NationMaster - Encyclopedia: Bohm interpretation

The Bohm interpretation of quantum mechanics, sometimes called the Bohmian Mechanics or Ontological interpretation is an interpretation postulated by David Bohm in 1952, which was an extension of the de Broglie-pilot-wave theory of 1927. Consequently it is sometimes called the de Broglie-Bohm theory. Bohm's interpretation is an example of a Hidden Variables theory. The hidden variables present provide an objective description which eliminates the "Schroedinger's Cat" measurement problem, and similar concerns. Further, the hidden variables permit determinism, as well. For a non-technical introduction to the topic, please see Introduction to Quantum mechanics. ... An interpretation of quantum mechanics is an attempt to answer the question, What exactly is quantum mechanics talking about? The question has its historical roots in the nature of quantum mechanics itself which was considered as a radical departure from previous physical theories. ... David Bohm. ...

1 Background
- 1.1 Two-slit experiment
- 1.2 Nonlocality
2 Mathematical foundation
3 Commentary
4 Benefits
5 Criticisms
6 See also
7 References

Background

The Bohm interpretation, of 1952, generalizes Louis de Broglie's pilot wave theory from 1927. It can be thought of as taking its cue from what one sees in the laboratory, say, in a two-slit experiment with electrons. We can see localized flashes whenever an electron is detected at some place on the screen. The overall pattern made by many such flashes is governed by a pattern closely matched by simple wave dynamics. Bohm and de Broglie posited that in the world of quantum phenomena, every kind of particle is accompanied by a wave which guides the motion of the particle, hence the term pilot wave. The pilot wave is just the wavefunction from conventional quantum mechanics, with an influence on the motion of the particles. In the absence of spin, we can formulate this influence using a wavefunction-derived-potential, called the quantum potential, which acts as upon the particles in a loose analogy to classical physics. Louis-Victor-Pierre-Raymond, 7th duc de Broglie, generally known as Louis de Broglie (August 15, 1892–March 19, 1987), was a French physicist and Nobel Prize laureate. ... In theoretical physics, pilot wave theory was the first known example of a hidden variable theory, presented by Louis de Broglie in 1927. ... 1927 (MCMXXVII) was a common year starting on Saturday (link will take you to calendar). ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ...

The wavefunction governs the motion of the particle, and evolves according to the Schrödinger equation. The interpretation assumes a single, nonsplitting universe (unlike the Everett many-worlds interpretation) and is both objective and deterministic (unlike the Copenhagen interpretation). It says the state of the universe evolves smoothly through time, without the collapsing of wavefunctions when a measurement occurs, as in the Copenhagen interpretation. In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, is the definition of energy of a quantum system. ... The many-worlds interpretation (or MWI) is an interpretation of quantum mechanics, based on Hugh Everetts relative-state formulation. ... The Copenhagen interpretation is an interpretation of quantum mechanics formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927. ...

Two-slit experiment

Thus, in this theory all fundamental entities, e.g. electrons, are particles. When one performs a double-slit experiment (see wave-particle duality), one is concerned with noting the positions on a screen at which electrons arrive. The usual Copenhagen interpretation is puzzling in that a single entity - the electron - seems to show characteristics of both a particle and a wave. The particle aspect is present in that we imagine that the electron traverses one slit or another, but never both. The wave aspect is present in that the positions at which the particles are detected form a pattern characteristic of wave-interference arising from a wave traversing both slits. The double-slit experiment consists of letting light diffract through two slits producing fringes on a screen. ... In physics, wave-particle duality holds that light and matter exhibit properties of both waves and of particles. ...

The Bohm interpretation seems to resolves the puzzle quite simply and naturally. Via Bohm, each particle goes through one slit rather than the other, while the wave traverses both slits. The electron's motion is guided - both in its choice of slits and its subsequent trajectory towards the screen - by the wave. The characteristic wave-interference pattern seen in the detection of the electrons arises by considering that the guiding wave itself will show interference in the familiar way one learns in the elementary physics of waves.

One might also note that what is measured in such an experiment - the position on the screen at which each electron arrives - is itself none other than the "hidden variable" the Bohm interpretation adds to the description, as we show in the formulation, below. Clearly the term "hidden variable" is not appropriate for the quantitity which is most conspicuously manifested in the experiment.

