De Broglie-Bohm pilot wave theory

Pilot-wave theory is the account of quantum mechanics that Louis de Broglie put to the Solvay conference of 1927 and David Bohm rediscovered and completed in 1952. It says that a particle has a position at every instant, whether or not anyone is looking, and that the wave function is not a list of probabilities but a field, obeying the Schrödinger equation, that guides the particle: each particle moves with a velocity given by the gradient of the wave function’s phase at the configuration of the whole system, times ℏ over its mass. Those two equations are the whole of the dynamics. Given one thing more, that the particles of an ensemble are distributed as |ψ|2, which is the Born rule, the rest of what the textbooks present as separate postulates follows: the collapse of the wave function at a measurement, which becomes an effective collapse of the wave function of the subsystem measured, and the uncertainty relations, as the statistical mechanics of particles whose motion is not classical because a wave steers them.

The double-slit experiment, in which Feynman found the only mystery, then has a plain description. Each electron passes through one slit; the wave passes through both; and the wave steers the electron into the bright fringes and away from the dark ones. The paths were first drawn in 1979 by Philippidis, Dewdney and Hiley, and the two-slit film below computes them again.

The theory is usually called an interpretation of quantum mechanics, as if it were a reading of the same physics. It is better regarded as a theory in its own right. It rests on different postulates, and in one regime it predicts differently. The Born rule, that the particles of an ensemble are distributed as |ψ|2, is not a law of the theory but a state of equilibrium, as a Maxwellian distribution of velocities is the equilibrium of a gas: the dynamics carries other distributions towards it on any coarse-grained view, when the wave is complicated enough to stir them, as the films show, and a system that had not yet reached it would do things that standard quantum mechanics forbids. Nothing now observed is out of this equilibrium, and whether the early universe was is a question for cosmology, which Antony Valentini has pursued for thirty years. Whether or not it describes nature, the theory is worth knowing, because it shows that the formalism of quantum mechanics is consistent with a world of particles that have positions at all times, which a great deal of what is said about quantum mechanics takes to be impossible.

Why it is on this site

Diffusion Monte Carlo, CASINO’s main method, moves electrons by a random walk in imaginary time to find the state of lowest energy. The walk is a device of calculation and nothing the electrons do. Pilot-wave trajectories are the motion of the electrons in real time, as the theory has it. The two have more to do with each other than one might expect. The drift that steers the walkers of DMC, the gradient of ln|Ψ| for the trial wave function, has the form of the osmotic velocity of Edward Nelson’s stochastic mechanics, a theory of 1966 in which particles diffuse at random, drifting with that velocity and the de Broglie-Bohm velocity together, and from which the Schrödinger equation is recovered, given a condition on the circulation of the flow that Wallstrom pointed out in 1994. Whether such connections can be made to improve quantum Monte Carlo is a question kept open here (Lecture 6 below is about calculating with quantum trajectories).

CASINO can also simply compute the trajectories. It evaluates its trial wave functions, with their gradients and local energies, at configurations read from a file (runtype pilot_eval), and the utility pilotdrive integrates de Broglie-Bohm trajectories in a superposition of such states for a real atom or molecule, following the relaxation of an ensemble towards quantum equilibrium by Valentini’s coarse-grained H-function; given a diffusion constant it integrates the stochastic dynamics of Bohm, Vigier and Nelson instead. The beryllium film on the Videos and simulations page, four electrons guided by a superposition of twelve of the atom’s states, was made this way. For model particles in a box there is LOUIS, a separate program of Mike Towler’s distributed with CASINO since 2013, which made the 2012 film and the Venus film below; the Relaxation film repeats its shipped example with a program written for the purpose and checked against it.

Relaxation to the Born rule: a particle in a box

Full screen: the button among the film’s controls, at its lower right in most browsers, or in Firefox, Chrome and Edge a double-click on the film; on a phone, tap the film, then that button, and turn the phone sideways. Or open the film by itself, alone in the browser’s window.

