Convergence to Bohmian mechanics in a de Broglie-like pilot-wave system

In his 1924 thesis [8], Louis de Broglie predicted that massive particles would behave like waves of the eponymous de Broglie wavenumber:

where $p=\gamma mu$ is the relativistic momentum of a particle of mass $m$ and velocity $u$. His vision deviated somewhat from the prevailing theories of Schrödinger, Bohr, and Heisenberg, however; in the subsequent years, he worked on his double-solution theory, which posited that a quantum particle consists of both particle and wave, evolving in unison [6, 7]. The wave field would contain a singularity at the particle’s position, though its form was unspecified by de Broglie, and the interplay between the particle and wave would give rise to emergent statistical behaviour in line with quantum theory. De Broglie famously presented a version of his theory at the 1927 Solvay conference, to mixed reception, and (temporarily) abandoned the pilot-wave approach shortly thereafter [9].

In 1952, David Bohm arrived at a different pilot-wave model of quantum mechanics [10, 11], providing an important counterexample to the “impossibility proofs” of von Neumann and others, which had suggested the impossibility of any hidden variable theory [12]. Alongside the Schrödinger wavefunction $\psi$, Bohm posits a deterministic, physical trajectory for each particle in the system [10, 11, 13]. Restricting to the non-relativistic, one-particle case for simplicity, the wavefunction remains that of Schrödinger,

and the particle position $q_{p}$ is guided along as

Notably, if Born’s rule (i.e., $P(q_{p}=q)\propto|\psi(q)|^{2}$) holds at one time, the trajectory equation (2) ensures that it holds at arbitrary times—as such, Bohmian predictions align exactly with those of standard quantum theory [10].

Bohm’s work inspired de Broglie to return to pilot-wave dynamics [14], and it also caught the attention of John Bell, who identified the mathematical gaps in von Neumann’s proof [15]. Bell also proved his own no-go theorem [16], demonstrating that any hidden variable approach to quantum mechanics must be inherently non-local; his work was corroborated by a famous experiment of Alain Aspect in 1982 [17], lending credence to Bohm’s non-local theory of particle trajectories.

A key property of the Bohmian approach, however, is that the particle’s Bohmian position has no influence on the wavefunction. The wavefunction evolution is prescribed by the system geometry, and the particle navigates its guiding wave without altering it. While this property has thus far been assumed necessary for agreement with quantum mechanics (with a few exceptions, e.g., [18, 19]), it marks a significant departure both from de Broglie’s earlier pilot-wave approach and from classical field theories, and it raises real consequences for how one can interpret the theory. As an example, one can see the one-way coupling as equivalent to the problem of “empty waves” in Bohmian mechanics; if the particle cannot affect the wave field, it is difficult to distinguish a field that is carrying a particle from one that is not.

In a different direction, there has been a great deal of recent interest in understanding classical pilot-wave systems, largely inspired by the “walking droplet” system of Couder and Fort [1]. In their system, a millimetric droplet moves about the surface of a vibrating fluid by interacting with a wave of its own creation. Walking droplets have been shown to engage in a variety of quantum-like behaviours: among many other examples [20], they have been found to undergo single-particle slit diffraction and interference [1, 21, 22], attain quantised orbits and stable spin states [23, 24, 25, 26], give rise to quantum-like statistics in an analogue of the quantum corral [27], and follow “surreal trajectories” similar to those predicted by Bohmian mechanics [28]. While the majority of these analogues are only qualitative, recent authors have attempted to construct classical field theories to achieve quantitative agreement with certain quantum results [3, 4], furthering the theoretical program initiated by Louis de Broglie [6, 7].

In fact, like de Broglie’s double-solution program before them, these classical systems are defined by their two-way coupling; the highly nonlinear interplay between particle and field gives rise to interesting dynamical behaviours, seemingly intractably at odds with the simple wave-guiding-particle approach of Bohmian mechanics.

In the present work, we work to connect these two classes of pilot-wave theory. We make use of the Lagrangian pilot-wave framework put forward by Darrow and Bush [5], which couples a Klein–Gordon field to a relativistic point particle. With a particular choice of particle-wave coupling, they recover many of the key behaviours hypothesised in de Broglie’s double-solution theory, as we describe in Section 2.

We investigate their framework with a different choice of coupling, recovering a distinct set of quantum-like behaviours. Namely, we show that, in the non-relativistic limit, the system converges exactly to single-particle Bohmian mechanics, where the guiding wave (less a particle-centred wavepacket) plays the role of the Schrödinger wavefunction. This result holds in a highly general setting of the pilot-wave framework, with arbitrary domain, initial conditions, and potential. Moreover, it extends the relation between Klein–Gordon and Schrödinger waves [29] to the case of a particle-wave system, and it provides a means by which Bohmian mechanics can emerge from a pilot-wave theory closer to de Broglie’s original double-solution program.

