is called quantum equilibrium. In quantum equilibrium, this theory is consistent with the results of standard quantum mechanics.

It is worth noting that, although this last relation is often presented as an axiom of theory, in the original article by Bohm from 1952 it was presented as a conclusion from statistical-mechanical arguments. This argument is supported by the work of Bohm from 1953 and confirmed by the work of Bohm and Vigier in 1954, in which they introduced stochastic fluid oscillations that control the process of asymptotic relaxation from a nonequilibrium quantum state to the state of quantum equilibrium (ρ → | ψ | ² ). ^[five]

Two Slit Experiment

Bomov trajectories for an electron passing through two slits. A similar picture was also extrapolated from weak measurements of single photons. ^[6]

The two-slit experiment illustrates wave-particle duality . In it, a beam of particles (for example, electrons) passes through a barrier that has two slots. If the detector screen is placed behind the barrier, the pattern of detected particles shows interference fringes characteristic of waves arriving at the screen from two sources (two slots). However, the interference pattern consists of individual points corresponding to particles that hit the screen. The system seems to demonstrate the behavior of both waves (interference fringes) and particles (dots on the screen).

If we change this experiment so that one slot is closed, no interference pattern is observed. Thus, the state of both slots affects the final result. We can also place a minimally invasive detector near one of the slots to detect which slit the particle passed through. When we do this, the interference pattern will disappear.

The Copenhagen interpretation claims that particles are not localized in space until they are detected, so if there is no detector on the slits, there is no information about which slits the particle passed through. If one of the slots is equipped with a detector, then the wave function instantly changes due to detection.

In de Broglie-Bohm theory, the wave function is determined for both slits, but each particle has a well-defined trajectory that passes exactly through one slit. The final position of the particle on the detector screen and the gap through which it passes is determined by the initial position of the particle. Such an initial position is unknowable or uncontrollable on the part of the experimenter, so that there is the appearance of randomness in the pattern of detection. In Bohm's 1952 work, he used the wave function to construct a quantum potential , which, when substituted into Newton’s equations, gives the trajectories of particles passing through two slits. As a result, the wave function interferes with itself and directs the particles through the quantum potential so that the particles avoid regions in which the interference is destructive and are attracted to regions in which the interference is constructive, as a result of which an interference pattern appears on the detector screen.

Theory

Ontology

The de Broglie-Bohm theory ontology consists of a configuration $q(t)\in Q$ ${\ displaystyle q (t) \ in Q}$ Universe and pilot waves $\psi (q,t)\in \mathbb {C}$ ${\ displaystyle \ psi (q, t) \ in \ mathbb {C}}$ . Configuration space $Q$ ${\ displaystyle Q}$ You can choose in different ways, as in classical mechanics and standard quantum mechanics.

Thus, the ontology of the wave-pilot theory contains as a trajectory $q(t)\in Q$ ${\ displaystyle q (t) \ in Q}$ that we know from classical mechanics as a wave function $\psi (q,t)\in \mathbb {C}$ ${\ displaystyle \ psi (q, t) \ in \ mathbb {C}}$ from quantum theory. So, at every moment of time, there is not only a wave function, but also a well-defined configuration of the entire Universe (that is, a system that is determined from the boundary conditions used to solve the Schrödinger equation). Compliance with our experience is made by identifying the configuration of our brain with some part of the configuration of the entire Universe $q(t)\in Q$ ${\ displaystyle q (t) \ in Q}$ as in classical mechanics.

While the ontology of classical mechanics is part of the de Broglie – Bohm theory ontology, the dynamics are very different. In classical mechanics, particle acceleration is caused directly by forces that exist in physical three-dimensional space. In de Broglie – Bohm theory, particle velocities are given by a wave function that exists in a 3N-dimensional configuration space, where N corresponds to the number of particles in the system ^[7] . Bohm suggested that each particle has a “complex and subtle internal structure” that provides the ability to respond to the information that the wave function provides through the quantum potential. ^[8] Also, unlike classical mechanics, physical properties (for example, mass, charge) are distributed in accordance with the wave function in the de Broglie – Bohm theory, and are not localized in the particle position. ^[9] ^[10]

The wave function, not the particles, determines the dynamic evolution of the system: particles do not affect the wave function. According to Bohm and Healy, “the Schrödinger equation for a quantum field has neither sources, nor any other way in which the state of particles can directly affect the field [...] Quantum theory allows the quantum field to be completely independent of particles” ^[11] P Holland considers the lack of particle interaction and wave function "one of the many non-classical properties shown by this theory." ^[12] It should be noted that Holland later called the lack of response obvious due to the incompleteness of the description of the theory. ^[13]

