SS symmetry
ArticleRevisiting de Broglie’s Double-Solution Pilot-Wave Theory witha Lorentz-Covariant Lagrangian Framework
David Darrow

and John W. M. Bush *Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA;ddarrow@mit.edu* Correspondence: bush@math.mit.edu

Citation: Darrow, D.; Bush, J.W.M.
Revisiting de Broglie’s
Double-Solution Pilot-Wave Theory
with a Lorentz-Covariant Lagrangian

Abstract: The relation between de Broglie’s double-solution approach to quantum dynamics andthe hydrodynamic pilot-wave system has motivated a number of recent revisitations and extensionsof de Broglie’s theory. Building upon these recent developments, we here introduce a rich familyof pilot-wave systems, with a view to reformulating and studying de Broglie’s double-solution
program in the modern language of classical field theory. Notably, the entire family is local andLorentz-invariant, follows from a variational principle, and exhibits time-invariant, two-way coupling
between particle and pilot-wave field. We first introduce a variational framework for generic pilotwave systems, including a derivation of particle-wave exchange of Noether currents. We then focus
on a particular limit of our system, in which the particle is propelled by the local gradient of its
pilot wave. In this case, we see that the Compton-scale oscillations proposed by de Broglie emerge
naturally in the form of particle vibrations, and that the vibration modes dynamically adjust to matchthe Compton frequency in the rest frame of the particle. The underlying field dynamically changes
its radiation patterns in order to satisfy the de Broglie relation p = h̄k at the particle’s position, evenas the particle momentum p changes. The wave form and frequency thus evolve so as to conformto de Broglie’s harmony of phases, even for unsteady particle motion. We show that the particle isalways dressed with a Compton-scale Yukawa wavepacket, independent of its trajectory, and that theassociated energy imparts a constant increase to the particle’s inertial mass. Finally, we see that theparticle’s wave-induced Compton-scale oscillation gives rise to a classical version of the Heisenberg
uncertainty principle.
Keywords: Klein–Gordon equation; hydrodynamic quantum analogues; pilot-wave theory;
Zitterbewegung; Heisenberg uncertainty principle; harmony of phases; Lagrangian mechanics

Framework. Symmetry 2024, 16, 149.
https://doi.org/10.3390/
sym16020149

1. Introduction
Academic Editor: Andrea Lavagno
Received: 19 December 2023
Revised: 18 January 2024
Accepted: 22 January 2024
Published: 26 January 2024

Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms andconditions of the Creative Commons
Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

In 1923, prior to the advent of our modern approach to quantum mechanics, de Broglie
proposed a physical picture of quantum dynamics [1,2]. This picture was based on afundamental symmetry argument. Specifically, at that time, light was understood to exhibit
both particle and wave aspects; he proposed that matter, too, must share this dual nature.
According to de Broglie’s so-called double-solution theory [3–5], microscopic quantum
particles have an internal vibration, which acts as a source of waves that serve to guide or‘pilot’ the particle. The wave-particle coupling was constrained by de Broglie’s harmonyof phases, a principle he referred to as a “grand loi de la nature”: the particle and wave
vibration are always in synchrony, locked in phase. The resulting pilot-wave dynamics wasthen posited, but never proven, to give rise to an emergent statistical behavior consistent
with the standard predictions of quantum mechanics.
De Broglie’s double-solution theory was not without its successes. On the basis of his
physical picture, he predicted electron diffraction, the validation of which by Davisson andGermer [6] led to de Broglie’s Nobel Prize in 1929, awarded him “for his discovery of the
wave nature of electrons”. His theory led to a number of cornerstones of modern quantum