Nonlocality

Now we must address the question of nonlocality. Within Bohm's interpretation, it can occur that events happening at one location in space can instantaneously influence other events which might be at large distances: thus we say that the theory fails to obey locality, i.e., it is non-local. This being the case, the response many physicists have to Bohm's theory is often related to how they regard this concept.

The question of nonlocality hinges upon the attitude one takes towards the Einstein-Podolsky-Rosen paradox ^[1] and Bell's theorem (see p.14 in ^[2]). There are often two camps into which people fall regarding the issue.

According to one camp, what has been shown is that quantum mechanics itself is nonlocal and that this cannot be avoided by appealing to any alternative interpretation. The same Bell responsible for Bell's theorem was a member of this group (p 196 in ^[3]): "It is known that with Bohm's example of EPR correlations, involving particles with spin, there is an irreducible nonlocality." If this is indeed the case, then the nonlocality of the Bohm interpretation can hardly be regarded as a strike against it.

Others see the consequences of "EPR" and Bell's theorem, in a different way. They regard the correct conclusion to be related not so much to quantum theory itself, but only to deterministic interpretations of the same (i.e., to hidden variable theories such as Bohm's interpretation). According to the people who think this way, what has been shown is that all deterministic theories must be nonlocal. This group would claim that retaining orthodox quantum mechanics - with its probabilitic character - would permit one to retain locality. Armed with such a viewpoint, these might tend to be less receptive to Bohm's interpretation.

Mathematical foundation

In a rough sense, one might conceptualize Bohm's theory by an analogy with a system of charged particles in motion due to electromagnetic fields. Within electrodynamics, one regards the electric and magnetic fields as functions defined for every position in space and for all times ${mathbf{E}(mathit{q}, t), mathbf{B}(mathit{q},t)}$ . The evolution of these fields is governed by maxwell's equations.

If we designate the particle position as Q, then the force on a particle of charge c at some time t is given by

$mathbf{F} = c mathbf{E}(mathit{Q}, t) + c mathbf{v} mathbf{x} mathbf{B}(mathit{Q}, t)$

where we note that the field strength is evaluated at the particle's position Q in each case.

It is important to note that the fields E and B generally exist throughout space, and hence are defined for all positions q. The influence they have on the particle depends upon the value they take at the particle's position Q.

One-particle formalism

The Schrödinger equation for one particle of mass m is In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, is the definition of energy of a quantum system. ...

$i hbar frac{partial psi(mathbf{x},t)}{partial t} = frac{-hbar^2}{2 m} nabla^2 psi(mathbf{x},t)+ V(mathbf{x}) psi(mathbf{x},t)$ ,

where the wavefunction $psi(mathbf{x},t)$ is a complex function of the spacial coordinate x and time t. This article discusses the concept of a wavefunction as it relates to quantum mechanics. ...

The probability density ρ(X,t) is a real function defined by

$rho(mathbf{X},t) = R(mathbf{X},t)^2 = |psi(mathbf{X},t)|^2 = psi^{*}(mathbf{X},t) psi(mathbf{X},t)$ .

Without loss of generality, we can express the wavefunction ψ in terms of a real probability density ρ = |ψ|² and a complex phase that depends on the real variable S, both of which are also functions of position and time, as Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = âˆ’1. ...

$psi = sqrt{rho} e^{i S / hbar}$ .

The Schrödinger equation can then be split into two coupled equations by taking the real and imaginary terms;

$-frac{partial rho}{partial t} = nabla cdot (rho frac{nabla S}{m}) qquad (1)$

$-frac{partial S}{partial t} = V + Q + frac{1}{2m}(nabla S)^2 qquad (2)$

where

$Q = -frac{hbar^2}{2 m} frac{nabla^2 R}{R} = -frac{hbar^2}{2 m} frac{nabla^2 sqrt{rho}}{ sqrt{rho}} = -frac{hbar^2}{2 m} left(frac{nabla^2 rho}{2 rho} -left(frac{nabla rho}{2 rho} right)^2 right)$

is called the quantum potential. The momentum of Bohm's "hidden variable" particle is defined by

$mathbf{p} = {m} mathbf{v} = nabla S qquad (3)$

and the particle's energy as $E = - partial S / partial t$ ; equation (1) is interpreted as simply the continuity equation for probability with All the examples of continuity equations below express the same idea; they are all really examples of the same concept. ...