Below we give the basics of the theory for a single particle moving in $\mathbb {R} ^{3}$ ${\ displaystyle \ mathbb {R} ^ {3}}$ and then extend it to the case $N$ ${\ displaystyle N}$ particles moving in 3 dimensions. In the first case, the configuration and real spaces coincide, and in the second, the real space is still $\mathbb {R} ^{3}$ ${\ displaystyle \ mathbb {R} ^ {3}}$ but the configuration space becomes $\mathbb {R} ^{3N}$ ${\ displaystyle \ mathbb {R} ^ {3N}}$ . While the positions of the particles are in real space, the velocity fields and the wave function are defined on the configuration space, which shows how the particles get entangled with each other in the framework of this theory.

Extensions to this theory include spin and more complex configuration spaces.

We use variations $\mathbf {Q}$ ${\ displaystyle \ mathbf {Q}}$ for the coordinates of the particles, while $\psi$ ${\ displaystyle \ psi}$ seems to be a complex-valued wave function given on the configuration space.

Control equation

For one spinless particle moving in $\mathbb {R} ^{3}$ ${\ displaystyle \ mathbb {R} ^ {3}}$ , the speed is set as

{\frac {d\mathbf {Q} }{dt}}(t)={\frac {\hbar }{m}}\operatorname {Im} \left({\frac {\nabla \psi }{\psi }}\right)(\mathbf {Q} ,t)

{\ displaystyle {\ frac {d \ mathbf {Q}} {dt}} (t) = {\ frac {\ hbar} {m}} \ operatorname {Im} \ left ({\ frac {\ nabla \ psi} {\ psi}} \ right) (\ mathbf {Q}, t)}

.

For many particles, we designate them as $\mathbf {Q} _{k}$ ${\ displaystyle \ mathbf {Q} _ {k}}$ for $k$ ${\ displaystyle k}$ th particles, and their velocities are given in the form

{\frac {d\mathbf {Q} _{k}}{dt}}(t)={\frac {\hbar }{m_{k}}}\operatorname {Im} \left({\frac {\nabla _{k}\psi }{\psi }}\right)(\mathbf {Q} _{1},\mathbf {Q} _{2},\ldots ,\mathbf {Q} _{N},t)

{\ displaystyle {\ frac {d \ mathbf {Q} _ {k}} {dt}} (t) = {\ frac {\ hbar} {m_ {k}}} \ operatorname {Im} \ left ({\ frac {\ nabla _ {k} \ psi} {\ psi}} \ right) (\ mathbf {Q} _ {1}, \ mathbf {Q} _ {2}, \ ldots, \ mathbf {Q} _ { N}, t)}

.

The main thing here is that this velocity field depends on the actual position of all $N$ ${\ displaystyle N}$ particles in the universe. As explained below, in most experimental situations, the effects of all these particles can be encapsulated in an effective wave function for the subsystem of the Universe.

Schrödinger equation

The one-particle Schrödinger equation determines the time evolution of a complex-valued wave function on $\mathbb {R} ^{3}$ ${\ displaystyle \ mathbb {R} ^ {3}}$ . The equation is a quantized version of the total energy of a classical system that evolves under the action of a real potential function $V$ ${\ displaystyle V}$ given on $\mathbb {R} ^{3}$ ${\ displaystyle \ mathbb {R} ^ {3}}$ :

i\hbar {\frac {\partial }{\partial t}}\psi =-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi +V\psi

{\ displaystyle i \ hbar {\ frac {\ partial} {\ partial t}} \ psi = - {\ frac {\ hbar ^ {2}} {2m}} \ nabla ^ {2} \ psi + V \ psi }

For many particles, the equation is the same, except that $\psi$ ${\ displaystyle \ psi}$ and $V$ ${\ displaystyle V}$ defined on the configuration space $\mathbb {R} ^{3N}$ ${\ displaystyle \ mathbb {R} ^ {3N}}$ .

i\hbar {\frac {\partial }{\partial t}}\psi =-\sum _{k=1}^{N}{\frac {\hbar ^{2}}{2m_{k}}}\nabla _{k}^{2}\psi +V\psi

{\ displaystyle i \ hbar {\ frac {\ partial} {\ partial t}} \ psi = - \ sum _ {k = 1} ^ {N} {\ frac {\ hbar ^ {2}} {2m_ {k }}} \ nabla _ {k} ^ {2} \ psi + V \ psi}

This is the same wave function from ordinary quantum mechanics.