A related program was considered by Sutherland [18, 19], who showed that a certain particle-wave coupling could be “gauged away” to reveal a match with Bohmian mechanics. However, the current work undertakes a different mechanism to decouple the wave field from particle feedback, and which cannot be added or removed with an appropriate gauge transformation. This difference ensures that, in limits where it does not converge to Bohmian mechanics, our system is able to undergo the same nonlinear evolution characteristic of the classical pilot-wave theories discussed above.

As a demonstration of the latter, we introduce an analogue of wavefunction collapse in our de Broglie-like pilot-wave system. Under an appropriately-defined analogue of a (single-particle) position measurement, we show that the wavefunction collapses exactly to its projection in the measured region containing the particle.

The present work helps to bridge the gap between two different perspectives on pilot-wave dynamics. Bohmian mechanics exactly recovers the predictions of quantum mechanics, but at the cost of a one-way coupling from wave to particle; classical pilot-wave systems maintain a two-way coupling, and they can recover convincing analogues of quantum dynamics in certain cases, but it is far from clear whether they could replicate even single-particle quantum predictions completely. Our work answers this latter question in the affirmative, and it demonstrates how the pilot-wave framework of Darrow and Bush neatly interpolates between Bohmian mechanics and classical—or at least de Broglie-like—dynamics, suggesting a stronger link between these systems than previously thought.

We briefly review the Lagrangian pilot-wave framework put forward by Darrow and Bush [5], before moving on to the specific limit under consideration. Using a joint particle-wave action, they couple a complex Klein–Gordon field $\phi$ to a relativistic particle of mass $m$ at the point $\vec{q}_{p}\in\mathbb{R}^{3}$. Extremising against both objects, they find that the particle trajectory and field satisfy the coupled equations

where $\vec{u}$ is the velocity of the particle, $\gamma=(1-(\frac{\vec{u}}{c})^{2})^{-1/2}$ its Lorentz factor, $k_{c}=mc/\hbar$ its Compton wavenumber, and $\sigma=\sigma(\phi)$ a real-valued coupling function¹¹1In fact, $\sigma$ should be a function of the continuous part ${\overline{\phi}}$, which we define shortly. This distinction does not play into the current discussion. between particle and wave.

Darrow and Bush focus on a particular limit of interest, where

for a real coupling parameter $b$. This limit formalises many of the properties de Broglie hypothesised for his double-solution program [6, 7], including a synchronised Compton-scale oscillation in both particle and field, the de Broglie relation $\vec{p}=\hbar\vec{k}$ in the vicinity of the particle, and a Compton-scale wavepacket that traces out the particle’s trajectory.

In the present work, we consider a new limit of (3), where the particle is coupled to the wave’s phase rather than its amplitude. Recall from Darrow and Bush [5] the definition of the continuous component of $\phi$:

Where $\phi$ is continuous, we find $\phi={\overline{\phi}}$; however, if

for a continuous function $\phi_{1}$ near a point $q_{p}$, we instead find ${\overline{\phi}}(q_{p})=\phi_{1}(q_{p})$. With this in mind, define the continuous wave phase to be

at the point $q_{p}$. This phase is well-defined (up to a period of $2\pi$) and finite.

Now, we define our particle-wave coupling by the following choice of sawtooth wave for $\sigma$:

where $\sigma/b<0$ for all $\theta$; we can ensure this condition by choosing $a$ and $b$ to have opposite signs with $|a|>2\pi|b|$.

We make this choice of $\sigma$ to ensure that dynamics are independent of global phase shifts, but much of our analysis can be extended to a generic function $\sigma=\sigma(\theta)$. We will briefly comment on alternate choices in the following section.

Finally, applying (3) to the coupling (5), we find

The means by which these equations reduce to (1) and (2) will be detailed in Section 3.

There are several immediate differences we can identify from the limit discussed by Darrow and Bush [5]. Two features are familiar from quantum mechanics: $\phi$ is now necessarily complex-valued, and the particle dynamics are now independent of field amplitude. For a more subtle distinction, recall that a Klein–Gordon field decomposes in the non-relativistic limit as $\phi\sim\psi e^{-i\omega_{c}t}$, taking on a global phase oscillation at the Compton frequency. In their limit, this oscillation imparts a particle vibration at the same frequency, as $\nabla\phi$ itself oscillates—this Zitterbewegung plays a critical role in recovering aspects of de Broglie’s double-solution theory. With the current choice of coupling, however, the force $\nabla\sigma\sim\nabla\theta$ no longer changes under a global change of phase, and we expect (and will demonstrate shortly) that the Zitterbewegung should disappear in the non-relativistic limit of the present system.