$mathbf{j} = rho mathbf{v} = rho frac{mathbf{p}}{m} = rho frac{nabla S}{m}$ ,

and equation (2) is a statement that total energy is the sum of potential energy, quantum potential and the kinetic energy. It is by no means accidental that S has the units and typical variable name of the action. In physics, the action principle is an assertion about the nature of motion, from which the trajectory of an object subject to forces can be determined. ...

Many-particle formalism

The many-particle Schrödinger equation is a straightforward generalisation of the one-particle example: In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, is the definition of energy of a quantum system. ...

$i hbar frac{partial psi(mathbf{x_1,x_2,...},t)}{partial t} = sum_i frac{-hbar^2}{2 m_i} nabla_i^2 psi(mathbf{x_1, x_2,..},t) + V(mathbf{x_1, x_2,..})psi(mathbf{x_1, x_2,...},t)$ ,

where the i-th particle has mass $mathbf{m_i}$ and position co-ordinate $mathbf{x_i}$ at time t. The wavefunction $psi(mathbf{x_1, x_2,...},t)$ is a complex function of the $mathbf{x_i}$ and time t. $nabla_i$ is the grad operator with respect to $mathbf{x_i}$ , i.e. of the i-th particle's position co-ordinate. As before the probability density $rho(mathbf{x_1, x_2,...},t)$ is a real function defined by This article discusses the concept of a wavefunction as it relates to quantum mechanics. ...

$rho(mathbf{x_1, x_2,..},t) = R(mathbf{x_1, x_2,..},t)^2 = |psi(mathbf{x_1, x_2,..},t)|^2 = psi^{*}(mathbf{x_1,x_2,...},t) psi(mathbf{x_1, x_2,...},t)$ .

The complex phase depends on the real variable $S(mathbf{x_1, x_2,...},t)$ so that we can define the same relationship as in the 1-particle example: Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = âˆ’1. ...

$psi = sqrt{rho} e^{i S / hbar}$ .

Again, the Schrödinger equation can be split into two coupled equations by taking the real and imaginary terms;

$-frac{partial rho}{partial t} = sum_i nabla_i cdot (rho frac{nabla_i S}{m_i}) qquad (1)$

$-frac{partial S}{partial t} = V + Q + sum_i frac{1}{2m_i}(nabla_i S)^2 qquad (2)$

where

$Q = -sum_i frac{hbar^2}{2 m_i} frac{nabla_i^2 R}{R} = -sum_ifrac{hbar^2}{2 m_i} left(frac{nabla_i^2 rho}{2 rho} -left(frac{nabla_i rho}{2 rho} right)^2 right)$

is the N-particle quantum potential.

The momentum of Bohm's i-th particle's "hidden variable" is defined by

$mathbf{p_i} = {m_i} mathbf{v_i} = nabla_i S qquad (3)$

and the particles' total energy as $E = - partial S / partial t$ ; equation (1) is the continuity equation for probability with All the examples of continuity equations below express the same idea; they are all really examples of the same concept. ...

$mathbf{j_i} = rho mathbf{v_i} = rho frac{mathbf{p_i}}{m_i} = rho frac{nabla_i S}{m_i}$ ,

and equation (2) is a statement that total energy is the sum of the potential energy, quantum potential and the kinetic energies.

Commentary on the formalism

Bohm's particle(s) are viewed as having definite positions and velocities at all times, with a probability distribution ρ that may be calculated from the wavefunction ψ. The wavefunction "guides" the particles by means of the quantum potential Q; alternately we can regard the particles' velocities defined by equation (3) -- these two approaches are equivalent.

There is a notable asymmetry with regard to the positions, $mathbf{x_i}$ , and velocities, $mathbf{v_i}$ , of the particles. Solving the Schrödinger equation solves for R and S, which immediately yields the particles' velocities, which are known precisely. By contrast the particles' positions are only known statistically, from R. As in classical mechanics successive observations of the particles' positions refines or pares away at the initial conditions. Thus, with succeeding observations, the initial conditions become more and more restricted. Yet this formalism is empirically indistinguishable from, and entirely consistent with, the Schrödinger equation, despite the hidden variable Bohm-particles following chaotic paths. It is this underlying chaotic behaviour of the hidden variables that allows the deterministic Bohm theory to generate the apparent indeterminacy associated with each measurement, and hence recover the Heisenberg uncertainty principle. In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, is the definition of energy of a quantum system. ... Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ... In quantum physics, the Heisenberg uncertainty principle, sometimes called the Heisenberg indeterminacy principle, expresses a limitation on accuracy of (nearly) simultaneous measurement of observables such as the position and the momentum of a particle. ...