Attitude to the Bourne rule

Bohm in his original works [Bohm 1952] considers how the results of measurements of ordinary quantum mechanics follow from de Broglie – Bohm theory. The basic idea is that this is done provided that the particle positions satisfy the statistical distribution given $|\psi |^{2}$ ${\ displaystyle | \ psi | ^ {2}}$ . Such a distribution is guaranteed to be true for all times thanks to the control equation, if the initial distribution of particles satisfies $|\psi |^{2}$ ${\ displaystyle | \ psi | ^ {2}}$ .

For this experiment, we can assume that the statement is true, and experimental verification will confirm this. This is disputed by Duer et al.: ^[14] such a distribution is characteristic of subsystems. They claim that $|\psi |^{2}$ ${\ displaystyle | \ psi | ^ {2}}$ by virtue of its equivariance under the influence of the dynamic evolution of the system, it is usually a suitable measure for the initial conditions of particle coordinates. They then prove that the vast majority of possible initial configurations are statistically subject to the Born rule (i.e. $|\psi |^{2}$ ${\ displaystyle | \ psi | ^ {2}}$ ) for the measurement results. As a result, in the Universe, under the control of de Broglie – Bohm dynamics, the Born rule is usually satisfied.

The situation is thus similar to the situation in classical statistical physics. An initial state with low entropy with an overwhelmingly high probability evolves into a state with a higher entropy: typical behavior that is consistent with the second law of thermodynamics. Of course, there are abnormal initial conditions that could lead to a violation of the second law. However, in the absence of detailed evidence supporting the actual implementation of one of these rare initial conditions, it would be unreasonable to expect anything other than the actually observed uniform increase in entropy. Similarly, in the de Broglie - Bohm theory, there are anomalous initial conditions that lead to a violation of the Born rule (i.e., in contradiction with the predictions of the standard quantum theory). But usually a theorem shows that, in the absence of special reasons to believe that one of these special initial conditions is realized, one should expect the Born rule to be satisfied.

The Bourne rule in de Broglie-Bohm theory is a theorem, and not an additional postulate (as in ordinary quantum theory).

It can be shown that the distribution of particles not distributed according to the Born rule (that is, the distribution “outside of quantum equilibrium”) and evolving in de Broglie – Bohm dynamics will in most cases develop into a state distributed as $|\psi |^{2}$ ${\ displaystyle | \ psi | ^ {2}}$ . ^{[15] An} electron density video in a 2D box under this process is available here .

Conditional wave function of a subsystem

In the formulation of the de Broglie - Bohm theory there is only the wave function of the entire Universe (which always evolves in accordance with the Schrödinger equation). However, it should be noted that the “Universe” is simply a system limited by the same boundary conditions that are used to solve the Schrödinger equation. However, as soon as the theory is formulated, it is convenient to introduce the concept of wave function also for subsystems of the Universe. We write the wave function of the Universe as $\psi (t,q^{\mathrm {I} },q^{\mathrm {II} })$ ${\ displaystyle \ psi (t, q ^ {\ mathrm {I}}, q ^ {\ mathrm {II}})}$ where $q^{\mathrm {I} }$ ${\ displaystyle q ^ {\ mathrm {I}}}$ denotes the configuration of variables associated with some subsystem (I) of the Universe and $q^{\mathrm {II} }$ ${\ displaystyle q ^ {\ mathrm {II}}}$ denotes the rest of the configuration variables. Denote, respectively, $Q^{\mathrm {I} }(t)$ ${\ displaystyle Q ^ {\ mathrm {I}} (t)}$ and $Q^{\mathrm {II} }(t)$ ${\ displaystyle Q ^ {\ mathrm {II}} (t)}$ the actual configuration of subsystem (I) and the rest of the universe. For simplicity, we consider here only the case with spinless particles. The conditional wave function of subsystem (I) is determined by the formula:

\psi ^{\mathrm {I} }(t,q^{\mathrm {I} })=\psi (t,q^{\mathrm {I} },Q^{\mathrm {II} }(t)).

{\ displaystyle \ psi ^ {\ mathrm {I}} (t, q ^ {\ mathrm {I}}) = \ psi (t, q ^ {\ mathrm {I}}, Q ^ {\ mathrm {II} } (t)).}

