What if the electron in the two slit experiment really is a particle, always at some definite place, and the wave is something extra that steers it? That simple idea has one of the strangest histories in physics: proposed, abandoned, wrongly declared impossible, rediscovered and championed by the man who proved the most famous theorem about entanglement.

De Broglie's pilot wave

At the Solvay conference in Brussels in 1927, the French physicist Louis de Broglie presented a "pilot wave" theory in which particles ride along guided by waves. The reception was poor, partly because of an objection from Wolfgang Pauli, and de Broglie soon set the idea aside. Then in 1932 the mathematician John von Neumann published a widely respected book that included a proof seeming to show that no theory of hidden properties could reproduce quantum mechanics. Most physicists took this as settled.

It was not. In 1935 the German mathematician and philosopher Grete Hermann pointed out that von Neumann's proof rested on an assumption that a hidden variable theory need not accept. Her criticism was largely ignored for decades. John Bell reached a similar conclusion independently in 1966.

Bohm's revival

In 1952 the American physicist David Bohm rediscovered and extended the pilot wave idea, showing that it reproduces all the predictions of standard quantum mechanics, including what happens in measurements. Bohm's personal circumstances were difficult. He had lost his position at Princeton after refusing to testify about colleagues before a congressional committee during the anticommunist investigations of the period, and he wrote the papers while preparing to leave for Brazil. The theory is now called de Broglie and Bohm theory, Bohmian mechanics or simply pilot wave theory.

How the theory works

The central claim is this: particles always have definite positions, and the ordinary wavefunction acts as a guiding wave that steers how they move. The wave obeys the usual smooth Schrödinger equation and never collapses. The particle follows a path set by the shape of the wave around it.

In the two slit experiment, each particle passes through one slit only, while its guiding wave passes through both. The two parts of the wave overlap beyond the slits and create a pattern of hills and valleys that funnels particles toward the bright bands. Run many particles and the interference pattern appears, built from individual trajectories.

The theory is fully deterministic. If you knew the exact starting position of every particle, you could predict everything. So where does quantum randomness come from? The randomness reflects our ignorance of the precise starting positions, which in practice are spread out according to the standard quantum odds. Bohmians call this the quantum equilibrium hypothesis and argue that it plays the same role as statistics in the theory of gases.

What about measurement? When a particle enters a detector, the wave splits into parts corresponding to different readings, and the particle ends up in just one part. The other parts still exist but no longer affect the particle's motion. No collapse is needed; the outcome was where the particle actually went.

The case for pilot waves

Defenders, including Bell, Sheldon Goldstein, Detlef Dürr and Tim Maudlin, argue that the theory gives a clear picture of reality with no role for observers, no vague cut and no branching worlds. It shows that the old impossibility proofs were wrong, and it makes the measurement problem simply disappear. Bell wrote that it was the theory that started him thinking about the questions behind his famous theorem.

Fair criticism

The theory has a price. It is openly nonlocal: the motion of one particle can depend instantly on where distant particles are, however far away. This cannot be used to send messages, and Bell's theorem, covered in QUAN 201, shows that any theory matching quantum predictions must have some nonlocal feature. Still, many physicists find the explicit instant influence hard to square with the spirit of relativity.

A related criticism is that the theory is hard to extend to relativity and to theories in which particles are created and destroyed, as they are in particle physics. Versions exist, but none is as natural as the original. Critics also point to the "empty" parts of the wave left behind after a measurement: they keep evolving, just as in many worlds. Deutsch has suggested that pilot wave theory is really many worlds with a reluctant refusal to admit it.

Worked example: bouncing droplets

In 2005 and 2006 the physicists Yves Couder and Emmanuel Fort in Paris found that oil droplets bouncing on a vibrating bath of oil can "walk" across the surface, guided by the ripples they create. These walking droplets showed effects that looked strikingly like quantum behavior, including patterns resembling two slit interference.

Did this prove pilot wave theory? No. A droplet is a classical object riding a real wave in ordinary space, while the quantum wavefunction lives in a far stranger mathematical space with one set of dimensions for each particle. Later experiments also failed to reproduce some of the early two slit results. The droplets are a valuable analogy: they show how a particle with a definite path can produce wave patterns. But an analogy is not a test.