Much of this 1-particle formalism was developed by Louis de Broglie; Bohm extended it from the case of a single particle to that of many particles, and also, by considering the particles in the measuring apparatus, re-interpreted the equations via an early form of quantum decoherence to include observation. Louis-Victor-Pierre-Raymond, 7th duc de Broglie, generally known as Louis de Broglie (August 15, 1892–March 19, 1987), was a French physicist and Nobel Prize laureate. ... In quantum mechanics, quantum decoherence is the process by which quantum systems in complex environments exhibit classical behavior. ...

Bohmian mechanics can also be extended to include spin, although the extension to relativistic conditions has not yet been successful.

Commentary

The Bohm interpretation is not popular among physicists for a number of scientific and sociological reasons that would be fascinating but long to study, but perhaps we can at least say here it is considered very inelegant by some (it was considered as "unnecessary superstructure" even by Einstein who dreamed about a deterministic replacement for the Copenhagen interpretation). Presumably Einstein, and others, disliked the non-locality of most interpretations of quantum mechanics, as he tried to show its incompleteness in the EPR paradox. The Bohm theory is unavoidably non-local, which counted as a strike against it; but this is now less so, now that non-locality has become more compelling due to experimental verification of Bell's Inequality. However the theory was used by others as the basis of a number of books such as The Dancing Wu Li Masters, which purport to link modern physics with Eastern religions. This, as well as Bohm's long standing philosophical friendship with J. Krishnamurti, may have led some to discount it. Albert Einstein, photographed in 1947 by Oren J. Turner. ... In quantum mechanics, the EPR paradox (Einstein-Podolsky-Rosen) is a thought experiment which challenged long-held ideas about the relation between, on the one hand the observed values of physical quantities and on the other, the values that can be accounted for by a physical theory. ... Bells theorem is the most famous legacy of the late John Bell. ... The Dancing Wu Li Masters (ISBN 055326382X) by Gary Zukav (pub. ... Jiddu Krishnamurti (May 11, 1895 Madanapalle, India - February 17, 1986 Ojai, California) was discovered as a young boy by C.W. Leadbeater in India on the private beach, that was part of the Theosophical headquarters in Adyar in Chennai. ...

Bohm's interpretation vs. Copenhagen (or quasi-Copenhagen as defined by Von Neumann and Paul Dirac) differs in crucial points: ontological vs. epistemological; quantum potential or active information vs. ordinary wave-particle and probability waves; nonlocality vs. locality wholeness vs. regular segmentary approach. Standard QM is also non-local; see EPR paradox. In his posthumous book The Undivided Universe, Bohm has (with Hiley, and, of course, in numerous previous papers) presented an elegant and complete description of the physical world. This description is in many aspects more satisfying than the prevailing one, at least to Bohm and Hiley. According to the Copenhagen interpretation, there is a classical realm of reality, of large objects and large quantum numbers, and a separate quantum realm. There is not a single bit of quantum theory in the description of "the classical world" - unlike the situation one encounters in Bohmian version of quantum mechanics. It also differs in a few matters that are experimentally tested with no consensus whether the Copenhagen, or other, interpretation has been proven inadequate; or the results are too vague to be interpreted unambiguously. The papers in question are listed at the bottom of the page, and their main contention is that quantum effects, as predicted by Bohm, are observed in the classical world - something unthinkable in the dominant Copenhagen version. A separate article covers Saint John Neumann, the American priest. ... Paul Adrien Maurice Dirac Paul Adrien Maurice Dirac, OM FRS (IPA: [dÉªrÃ¦k]) (August 8, 1902 â€“ October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... In physics, the principle of locality is that distant objects cannot have direct influence on one another: an object is influenced directly only by its immediate surroundings. ... In quantum mechanics, the EPR paradox (Einstein-Podolsky-Rosen) is a thought experiment which challenged long-held ideas about the relation between, on the one hand the observed values of physical quantities and on the other, the values that can be accounted for by a physical theory. ...

