Intuitive

Trajectories of a particle in the famous double slit experiment in Bohmian mechanics. This image shows nicely how dependent the paths are on tiny changes in the initial position, i.e. where exactly the particle ends up in the double slit.

The main idea of Bohmian mechanics is that all strange effects of quantum mechanics that we observe in experiments are simply the result of a new force. This force arises from a new potential, called "quantum potential". The reason that quantum effect often seem so strange is explained in the Bohmian framework by the observation that the quantum potential is extremely sensitive to tiny changes. Hence a tiny change in the initial position can have a large effect on the final position.

When elementary particles fly through space they constantly interact with the quantum potential. The quantum potential determines their path and the path in turn changes the quantum potential.

One way to understand this is to visualize a ball bouncing of an elastic surface. Every time the ball bounces it causes ripples in the surface. These ripples in turn influence how the ball bounces.

This is nicely visualized this video by Veritasium.

• See also the visualizations of Bohmian mechanics by the university of Innsbruck here.

Concrete

In Bohmian mechanics, the wave equation of ordinary Quantum Mechanics, e.q., the Schrödinger equation is rewritten, using the ansatz \begin{equation} \Psi= R \mathrm{e}^{i S}. \end{equation}

This results in two equation, one continuity equation, and one equation that looks exactly like the Hamilton-Jacobi equation of classical mechanics plus an extra term. A very natural interpretation of this extra term is as an extra potential. This extra potential is responsible for the quantum-like behaviour and makes all the difference between quantum and classical mechanics. This is often regarded to as "Pilot-Wave", which is in some sense analogous to the bouncing droplet wave.

Because both formulations, Canonical Quantum Mechanics and Bohmian Mechanics, originate in the same equation, their experimental predictions are the same. The difference lies in the interpretation.

Recommended Resources

• On Bohmian Mechanics - an email exchange by Sheldon Goldstein & Steven Weinberg
• See also Why isn’t every physicist a Bohmian? by Oliver Passon
• The Pilot-Wave Perspective on Quantum Scattering and Tunneling by Travis Norsen
• De Broglie-Bohm pilot-wave theory and the foundations of quantum mechanics - graduate lecture course by by Mike Towler
• www.bohmian-mechanics.net/ - "the homepage of the international research network on Bohmian Mechanics"

Abstract

Why is it interesting?

Bouncing oil droplets behave in many experiments (including the double-slit experiment) just like mysterious quantum objects, although their behavior can be captured on video and is by every definition classical. The bouncing of the droplets generate a wave on the liquid surface they are bouncing on and this wave determines the further behavior of the droplets. This wave is real. It can be captured on film! Nevertheless, it is responsible for the wave-like behavior of the droplets. A very good overview article about those new experiments can be found here.

In quantum mechanics, the wave responsible for the wave-like behavior of particles is interpreted as an abstract probability wave. (Which is in some way real, too, but could of course never be captured on film.)

For a very long time the Quantum (Copenhagen) interpretation dominated. Most people believing in and working on Bohmian mechanics weren't taken seriously. The discussions were mostly philosophical and kind of boring. Quantum mechanics was a further developed and therefore easier to handle theory. Everyone learns it in university. Bohmian mechanics was regarded as a lost cause.

Surprisingly, the new experiments indicate that Bohmian Mechanics may be closer to reality than the Copenhagen interpretation. Of course, at this stage, both theories are still equivalent regarding experimental predictions. Therefore, many physicist no-longer call Bohmian mechanics wrong, but simply, not useful, because everyone is an expert in the quantum framework. Why should one learn something new, that offers nothing new?

One reason could be that a new perspective, a new interpretation, may predict something new if enough people are thinking this way. Furthermore, some hope for advances in bringing quantum mechanics and gravity together. All attempts trying to incorporate gravity into the quantum framework failed. The other way round, incorporating quantum mechanics into the classical mechanics' framework, could be a more successful route.

It took many years and the ideas of many brilliant minds making quantum theory (and of course quantum field theory) the theory it is today and no one can tell if working intensively on Bohmian mechanics would be a fruitful endeavor. Nevertheless, the bouncing droplet experiments may motivate some to take the risk.

The Bohmian approach is the best way of understanding the particle-wave dichotomy, with its local and non-local aspects. It ran into difficulties with quantum-field theory, but with new ideas I think the difficulties can be circumvented. I predict that Bohm will be seen as far ahead of his time.

M. F. Atiyah

What I like about Bohmian mechanics is that it is by far the simplest formulation of nonrelativistic quantum mechanics satisfying the requirements of coherence and objectivity (in the above sense). Two equations, Schrödinger’s and the de Broglie-Bohm guiding equation, completely express the theory. From these simple equations the rest of the quantum formalism flows. Thus to the extent that “Bohm’s quantum mechanics uses the same formalism as ordinary quantum mechanics, including a wave function that satisfies the Schrödinger equation,” it is only the Schrödinger equation part of that formalism that (along with the guiding equation) is fundamental in Bohmian mechanics, with the rest of the formalism arising as a consequence.

