A Bohmian Rhapsody

A Bohmian Rhapsody

Mrittunjoy Guha Majumdar talks Bohmian mechanics, the ‘causal interpretation’ of the strange world of quantum mechanics.

Fluid droplets bounce when placed on the surface of a vibrating fluidic bath. A student working at the Matter and Complex Systems Laboratory, National Centre for Scientific Research (CNRS) in France discovered this using oil droplets and an oil bath in 2005. The bouncing of the droplets seemed to be guided by an unseen wave – a guiding, or “pilot”, wave. If that struck you as bizarre, there’s more – what if one were to posit that nature is fundamentally like those bouncing droplets riding waves? What if such pilot-waves could explain the peculiar and fairly counter-intuitive behaviour of particles on the quantum level?

In the early twentieth century, it was discovered that the laws of Physics that govern macroscopic objects don’t quite apply to microscopic realms. For instance, in these realms, the act of observing physical phenomena actually influences the phenomena taking place. (Thankfully, this doesn’t occur on macroscopic levels – one would imagine life would be quite strange otherwise.) On the quantum level, waves can also act like particles and particles can act like waves. Matter can also go, or ‘tunnel’, from one spot to another without moving through the intervening space. And if that were not enough, information can move across vast distances instantaneously in what Einstein best described as ‘spooky action at a distance’. Outlandish!

For most of the past century, the predominant explanation for why a quantum particle sometimes behaves like a wave is called the `Copenhagen interpretation’, which states that a single particle is like a wave that is smeared out across the universe, and that smear collapses into a certain position only when observed. (To quantum physicists, such a smear is known as a ‘superposition’). One could thus say that a quantum object, like a particle, is always is in a superposition of states, until one collapses it into just one of these states. This collapse is found to be depend on the laws of probability. However, what if we could explain this wave-particle behaviour deterministically, such that the result isn’t simply a result of chance?

Enter Louis De-Broglie and David Bohm, and their alternative interpretation, known as the ‘pilot-wave theory’, which posits that quantum particles are borne along on pilot-waves, just like how the oil drops were borne along pilot-waves when bouncing as observed at CNRS. Unlike other interpretations of quantum mechanics, such as the Copenhagen Interpretation or Many-Worlds theories, the Bohmian interpretation (which, in fact, precedes the Copenhagen Interpretation) does not consider observers or the act of observation as necessary for the predictions of quantum mechanics to hold true. It is a ‘quantum theory without observers’, if you will.

In the Bohmian formulation, an individual quantum system is formed by a point particle and a guiding wave. Wavefunctions are quantities that mathematically describe the wave characteristics of a particle. While most quantum theories suggest you can describe a system by wavefunctions alone, Bohmian mechanics states that a quantum system is fundamentally about the behaviour of particles. The particle nature of matter becomes primary while the wavefunction is secondary. These particles can be described by their positions, and Bohmian mechanics discusses how they change with time.

Here is a simplistic overview of what Bohmian mechanics entails. The state of a system of N particles is described by its wavefunction, a complex function that varies based on the possible configurations of the system, and its actual configuration, defined by the actual positions of its particles. The behaviour of the system is then described by two evolution equations, the famous Schrödinger’s Equation, which describes how the wavefunction changes with time, and a Guiding Equation, which describes how the position of the particles changes.

Researchers who have spent time analysing the Bohmian idea with scientific rigour showed that Bohmian mechanics agrees with most, if not all, experiments in the quantum realm carried out up to now. Some of the best proofs of Bohmian mechanics have arisen from studying the characteristics of the particle and its guiding pilot-wave, and relating them with empirical evidence.

More fascinatingly, Bohmian mechanics is an example of a hidden-variable theory. Hidden-variable theories are those that regard the Universe as inherently deterministic (“A must cause B and not C”) and only seemingly probabilistic (“A could cause B or C”) due to variables that we are not aware of – variables that are hidden.

In Bohmian mechanics, the variable hidden from us is the position of the particle. Hidden-variable theories predict that the gradient-field of the hidden variable should be observable as a weak measurement, that is, a measurement that would not greatly affect the system by the act of measurement itself. In the case of Bohmian mechanics, this corresponds to the particle’s velocity. As such, weak measurements of particle velocities have been used in quantum experiments to track the trajectories of single photons.

Although Bohmian mechanics resolves the issues of quantum wavefunction collapse and measurement quite nicely, it has a fair share of criticism. Many quantum physicists believe that Bohmian mechanics is useful for research.

The Nobel laureate Steven Weinberg, in a private exchange of letters with colleague Sheldon Goldstein, wrote: “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.”

Tomas Bohr, fluid physicist at the Technical University of Denmark and grandson of the famous physicist Neils Bohr, recently also gave a strong argument against Bohmian mechanics in a thought experiment that could be its downfall.

Nonetheless, for now, Bohmian mechanics remains one of the most fascinating interpretations of quantum mechanics today and is one of the last hidden-variable theories to survive the test of time. As I like to say: harmonious, this Bohmian rhapsody wafts along!

Mrittunjoy Guha Majumdar is a postdoctoral fellow at the Cavendish Laboratory, working under Nobel laureate Prof. Brian Josephson.