The Bohmian interpretation of Quantum Mechanics is characterized by the following features:

It is based on concepts of non-local quantum potential and active information. Just as an aside, one should keep in mind that the Bohmian approach is not new with regard to mathematical formalism, but is a reinterpretation of the ordinary quantum mechanical Schrödinger equation (which under a certain approximation is the same as the classical Hamilton-Jacobi equation), that simply, in the process of calculation, gives an additional term that Bohm interprets as a quantum potential Q acting on the particles. Therefore, Bohm's interpretation is not an original mathematical formalism (it's just a wave function with the Schrödinger equation applied to it) but an interpretation that denies central features of ordinary quantum mechanics: no wave-particle dualism (electron is a real particle guided by a real quantum potential field), and no epistemological approach (i.e., quantum realism and ontology).

Perhaps the most interesting part about Bohm's approach is its formalism: it gives a new version of the microworld, not only a new (albeit radical) interpretation. It describes a world where concepts such as causality, position and trajectory have concrete physical meanings. Putting aside possible objections with regard to non-locality, and possible triumphs of Bohmian view (for instance, no need for anything like a complementarity principle) - one is left with the impression that what Bohm offers is perhaps a new paradigm and absolutely a boldly rephrased version of the old and established quantum mechanics.

Bohm emphasized that experiment and experimenter comprise an undivided whole. There is nothing separate from this undivided whole. The quantum potential Q does not go to zero at infinity.

In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, is the definition of energy of a quantum system. ... The Hamilton-Jacobi-Bellman (HJB) equation is a partial differential equation which is central to optimal control theory. ... In physics, complementarity is a basic principle of quantum theory, and refers to effects such as the wave-particle duality, in which different measurements made on a system reveal it to have either particle-like or wave-like properties. ...

Benefits

For supporters, Bohm's interpretation is the better formulation of Quantum Mechanics, because it is defined more precisely than the Copenhagen interpretation which is based on theorems which are not expressed in precise mathematical terms but in natural words, like "when measuring". The Copenhagen interpretation is an interpretation of quantum mechanics formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927. ...

Indeed, Bohm's interpretation subsumes the quantum concepts of measurement, complementarity, decoherence, and entanglement into mathematically precise guidance conditions and position variables. The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ... In physics, complementarity is a basic principle of quantum theory, and refers to effects such as the wave-particle duality, in which different measurements made on a system reveal it to have either particle-like or wave-like properties. ... Quantum decoherence is the general term for the consequences of irreversible quantum entanglement. ... Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. ...

The minimum benefit of Bohm's interpretation - independently from the debate whether it is the preferable formulation - is a disproof of the claim that quantum mechanics implies that particles cannot exist before being measured. For a non-technical introduction to the topic, please see Introduction to Quantum mechanics. ...

Bohm's interpretation gives non-mystical explanations of famous experiments of Quantum Mechanics. For example, in the Double-slit experiment for electrons, each electron just travels through only one slit, but the wave function causes the interference pattern. Not only the wave function, but also the trajectory of each electron can be calculated back when knowing the position where the electron hit the screen. The double-slit experiment consists of letting light diffract through two slits producing fringes on a screen. ...

Bohm's interpretation gives natural answers to such philosophical questions. For example, every particle exists all the time and has a unique position, also when not being measured at the moment.

Criticisms

The main points of critics, together with the responses of Bohm-interpretation advocates, are summarized in the following points:

The main weakness of Bohm's theory is that it looks contrived — and gives the same measurable predictions which are in all details identical to conventional quantum mechanics, so it is not really a scientific theory.

Response: It was deliberately designed to give the same predictions as other the conventional approach. Bohm's aim was not to make a serious counterproposal but simply to demonstrate that hidden-variables theories are indeed possible, contrary to the earlier belief (due to von Neuman that this was not possible. Bohm's hope was that this could lead to new insights and experiments that would lead beyond the current quantum theories.

The wavefunction must "disappear" or "collapse" after the measurement, and this process seems highly unnatural in the Bohmian models.

Response: The von Neumann theory of quantum measurement combined with the Bohmian interpretation explains why physical systems behave as if the wavefunction "disappeared", despite the fact that there is no true "disappearance" or "collapse". This is called decoherence.