In parallel with the non-relativistic limit of the classical Klein–Gordon theory [29], we introduce an oscillating ansatz of the form

with $\omega_{c}=mc^{2}/\hbar$ the Compton frequency of our particle, $k_{c}=mc/\hbar$ the corresponding wavenumber, and $b$ as in (5). Notably, the continuous component of our wave is simply ${\overline{\phi}}=\psi e^{-i\omega_{c}t}$. We show here that, in the non-relativistic limit, the evolution of $\psi$ approaches that of a Schrödinger wavefunction, and the evolution of $q_{p}$ approaches that of a Bohmian particle guided by $\psi$.

Specifically, we demonstrate the following result:

To show this result, we instead demonstrate that such a convergence occurs for a short time, with $t\in(0,\tau]$ for some

In particular, if we define

we argue that $\tau$ is a strictly decreasing function of $M$. In Bohmian mechanics, however, all of these terms are locally uniformly bounded, since the particle never intersects the $\psi=0$ locus; that is, for any $T>0$, we can fix

and choose a corresponding $\tau_{T}=\tau_{T}(M_{T})$ that ensures convergence for any subprocess $t\in(t_{0},t_{0}+\tau]\subset(0,T]$. Piecing these processes together, we see that the process itself converges over the full interval—and since $T$ was arbitrary, that it converges for all time.

We have demonstrated how, with an appropriate choice of particle-field coupling, the relativistic model of Darrow and Bush [5] reduces to traditional Bohmian mechanics in the non-relativistic limit. In this limit, the particle continues to carry a Compton-scale wavepacket, as noted by Darrow and Bush, but this wavepacket no longer radiates energy into the rest of the field.

We can force the system to deviate from Bohmian mechanics by introducing a thermodynamic element to the system’s evolution, however. In this section, we show how coupling the wave field to a set of heat sinks gives rise to a compelling analogue of wavefunction collapse. Along with greater insight into the energetics of the present pilot-wave system, this argument suggests how an experiment with, e.g., the walking droplet system of Couder and Fort [1] might give a macroscopic analogue of quantum measurements.

We model a position measurement as follows:

1. 1.

   We fix a partition of the domain $\Omega\subset\mathbb{R}^{3}$ into open cells $U_{\alpha}\subset\Omega$. During the course of measurement, we enforce that the particle cannot cross from one grid cell to the next, and that field dynamics are decoupled between neighbouring cells.

2. 2.

   We let the pilot-wave system undergo an energy-minimising flow in each cell, corresponding to a heat sink attached to each cell.

3. 3.

   Post-measurement, we lift the partition and allow the system to continue evolving according to (6).

The above process is depicted in Fig. 1.

To make the measurement process precise, we first derive an expression for the cell-averaged energy.

There are several strong similarities between Bohmian mechanics and de Broglie’s 1927 proposal [14]; in both, a pilot-wave guides a point-like particle, and Born’s probability rule emerges dynamically. Even still, de Broglie eventually decided that Bohmian mechanics deviated too far from his original picture; he believed that the underlying theory should be relativistic, that the pilot-wave should be a “physical one… which cannot be arbitrarily normed, and which is distinct from the [Schrödinger wavefunction],” and that the particle must “form on [the pilot-wave] a small region of high energy concentration, which may be likened… to a moving singularity” [33].

While de Broglie’s double-solution program was never able to fully explain quantum mechanics, it provided a striking physical picture that presaged some of the classical pilot-wave systems studied today. These systems—such as the “walking droplet” system of Couder and Fort [1] and its various abstractions [2, 3, 4, 5]—give rise to a variety of novel, classical behaviours, and they offer dynamical analogues of several key results from quantum theory [20]. However, it remains unclear whether such systems could replicate even single-particle quantum predictions completely.

Here, we have answered the latter question in the affirmative. Specifically, we have re-examined the de Broglie-like pilot-wave framework of Darrow and Bush [5], in the case in which a point-particle is coupled to the phase of a Klein–Gordon field. We have here demonstrated that its non-relativistic limit reduces exactly to Bohmian mechanics, in a highly general setting. In so doing, we see that de Broglie’s double-solution theory—modernised and formalised in the system studied by Darrow and Bush—can be connected to Bohmian mechanics using a single Lagrangian framework. Moreover, by pairing this system with an abstracted version of a measuring apparatus, we have shown that it can give rise to a compelling analogue of wavefunction collapse.

The present work further demonstrates the richness of the classical field theory proposed by Darrow and Bush [5], its potential for connecting different classes of pilot-wave systems, and its potential for capturing new classical quantum analogues.

Acknowledgments

I would like to express my sincere gratitude to Professor John Bush (MIT) for his invaluable guidance, support, and encouragement. His constructive dissent has consistently played a crucial role in shaping the direction of my research.