A single particle in a square box is guided by a wave made of sixteen of the box’s standing waves, with random phases. The smoke is an ensemble of such particles: each moves on its own path, a de Broglie-Bohm trajectory set by the wave, and together they start in the wrong place, not where the wave’s intensity says they should be. Where the wave vanishes its phase turns, and the flow circles round those points like small tornadoes, which draw the smoke into ever finer filaments. Seen at any fixed scale, the smoke comes to follow the wave’s intensity: within one period of the wave the ensemble has largely forgotten where it began, and the Born rule emerges from the motion itself. The 2012 film of Towler, Russell and Valentini computed again from the same wave, the input of LOUIS’s shipped example, now on a grid of 2048 x 2048 points with four trajectories to a point and to the wave’s full period, by a program written for the purpose and checked against LOUIS; the sound follows the fall of the ensemble’s H-function. Idea for the film was by student Nick Russell (what happened to him?).
The 2012 film: the density of an ensemble of particles in a box relaxing to the Born rule
The film as it was published with the 2012 paper: the same box and the same wave, the ensemble’s density drawn on a coarser grid, one period of the wave. The paper’s figure 2 below it shows |ψ|2 (top) and the ensemble’s density (bottom) at the start, half way and at the end of the period.
Figure 2 of Towler, Russell and Valentini (2012): |psi|^2 and the density at t = 0, 2 pi and 4 pi

Relaxation in three dimensions, where the wave vanishes along lines

Full screen: the button among the film’s controls, at its lower right in most browsers, or in Firefox, Chrome and Edge a double-click on the film; on a phone, tap the film, then that button, and turn the phone sideways. Or open the film by itself, alone in the browser’s window.

A particle in a cubic box is guided by a wave made of sixty-four of the box’s standing waves with random phases, and an ensemble of such particles, started away from the wave’s intensity, is stirred to equilibrium by the wave’s nodes, which in three dimensions are lines that move, stretch and reconnect.

The Birth of Venus, out of quantum equilibrium

Full screen: the button among the film’s controls, at its lower right in most browsers, or in Firefox, Chrome and Edge a double-click on the film; on a phone, tap the film, then that button, and turn the phone sideways. Or open the film by itself, alone in the browser’s window.

Botticelli’s Birth of Venus, taken as an initial distribution out of quantum equilibrium for a particle in the same two-dimensional box and the same wave as above: the brightness of the painting is the ratio of the ensemble’s density to |ψ|2, which de Broglie-Bohm dynamics keeps constant along every trajectory, so that each point of the picture is carried with the particle that started there. The de Broglie-Bohm flow stirs the painting into ever finer filaments while the coarse-grained density relaxes to |ψ|2, the two small panels showing the two coming to agree and the curve the coarse-grained H-function falling. One period of the wave function, 262,144 trajectories a frame, computed with LOUIS. The painting: Sandro Botticelli, about 1485, Uffizi, Florence (the Google Art Project reproduction, public domain). Film by Mike Towler.

Two slits

Full screen: the button among the film’s controls, at its lower right in most browsers, or in Firefox, Chrome and Edge a double-click on the film; on a phone, tap the film, then that button, and turn the phone sideways. Or open the film by itself, alone in the browser’s window.

Electrons of 45 keV leave two slits a micrometre apart. The glowing curve moving away from the slits is the probability density across the beam, its peak held at a constant height: two separate beams at first, which spread, meet and break into fringes. Behind it the landscape of the quantum potential rises, and the threads drawn across it are the paths of individual electrons, each through one slit, guided into the bright fringes. The calculation is that of C. Philippidis, C. Dewdney and B. J. Hiley, Il Nuovo Cimento B 52, 15 (1979), with the slits and distances of Jönsson’s experiment of 1961. Film by Mike Towler.

Another theory of trajectories: Nelson’s stochastic mechanics

For comparison, electrons moving as a different theory says they move. In Nelson’s stochastic mechanics every particle diffuses at random, with a drift that quantum Monte Carlo would recognise, and the Schrödinger equation is recovered rather than assumed; whether its predictions for measurements made at two different times agree with quantum mechanics’ has been argued over since the 1980s.

Full screen: the button among the film’s controls, at its lower right in most browsers, or in Firefox, Chrome and Edge a double-click on the film; on a phone, tap the film, then that button, and turn the phone sideways. Or open the film by itself, alone in the browser’s window.

Electrons moving as Edward Nelson’s stochastic mechanics of 1966 says they move, computed with CASINO: the hydrogen atom’s electron in its exact 1s state, then the electrons of 2000 hydrogen atoms drawn about one proton, then caffeine’s 74 outer electrons in CASINO’s approximation to the molecule’s ground state, one configuration and then up to 256, every electron drifting and diffusing by Nelson’s equation for these states, dX = ∇ln|Ψ| dt + dW, integrated in steps of 0.0005 atomic units of time, the drift capped near the wave function’s nodes in 0.9 per cent of caffeine’s steps. The trails are computed paths, the glow the electron positions shown so far, added up, and the clock the theory’s time, a second of film being 10.9 attoseconds. The jitter is Nelson’s postulate, neither the electron’s zero-point motion nor the jitter of quantum electrodynamics, and quantum mechanics itself gives the electron no path. Film by Mike Towler.