The theory artificially picks privileged observables: while orthodox quantum mechanics admits many observables on the Hilbert space that are treated almost equivalently (much like the bases composed of their eigenvectors), Bohm's interpretation requires one to pick a set of "privileged" observables that are treated classically - namely the position. There is no experimental reason to think that some observables are fundamentally different from others.

Response: Every physical theory can be rewritten based on different fundamental variables without being different empirically. The Hamilton-Jacobi equation formulation of the classical mechanics is an example. Positions may be considered as a natural choice for the selection because positions are most directly measurable.

The Bohmian models are truly non-local: this non-locality is likely to violate the Lorentz invariance; contradictions with special relativity are therefore expected; they make it highly nontrivial to reconcile the Bohmian models with up-to-date models of particle physics, such as quantum field theory or string theory, and with some very accurate experimental tests of special relativity, without some additional explanation. On the other hand, other interpretations of quantum mechanics - such as Consistent Histories or Many-worlds interpretation allow us to explain the experimental tests of quantum entanglement without any non-locality whatsoever.

Response: Non-locality and Lorentz invariance are not in contradiction. An example of a non-local Lorenz-invariant theory is the Feynman-Wheeler theory of electromagnetism.

Furthermore, it is questionable whether other interpretations of quantum theory are in fact local, or simply less explicit about non-locality. Recent tests of Bell's Theorem add weight to the belief that all quantum theories must either abandon the principle of locality or counterfactual definiteness.

That said, it is true that finding a Lorentz-invariant expression of the Bohm interpretation (or any similar nonlocal hidden-variable theory) has proved very difficult, and it remains an open question for physicists today whether such a theory is possible and how it would be achieved.

The Bohmian interpretation has subtle problems to incorporate spin and other concepts of quantum physics: the eigenvalues of the spin are discrete, and therefore contradict rotational invariance unless the probabilistic interpretation is accepted.

Response: This criticism is based on the wrong assumption that the particle position variables in Bohm's equations must carry spin. There are natural variants of the Bohm interpretation in which such problems do not appear: Spin is only a property of the wave function as in the Schrödinger equation, but the particle variables itself have no spin in the mathematical formulation, spin being a measurable result of the wave function.

The Bohmian interpretation also seems incompatible with modern insights about decoherence that allow one to calculate the "boundary" between the "quantum microworld" and the "classical macroworld"; according to decoherence, the observables that exhibit classical behavior are determined dynamically, not by an assumption.

Response: When the Bohm interpretation is treated together with the von Neumann theory of quantum measurement, no incompatibility with the insights about decoherence remains. On the contrary, the Bohm interpretation may be viewed as a completion of the decoherence theory, because it provides an answer to the question that decoherence by itself cannot answer: What causes the system to pick up a single definite value of the measured observable?

Another possible route to new measureable predictions is opened up by current developments in quantum chaos. In this theory, there exist quantum wave functions that are fractal and thus differentiable nowhere. While such wave functions can be solutions of the Schrödinger equation, taken in its entirety, they would not be solutions of Bohm's coupled equations for the polar decompsition of ψ into ρ and S, given above. The breakdown occurs when expressions involving ρ or S become infinite (due to the non-differentiability), even though the average energy of the system stays finite, and the time-evolution operator stays unitary. As of 2005, it does not appear that experimental tests of this nature have been performed.

The Bohm interpretation involves reverse-engineering of quantum potentials and trajectories from standard QM. Diagrams in Bohm's book are constructed by forming contours on standard QM interference patterns and are not calculated from his "mathematical" formulation. Recent experiments with photons arXiv:quant-ph/0206196 v1 28 Jun 2002 favor standard QM over Bohm's trajectories.

Response: The Bohm interpretation takes the Schrödinger equation even more seriously than does the conventional interpretation. In the Bohm interpretation, the quantum potential is a quantity derived from the Schrödinger equation, not a fundamental quantity. Thus, the interference patterns in the Bohm interpretation are identical to those in the conventional interpretation. As shown in [1] and [2], the experiments above only disprove an incorrect misinterpretation of the Bohm interpretation, not the Bohm interpretation itself.

The Bohm particle(s) are not observable entities in the sense that we can remove them from the theory and still account for all our observations, since Bohm regards the universal wavefunction as a complex-valued but real field that never collapses. This was first noted by Hugh Everett whilst developing his many worlds interpretation of quantum mechanics, who showed that the wavefunction alone is sufficient explanation for all our observations; Bohm accepts that the particles can never be observed directly; Everett (section 6.c of The Theory of the Universal Wavefunction) claimed that they couldn't be observed at all, directly or indirectly.