[…]

You should therefore appreciate why others who agree with us on this, and who are not aware of any other adequate alternatives to the Copenhagen interpretation, might be attracted to Bohmian mechanics: they want to make sense of quantum mechanics, something that the Copenhagen interpretation manifestly does not do and that Bohmian mechanics manifestly does.

Sheldon Goldstein in http://inference-review.com/article/on-bohmian-mechanics

What is not widely understood, even amongst physicists, is that a belief in the mystical aspects of the theory is a choice that one makes, rather than something inevitable. One formulation of quantum mechanics - long ignored or derided by just about everyone - which makes this particularly clear is the pilot-wave theory (also known as Bohmian mechanics, de Broglie-Bohm theory, the causal or ontological interpretation of QM). In this theory, wave particle duality is explained through the startlingly sensible notion of having both waves and particles (think about how that makes the double slit experiment intelligible!). So unlike in orthodox QM - where the wave function is all there is - the particles have an objectively real existence and they move along trajectories, guided by the waves. In such a formalism the standard paradoxes related to measurement, observation or wave function collapse (Schrödinger's cat, and so on) simply evaporate. The classical limit emerges out of the theory, rather than being presupposed. All the 'talk' is replaced by sharply-defined mathematics, it becomes possible to 'visualize' the reality of most quantum events, and - most importantly - the theory is completely consistent with the full range of QM predictive-observational data. http://www.tcm.phy.cam.ac.uk/~mdt26/pilot_waves.html

FAQ

What are problems of Bohmian Mechanics?

Unfortunately, Bohmian mechanics is a nonrelativistic theory, and so it is of value primarily for the lessons it conveys about finding a sensible interpretation of quantum mechanics that is relativistic, rather than for the specific details of the theory itself. Now the question you raised about pair creation is, of course, very important. Bohmian mechanics itself is not a theory with particle creation or annihilation. However, I see no reason why some Bohm-type theory should not permit these things. Here are two ways they could arise: What might be called the Bohm motion for the Klein-Gordon equation permits trajectories that can reverse time direction, which can naturally be regarded as describing pair annihilation or creation depending on the sense of the reversal. Also, if the basic variables were field configurations, then a Bohm-type evolution for such configurations that permits the number of particle-like elements (solitons or what have you) of the field configuration to change during the course of the evolution might well exist.

Sheldon Goldstein

For a possible solution, see section 4.3 here. To quote from it:

Bohmian mechanics is very successful for non-relativistic QM, but it has two major problems in attempts of generalization to relativistic quantum field theory (QFT). First, how can Bohmian non-locality be compatible with relativity? Second, how can continuous particle trajectories be compatible with particle creation and destruction? Here I propose a simple approach for dealing with these problems. The key is the analogy with the Bohmian interpretation of phonons. [..] But phonons are not fundamental. From condensed-matter perspective, fundamental particles are electrons and nuclei described by non-relativistic QM. Hence, from condensed-matter perspective, there is no creation and destruction of fundamental particles. Lorentz invariance and QFT are emergent, i.e. derived from non-relativistic QM. Phonon is a quasi-particle, not a “true” particle. Hence, in the Bohmian interpretation of phonons, there are no trajectories for phonons. For a Bohmian interpretation of phonons it is sufficient to have non-relativistic trajectories of electrons and nuclei. In this sense, non-relativistic Bohmian mechanics is fundamental.https://arxiv.org/pdf/1703.08341.pdf

What are arguments against Bohmian Mechanics?

In any case, the basic reason for not paying attention to the Bohm approach is not some sort of ideological rigidity, but much simpler—it is just that we are all too busy with our own work to spend time on something that doesn’t seem likely to help us make progress with our real problems.

Steven Weinberg in http://inference-review.com/article/on-bohmian-mechanics

History

In 1951, egged on by Einstein, Bohm found that the seeming randomness of the quantum world could be a result of invisible waves traveling faster than the speed of light. This was completely at odds with the rest of physics at the time. Bohm’s ideas about “pilot waves” guiding the motion of quantum particles were similar to work done by the physicist Louis de Broglie 25 years earlier. But Bohm took his ideas further and turned them into a full theory. That theory could account for all the usual results of quantum mechanics — in fact, it was mathematically equivalent to the standard theory — but without making strange pronouncements about cats being both dead and alive until you look. But before Bohm could defend his revolutionary ideas about quantum physics, he was swept up in the anti-Communist hysteria of the McCarthy era. Bohm ended up blacklisted and trapped in Brazil, in exile from the rest of the physics world. 13 years later, Bohm’s ideas served as inspiration for another physicist, John Bell. As a direct result of reading Bohm’s work, Bell discovered a theorem that’s been called “the most profound discovery of science.” Adam Becker

• What is real? The Unfinished Quest for the Meaning of Quantum Physics by Adam Becker is a great book about the history of Bohmian mechanics.