De Broglie–Bohm theory

The Bohm interpretation of quantum mechanics, sometimes called the Causal interpretation, the Ontological interpretation, the de Broglie-Bohm theory, or the hidden variables interpretation is an interpretation postulated by David Bohm. The hidden variables present render this a deterministic interpretation.

Background

The Bohm interpretation 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. The founding fathers, when considering such systems, began to address "the wave/particle duality", i.e., they were postulating that the same entity seemed to be showing both wave-like and particle-like characteristics. De Broglie changed the picture rather fundamentally. He allowed the electrons (and all other quantum entities) to return to being localized particles. But he posited that in the world of quantum phenomena, every kind of particle is accompanied by a wave.

This wave 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 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.

Hidden variables, 'impossiblity proofs' and Nonlocality

The Bohm interpretation is an example of a so-called 'hidden variables' theory. According to this brand of thinking, one regards the orthodox quantum theory as 'incomplete', i.e., its description of state (the wave function) fails to account for the full physical reality of the phenomena (whose mathematical description would require also the hidden variables). The probabilistic nature of the orthodox quantum is then to be traced to this incompletness. The fact that one can find varying results for systems with the same wave-function could be said to arise from variations in the hidden variables (Mathematically, the hidden variables provide additional degrees of freedom).

It is impossible to fully address hidden variables and the Bohm interpretation, without bringing in the analysis and arguments of John Stewart Bell, who one might claim single-handedly changed the entire nature of foundations of quantum mechanics. Not only did Bell derive his famous "Bell's inequality" with its own implications for quantum physics, but he also refuted the so called 'no-hidden variables' proofs.

What motivated Bell's interest in hidden variables? Despite the fact the such schemes are often associated with the issue of indeterminism, or uncertainty, Bell was instead concerned with the fact that orthodox quantum mechanics is a subjective theory, and the concept of 'measurement' figures prominently in its formulation. It was not that Bell found 'measurement' unacceptable in itself, but he objected to its appearance at quantum mechanics' most fundamental theoretical level - which he insisted must be concerned only with sharply-defined mathematical quantities and unambiguous physical concepts.

Bell was impressed that within Bohm’s hidden variables theory, reference to this concept was not needed, and it was this which sparked his interest in the field of research.

But if he were to thoroughly explore the viability of Bohm's theory, Bell needed to answer the challenge of the so-called 'impossibility proofs' against hidden variables. Bell addressed these in a paper entitled 'On the problem of hidden variables in quantum mechanics' . Here he showed that von Neumann’s argument does not prove impossibility, as it claims. The argument fails in this regard due to its reliance on a physically unreasonable assumption. In this same work, Bell showed that a stronger effort at such a proof (based upon Gleason's theorem) also fails to eliminate the hidden variables program. (Interestingly, the flaw in von Neumann's proof was previously discovered by Grete Hermann in 1935, but did not become common knowledge until rediscovered by Bell.)

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. Therefore the response many physicists have to Bohm's theory is often related to how they regard this concept.

This question hinges upon the attitude one takes towards the Einstein-Podolsky-Rosen paradox <><> and Bell's theorem <><>. There are often two camps to be seen here. Some concur with Bell that (p 196 in ) "It is known that with Bohm's example of EPR correlations, involving particles with spin, there is an irreducible nonlocality." For those who share this view, 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 conclusion to be the demise of "local realism". This group would claim that retaining quantum orthodoxy would permit one to avoid non-locality. They might tend to be less receptive to Bohm's interpretation.

Mathematical foundation

One-particle formalism

The Schrödinger equation for one particle of mass m is

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

where the wavefunction $\psi (\mathbf {x} ,t)$ is a complex function of the particle's position x and time t. 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

$\psi ={\sqrt {\rho }}e^{iS/\hbar }$.

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

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

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

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

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.

Many-particle formalism

The many-particle Schrödinger equation is a straightforward generalisation of the one-particle example:

$i\hbar {\frac {\partial \psi (\mathbf {x_{1},x_{2},...} ,t)}{\partial t}}=\sum _{i}{\frac {-\hbar ^{2}}{2m_{i}}}\nabla _{i}^{2}\psi (\mathbf {x_{1},x_{2},..} ,t)+V(\mathbf {x_{1},x_{2},..} )\psi (\mathbf {x_{1},x_{2},...} ,t)$,

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

$\rho (\mathbf {x_{1},x_{2},..} ,t)=R(\mathbf {x_{1},x_{2},..} ,t)^{2}=|\psi (\mathbf {x_{1},x_{2},..} ,t)|^{2}=\psi ^{*}(\mathbf {x_{1},x_{2},...} ,t)\psi (\mathbf {x_{1},x_{2},...} ,t)$.

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:

$\psi ={\sqrt {\rho }}e^{iS/\hbar }$.

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

$Q=-\sum _{i}{\frac {\hbar ^{2}}{2m_{i}}}{\frac {\nabla _{i}^{2}R}{R}}=-\sum _{i}{\frac {\hbar ^{2}}{2m_{i}}}\left({\frac {\nabla _{i}^{2}\rho }{2\rho }}-\left({\frac {\nabla _{i}\rho }{2\rho }}\right)^{2}\right)$

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

$\mathbf {j_{i}} =\rho \mathbf {v_{i}} =\rho {\frac {\mathbf {p_{i}} }{m_{i}}}=\rho {\frac {\nabla _{i}S}{m_{i}}}$,

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.

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.

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.

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.

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.

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".

Indeed, Bohm's interpretation subsumes the quantum concepts of measurement, complementarity, decoherence, and entanglement into mathematically precise guidance conditions and position variables.

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.

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.

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 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?

• The Bohm interpretation does not lead to new measurable predictions, so it is not really a scientific theory.

Response: In the domain in which the predictions of the conventional interpretation of quantum mechanics are unambiguous, the predictions of the Bohm interpretation are identical to those of the conventional interpretation. However, in the domain in which the conventional interpretation is ambiguous, such as the question of the time-observable in non-relativistic quantum mechanics and the position-observable in relativistic quantum mechanics, the Bohm interpretation leads to new unambiguous measurable predictions.

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.

See also

• David Bohm
• Holomovement
• Implicate and Explicate Order
• Interpretation of quantum mechanics

References

• Albert, David Z. (1994). "Bohm's Alternative to Quantum Mechanics". Scientific American. {{cite journal}}: Unknown parameter |month= ignored (help)
• Barbosa, G. D. (2004). "A Bohmian Interpretation for Noncommutative Scalar Field Theory and Quantum Mechanics". Physical Review D. 69: 065014. arXiv:hep-th/0304105. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• 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 (1993). The Undivided Universe: An ontological interpretation of quantum theory. London: Routledge. ISBN 0-415-12185-X. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
• Durr, Detlef (2004). "Bohmian Mechanics" (PDF). {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |month= ignored (help)
• 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. {{cite journal}}: Cite journal requires |journal= (help) (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 0521485436.
• Nikolic, H. (2004). "Relativistic quantum mechanics and the Bohmian interpretation". arXiv:quant-ph/0406173. {{cite journal}}: Cite journal requires |journal= (help)
• Passon, Oliver (2004). "Why isn't every physicist a Bohmian?". arXiv:quant-ph/0412119. {{cite journal}}: Cite journal requires |journal= (help)
• Sanz, A. S. (2003). "A Bohmian view on quantum decoherence". arXiv:quant-ph/0310096. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)
• 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 2006-06-25.
• Valentini, Antony (2004). "Dynamical Origin of Quantum Probabilities". arXiv:quant-ph/0403034. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |coauthors= ignored (|author= suggested) (help)