The word theory has a number of distinct meanings in different fields of knowledge, depending on the context and their methodologies. ... John von Neumann in the 1940s. ... In certain interpretations of quantum mechanics, wavefunction collapse is one of two processes by which quantum systems apparently evolve according to the laws of quantum mechanics. ... A separate article covers Saint John Neumann, the American priest. ... The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ... Collapse is a puzzle game published in 1999 by the software company GameHouse. ... Quantum decoherence is the general term for the consequences of irreversible quantum entanglement. ... For a non-technical introduction to the topic, please see Introduction to Quantum mechanics. ... In mathematics, a Hilbert space is a generalization of Euclidean space which is not restricted to finite dimensions. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... The Hamilton-Jacobi-Bellman (HJB) equation is a partial differential equation which is central to optimal control theory. ... Lorentz covariance is a term in physics for the property of space time, that in two different frames of reference, located at the same event in spacetime but moving relative to each other, all non-gravitational laws must make the same predictions for identical experiments. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. ... Particles explode from the collision point of two relativistic (100 GeV per nucleon) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ... Quantum field theory (QFT) is the application of quantum mechanics to fields. ... Interaction in the subatomic world: world lines of pointlike particles in the Standard Model or a world sheet swept up by closed strings in string theory String theory is a model of fundamental physics whose building blocks are one-dimensional extended objects (strings) rather than the zero-dimensional points (particles... In quantum mechanics, the consistent histories approach is intended to give a modern interpretation of quantum mechanics, generalising the conventional Copenhagen interpretation and providing a natural interpretation of quantum cosmology. ... // The many-worlds interpretation (or MWI) is an interpretation of quantum mechanics that rejects the non-deterministic and irreversible wavefunction collapse associated with measurement in the Copenhagen interpretation in favor of a description in terms of quantum entanglement and reversible time evolution of states. ... Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. ... Lorentz covariance is a term in physics for the property of space time, that in two different frames of reference, located at the same event in spacetime but moving relative to each other, all non-gravitational laws must make the same predictions for identical experiments. ... In physics, Feynman-Wheeler theory is a theory of electromagnetism that uses both retarded and advanced waves. ... Lorentz covariance is a term in physics for the property of space time, that in two different frames of reference, located at the same event in spacetime but moving relative to each other, all non-gravitational laws must make the same predictions for identical experiments. ... Bells theorem is the most famous legacy of the late John Bell. ... In physics, the principle of locality is that distant objects cannot have direct influence on one another: an object is influenced directly only by its immediate surroundings. ... Counterfactual definiteness or CFD is a property of some interpretations of quantum mechanics but not others. ... In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is generated by the motion of its center of mass about an external point. ... This article or section does not cite its references or sources. ... In physics, the SchrÃ¶dinger equation, proposed by the Austrian physicist Erwin SchrÃ¶dinger in 1925, is the definition of energy of a quantum system. ... Quantum decoherence is the general term for the consequences of irreversible quantum entanglement. ... A separate article covers Saint John Neumann, the American priest. ... The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ... Quantum decoherence is the general term for the consequences of irreversible quantum entanglement. ... Quantum chaos is an interdisciplinary branch of physics, arising from so-called semi-classical models. ... 2005 is a common year starting on Saturday of the Gregorian calendar. ... The Universal Wavefunction is a term introduced by Hugh Everett in his Princeton PhD Thesis[1], entitled The Theory of the Universal Wavefunction and forms a core concept in the relative state interpretation[2][3] or many-worlds interpretation[4][5] of quantum mechanics. ... Hugh Everett III (November 11, 1930 â€“ July 19, 1982) was an American physicist who first proposed the many-worlds interpretation of quantum physics, which he called his relative state formulation. ... This article may be too technical for most readers to understand. ... The Universal Wavefunction is a term introduced by Hugh Everett in his Princeton PhD Thesis[1], entitled The Theory of the Universal Wavefunction and forms a core concept in the relative state interpretation[2][3] or many-worlds interpretation[4][5] of quantum mechanics. ...