The lectures

Mike Towler gave an eight-lecture graduate course on pilot-wave theory in the Cavendish Laboratory in 2009, and a two-hour public lecture on it to the Cambridge University Physical Society on 21 October 2009, whose title the student organisers asked to be made controversial. I think I succeeded, do you? [wink emoji] The slides are here, with the films shown in the lectures, now in a form a browser plays.

The return of pilot waves, or why Bohr, Heisenberg, Pauli, Born, Schrödinger, Oppenheimer, Feynman, Wheeler, von Neumann and Einstein were all wrong about quantum mechanics (slides, 64 pages; the poster).

Pilot-wave theory, Bohmian metaphysics, and the foundations of quantum mechanics, the course of 2009:

The last lecture is not about de Broglie-Bohm theory but about what Bohm was doing in his last two decades, which has since found a mathematical expression in the work of his closest colleague, Basil Hiley; the slides of Hiley’s talk to the Cambridge Electronic Structure Discussion Group in February 2012, Weak measurements: Wigner-Moyal and Bohm in a new light, are here too.

MDT is currently preparing a new 15 pilot-wave lecture series, rewritten from the ground up without particular reference to the originals, which will extend into the relativistic and quantum field theory arena, which therefore requires research

The films shown in the lectures

Two-path interference built up one electron at a time: Akira Tonomura’s film of 1989 (Hitachi), made with an electron biprism in place of two slits, in which the pattern builds up from single arrivals. Shown in the public lecture and in Lecture 1.
Pilot-wave trajectories through a single slit, and below, particles meeting an obstacle and the hydrogen atom in a stationary and in a non-stationary state: the animations of the Bohmian mechanics pages of the Institute for Theoretical Physics, University of Innsbruck (Sabine Kreidl). Shown in Lecture 1.
Particles meeting an obstacle (Innsbruck). Shown in the public lecture.
The hydrogen atom in a stationary state (Innsbruck). Shown in Lecture 3.
The hydrogen atom in a non-stationary state (Innsbruck). Shown in Lecture 3.
Particle trajectories near a moving node of the wave function (the star), with the standard guidance equation; below, the same with the spin term added to the guidance equation. Samuel Colin. Shown in the public lecture.
Trajectories near a moving node, guidance equation with the spin term. Samuel Colin. Shown in the public lecture.
The 1927 Solvay conference, on film, with a spoken commentary. Shown in the public lecture and in Lecture 7.

Also shown in the public lecture: career advice from Antony Valentini.

The 2010 meeting

The participants of the 2010 meeting at Vallico Sotto

In the summer of 2010 most of the world’s de Broglie-Bohm researchers came to Vallico Sotto for 21st-century directions in de Broglie-Bohm theory and beyond, 28 August to 4 September, the largest meeting the Apuan Alps Centre for Physics has held. The report of the meeting is on the Towler Institute’s site, with all the talks; the two introductory talks that set its framework are Mike Towler’s, on the basics, and Antony Valentini’s, on current research.

LOUIS and the 2012 paper

LOUIS is the program Mike Towler wrote to study the approach to quantum equilibrium: a particle in a box, guided by a superposition of the box’s standing waves, and ensembles of them, with the time-dependent density and the H-function. It is in the CASINO distribution, under utils/louis, with its own manual; it made the 2012 film and the Venus film above, and the Relaxation film repeats its example. It led to the paper

M. D. Towler, N. J. Russell and A. Valentini, Time scales for dynamical relaxation to the Born rule, Proc. R. Soc. A 468, 990 (2012), doi:10.1098/rspa.2011.0598,

which showed, for a particle in a box, how an ensemble that starts away from the Born rule, normally taken as a postulate of quantum mechanics, is carried towards it by the motion on de Broglie-Bohm trajectories, and on what time scale.

Reading

  • D. Bohm, Phys. Rev. 85, 166 (1952) and Phys. Rev. 85, 180 (1952): the two papers in which the theory was completed.
  • P. R. Holland, The Quantum Theory of Motion (Cambridge University Press, 1993): the textbook.
  • D. Bohm and B. J. Hiley, The Undivided Universe (Routledge, 1993).
  • C. Philippidis, C. Dewdney and B. J. Hiley, Il Nuovo Cimento B 52, 15 (1979): the two-slit trajectories.
  • A. Valentini and H. Westman, Proc. R. Soc. A 461, 253 (2005): relaxation to quantum equilibrium.
  • E. Nelson, Phys. Rev. 150, 1079 (1966): stochastic mechanics.
  • T. C. Wallstrom, Phys. Rev. A 49, 1613 (1994): why recovering the Schrödinger equation from a flow needs a condition on its circulation.

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