This follows immediately from the fact that $Q(t)=(Q^{\mathrm {I} }(t),Q^{\mathrm {II} }(t))$ ${\ displaystyle Q (t) = (Q ^ {\ mathrm {I}} (t), Q ^ {\ mathrm {II}} (t))}$ satisfies the governing equation. He is also satisfied with the configuration $Q^{\mathrm {I} }(t)$ ${\ displaystyle Q ^ {\ mathrm {I}} (t)}$ identical to that presented in the formulation of the theory, but with a universal wave function $\psi$ ${\ displaystyle \ psi}$ replaced by a conditional wave function $\psi ^{\mathrm {I} }$ ${\ displaystyle \ psi ^ {\ mathrm {I}}}$ . In addition, the fact that $Q(t)$ ${\ displaystyle Q (t)}$ is random with probability density given by the squared modulus $\psi (t,\cdot )$ ${\ displaystyle \ psi (t, \ cdot)}$ suggests conditional probability densities $Q^{\mathrm {I} }(t)$ ${\ displaystyle Q ^ {\ mathrm {I}} (t)}$ this $Q^{\mathrm {II} }(t)$ ${\ displaystyle Q ^ {\ mathrm {II}} (t)}$ is given by the squared modulus of the vector (normalized) conditional wave function $\psi ^{\mathrm {I} }(t,\cdot )$ ${\ displaystyle \ psi ^ {\ mathrm {I}} (t, \ cdot)}$ (in the terminology of Dura et al. ^[16] this fact is called the fundamental formula of conditional probability ).

Unlike the universal wave function, the conditional wave function of a subsystem does not always (but often) evolve in accordance with the Schrödinger equation. For example, if the universal wave function decomposes into a product as:

\psi (t,q^{\mathrm {I} },q^{\mathrm {II} })=\psi ^{\mathrm {I} }(t,q^{\mathrm {I} })\psi ^{\mathrm {II} }(t,q^{\mathrm {II} })

{\ displaystyle \ psi (t, q ^ {\ mathrm {I}}, q ^ {\ mathrm {II}}) = \ psi ^ {\ mathrm {I}} (t, q ^ {\ mathrm {I} }) \ psi ^ {\ mathrm {II}} (t, q ^ {\ mathrm {II}})}

then the conditional wave function of subsystem (I) up to an irrelevant scalar factor is $\psi ^{\mathrm {I} }$ ${\ displaystyle \ psi ^ {\ mathrm {I}}}$ (this is what standard quantum theory will consider as a wave function of subsystem (I)). If, in addition, the Hamiltonian does not contain an interaction between subsystems (I) and (II), then $\psi ^{\mathrm {I} }$ ${\ displaystyle \ psi ^ {\ mathrm {I}}}$ satisfies the Schrödinger equation. In a more general sense, suppose that the universal wave function $\psi$ ${\ displaystyle \ psi}$ written as:

\psi (t,q^{\mathrm {I} },q^{\mathrm {II} })=\psi ^{\mathrm {I} }(t,q^{\mathrm {I} })\psi ^{\mathrm {II} }(t,q^{\mathrm {II} })+\phi (t,q^{\mathrm {I} },q^{\mathrm {II} }),

{\ displaystyle \ psi (t, q ^ {\ mathrm {I}}, q ^ {\ mathrm {II}}) = \ psi ^ {\ mathrm {I}} (t, q ^ {\ mathrm {I} }) \ psi ^ {\ mathrm {II}} (t, q ^ {\ mathrm {II}}) + \ phi (t, q ^ {\ mathrm {I}}, q ^ {\ mathrm {II}} ),}

Where $\phi$ ${\ displaystyle \ phi}$ solves the Schrödinger equation and $\phi (t,q^{\mathrm {I} },Q^{\mathrm {II} }(t))=0$ ${\ displaystyle \ phi (t, q ^ {\ mathrm {I}}, Q ^ {\ mathrm {II}} (t)) = 0}$ for all $t$ ${\ displaystyle t}$ and $q^{\mathrm {I} }$ ${\ displaystyle q ^ {\ mathrm {I}}}$ . Further, again, the conditional wave function of subsystem (I) up to an irrelevant scalar factor is $\psi ^{\mathrm {I} }$ ${\ displaystyle \ psi ^ {\ mathrm {I}}}$ and, if the Hamiltonian does not contain an interaction between subsystems (I) and (II), $\psi ^{\mathrm {I} }$ ${\ displaystyle \ psi ^ {\ mathrm {I}}}$ satisfies the Schrödinger equation.

The fact that the conditional wave function of a subsystem does not always evolve in accordance with the Schrödinger equation is due to the fact that the usual reduction rule in standard quantum theory arises from the Bomov formalism when considering the conditional wave functions of subsystems.