References

Albert, David Z. (May 1994). "Bohm's Alternative to Quantum Mechanics". Scientific American.
Barbosa, G. D., N. Pinto-Neto (2004). "A Bohmian Interpretation for Noncommutative Scalar Field Theory and Quantum Mechanics". Physical Review D 69: 065014. arXiv:hep-th/0304105.
Bohm, David (1952). "A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables" I". Physical Review 84: 166-179.
Bohm, David (1952). "A Suggested Interpretation of the Quantum Theory in Terms of "Hidden Variables", II". Physical Review 85: 180-193.
Bohm, David (1990). "A new theory of the relationship of mind and matter". Philosophical Psychology 3 (2): 271-286.
Bohm, David; B.J. Hiley (1993). The Undivided Universe: An ontological interpretation of quantum theory. London: Routledge. ISBN 0-415-12185-X.
Durr, Detlef, Sheldon Goldstein, Roderich Tumulka and Nino Zangh (December 2004). "Bohmian Mechanics".
Goldstein, Sheldon (2001). "Bohmian Mechanics". Stanford Encyclopedia of Philosophy.
Hall, Michael J.W. (2004). "Incompleteness of trajectory-based interpretations of quantum mechanics". arXiv:quant-ph/0406054. (Demonstrates incompleteness of the Bohm interpretation in the face of fractal, differentialble-nowhere wave functions.)
Holland, Peter R. (1993). The Quantum Theory of Motion : An Account of the de Broglie-Bohm Causal Interpretation of Quantum Mechanics. Cambridge: Cambridge University Press. ISBN 0-521-48543-6.
Nikolic, H. (2004). "Relativistic quantum mechanics and the Bohmian interpretation". arXiv:quant-ph/0406173.
Passon, Oliver (2004). "Why isn't every physicist a Bohmian?". arXiv:quant-ph/0412119.
Sanz, A. S., F. Borondo (2003). "A Bohmian view on quantum decoherence". arXiv:quant-ph/0310096.
Sanz, A.S. (2005). "A Bohmian approach to quantum fractals". J. Phys. A: Math. Gen. 38. (Describes a Bohmian resolution to the dilema posed by non-differentiable wave functions.)
Streater, Ray F. (2003). Bohmian mechanics is a "lost cause". Retrieved on 2006-06-25.
Valentini, Antony, Hans Westman (2004). "Dynamical Origin of Quantum Probabilities". arXiv:quant-ph/0403034.

Category: Quantum measurement arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and biology which can be accessed via the internet. ... The Stanford Encyclopedia of Philosophy (hereafter SEP) is a free online encyclopedia of philosophy run and maintained by Stanford University. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and biology which can be accessed via the internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and biology which can be accessed via the internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and biology which can be accessed via the internet. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and biology which can be accessed via the internet. ... Ray Streater is a physicist known for his contributions to axiomatic quantum field theory and his interest in this approach to quantum field theory. ... 2006 (MMVI) is a common year starting on Sunday of the Gregorian calendar. ... June 25 is the 176th day of the year (177th in leap years) in the Gregorian Calendar, with 189 days remaining. ... arXiv (pronounced archive, as if the X were the Greek letter Ï‡) is an archive for electronic preprints of scientific papers in the fields of physics, mathematics, computer science and biology which can be accessed via the internet. ...

Results from FactBites:

Bohm interpretation - Wikipedia, the free encyclopedia (2786 words)
	The Bohm interpretation of quantum mechanics, sometimes called the Ontological interpretation, or Bohmian Mechanics is an interpretation postulated by David Bohm in 1952, which was an extension of the de Broglie-pilot-wave theory of 1927.
	The interpretation assumes a single, nonsplitting universe (unlike the Everett many-worlds interpretation) and is deterministic (unlike the Copenhagen interpretation).
	The minimum benefit of Bohm's interpretation - independently from the debate whether it is the preferable formulation - is a disproof of the claim that quantum mechanics implies that particles cannot exist before being measured.

David Bohm (555 words)
	Bohm was born in 1917 at Wiles-Barre, Pennsylvania[?].
	Bohm's colleagues sought to have his position at Princeton reinstated, and Einstein reportedly wanted Bohm to serve as his assistant, but Bohm's contract with the university was not renewed.
	Bohm made a number of significant contributions to physics, particularly in the area of quantum mechanics and relativity theory.

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