Notes

↑ Bohm, David (1952).
↑ F. David Peat, Infinite Potential: The Life and Times of David Bohm (1997), p. 133
↑ David Bohm and Basil J. Hiley, The Undivided Universe - An Ontological Interpretation of Quantum Theory appreared after Bohm's death, in 1993; reviewed by Sheldon Goldstein in Physics Today (1994)
↑ John WM Bush . "Quantum mechanics writ large" .
↑ Publications of D. Bohm in 1952 and 1953 and of J.-P. Vigier in 1954 as cited in Antony Valentini; Hans Westman (January 8, 2005).
↑ "Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer"
↑ David Bohm (1957).
↑ D. Bohm and B. Hiley: The undivided universe: An ontological interpretation of quantum theory , p. 37.
↑ HR Brown, C. Dewdney and G. Horton: "Bohm particles and their detection in the light of neutron interferometry", Foundations of Physics , 1995, Volume 25, Number 2, pp. 329–347.
↑ J. Anandan, "The Quantum Measurement Problem and the Possible Role of the Gravitational Field", Foundations of Physics , March 1999, Volume 29, Issue 3, pp. 333–348.
↑ D. Bohm and B. Hiley: The undivided universe: An ontological interpretation of quantum theory , p. 24
↑ Peter R. Holland: The Quantum Theory of Motion: An Account of the De Broglie-Bohm Causal Interpretation of Quantum Mechanics , Cambridge University Press, Cambridge (first published 25 June 1993), ISBN 0-521-35404-8 hardback, ISBN 0-521-48543-6 paperback, transferred to digital printing 2004, Chapter I. section (7) "There is no reciprocal action of the particle on the wave", p. 26
↑ P. Holland: "Hamiltonian theory of wave and particle in quantum mechanics II: Hamilton-Jacobi theory and particle back-reaction", Nuovo Cimento B 116, 2001, pp. 1143–1172, full text preprint p. 31 )
↑ Dürr, D., Goldstein, S., and Zanghì, N., "Quantum Equilibrium and the Origin of Absolute Uncertainty" , Journal of Statistical Physics 67: 843–907, 1992.
↑ Towler, MD; Russell, NJ; Valentini A., pbs., "Timescales for dynamical relaxation to the Born rule" quant-ph / 11031589
↑ "Quantum Equilibrium and the Origin of Absolute Uncertainty" , D. Dürr, S. Goldstein and N. Zanghì, Journal of Statistical Physics 67, 843–907 (1992).

[1] Bohm, David (1952).

[2] F. David Peat, Infinite Potential: The Life and Times of David Bohm (1997), p. 133

[3] David Bohm and Basil J. Hiley, The Undivided Universe - An Ontological Interpretation of Quantum Theory appreared after Bohm's death, in 1993; reviewed by Sheldon Goldstein in Physics Today (1994)

[4] John WM Bush . "Quantum mechanics writ large" .

[5] Publications of D. Bohm in 1952 and 1953 and of J.-P. Vigier in 1954 as cited in Antony Valentini; Hans Westman (January 8, 2005).

[6] "Observing the Average Trajectories of Single Photons in a Two-Slit Interferometer"

[7] David Bohm (1957).

[8] D. Bohm and B. Hiley: The undivided universe: An ontological interpretation of quantum theory , p. 37.

[9] HR Brown, C. Dewdney and G. Horton: "Bohm particles and their detection in the light of neutron interferometry", Foundations of Physics , 1995, Volume 25, Number 2, pp. 329–347.

[10] J. Anandan, "The Quantum Measurement Problem and the Possible Role of the Gravitational Field", Foundations of Physics , March 1999, Volume 29, Issue 3, pp. 333–348.

[11] D. Bohm and B. Hiley: The undivided universe: An ontological interpretation of quantum theory , p. 24

[12] Peter R. Holland: The Quantum Theory of Motion: An Account of the De Broglie-Bohm Causal Interpretation of Quantum Mechanics , Cambridge University Press, Cambridge (first published 25 June 1993), ISBN 0-521-35404-8 hardback, ISBN 0-521-48543-6 paperback, transferred to digital printing 2004, Chapter I. section (7) "There is no reciprocal action of the particle on the wave", p. 26

[13] P. Holland: "Hamiltonian theory of wave and particle in quantum mechanics II: Hamilton-Jacobi theory and particle back-reaction", Nuovo Cimento B 116, 2001, pp. 1143–1172, full text preprint p. 31 )

[dgz92-14] Dürr, D., Goldstein, S., and Zanghì, N., "Quantum Equilibrium and the Origin of Absolute Uncertainty" , Journal of Statistical Physics 67: 843–907, 1992.

[15] Towler, MD; Russell, NJ; Valentini A., pbs., "Timescales for dynamical relaxation to the Born rule" quant-ph / 11031589

[16] "Quantum Equilibrium and the Origin of Absolute Uncertainty" , D. Dürr, S. Goldstein and N. Zanghì, Journal of Statistical Physics 67, 843–907 (1992).