Symmetry 2024, 16, 149. https://doi.org/10.3390/sym16020149

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mechanics. For example, the frequency of particle vibration was deduced by equating theparticle’s rest mass energy E = mc2 with wave energy h̄ω, yielding the so-called Einstein-de
Broglie relation and the Compton frequency ωc = mc2 /h̄. His physical picture also led
naturally to the deduction of the de Broglie relation, p = h̄k, as follows from equating theparticle velocity with the group velocity of its Klein–Gordon pilot-wave field. Despite its
early successes, the theory was never satisfactorily completed; consequently, it never took
center stage in the development of quantum theory.
It is important to clarify the distinction between de Broglie’s double-solution theory
and the relatively well-known Bohmian mechanics, or de Broglie–Bohm pilot-wave theory [7–9].
According to the latter, quantum particles are guided by the standard wave function;
however, the particles are not seen as sources of that wave, whose form is unaltered by theparticle position. Recent advances in Bohmian mechanics include the Lagrangian theories
of Sutherland [10] and Holland [11]. These Lagrangian approaches bear some similarity tothe present work; indeed, both theories fall under the purview of the variational results
to be developed in Section 2. However, our focus is on a classical pilot-wave dynamics ofthe form envisaged in de Broglie’s double-solution theory and engendered in the walkingdroplet system, where the particle responds exclusively to a wave of its own making.
De Broglie’s double-solution theory had two principle shortcomings. First, the physical nature of the wave was never specified. A number of possibilities have since been
proposed and explored. The most highly developed pilot-wave theory of the form proposed by de Broglie may be found in stochastic electrodynamics [12,13], in the work of dela Peña and Cetto [14,15], who seek matter waves in the electromagnetic quantum vacuum.
The possibility of matter waves being of gravitational origin has also been explored, inwhich case they would represent undulations in the fabric of spacetime [16,17]. The second shortcoming of de Broglie’s double-solution theory is that the manner in which theproposed pilot-wave dynamics could give rise to statistics of the form arising in quantum
systems was never made clear. However, evidence of quantum-like statistics emerging
from classical pilot-wave dynamics has come from recent advances in fluid mechanics.
In 2005, Couder and Fort discovered a hydrodynamic pilot-wave system in whicha millimetric droplet self-propels over the surface of a vibrating bath, piloted through aresonant interaction with its own wave field [18]. The resulting ‘walker’ comprises both
the droplet and its quasi-monochromatic wave field, and so represents a macroscopic realization of wave-particle duality [18,19]. Remarkably, this system has captured a number offeatures of quantum systems previously thought to be unique to the microscopic realm [20].
Consequently, it has launched the field of hydrodynamic quantum analogues [21], the goalof which is to revisit and redefine the boundary between classical and quantum behaviours.
The map between de Broglie’s double solution theory and the walking-droplet system isdirect: the bouncing of the droplet at the Faraday frequency plays the role of de Broglie’s
particle vibration at the Compton frequency, and the Faraday wavelength the role of the de
Broglie wavelength [20].
The hydrodynamic pilot-wave system has made clear the richness and complexity ofclassical pilot-wave dynamics, as could not have been anticipated by physicists a century
ago. In particular, it has shown how classical pilot-wave dynamics of the form proposed byde Broglie may give rise to emergent statistics of the form described by quantum mechanics. Salient examples include the hydrodynamic analogues of single-particle diffraction
and interference [22–24], quantised orbits [18,25–29], the quantum corral [30], statisticalprojection effects [31], Friedel oscillations [32], Anderson localisation [33], and surreal trajectories [34]. A remarkable feature of the hydrodynamic system is that the three fundamental
timescales in the problem, that of particle vibration (0.01 s), particle translation (1 s), andstatistical convergence (1 h), may all be readily resolved in the laboratory [20,21].
The successes of pilot-wave hydrodynamics in capturing quantum phenomena havemotivated and informed the exploration of a broader class of classical pilot-wave systems [35–38]. Moreover, they have motivated a revisitation of de Broglie’s double-solution
theory [39–42]. In their recent research program, the so-called hydrodynamic quantum fieldSymmetry 2024, 16, 149

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theory (HQFT), Dagan and Bush [43], Durey and Bush [44], and Dagan [45] consideredparticles with an intrinsic vibration generating, then moving in response to, a Klein–Gordon
field. This pilot-wave field was periodically forced at twice the Compton frequency in a
finite region adjoining the particle. For the sake of simplicity, particle inertia was neglected:
the particle velocity was taken to be proportional to the local wave gradient and the trajectory equation was first-order in time. The coupling constant between particle velocity
and wave gradient was determinant in the resulting dynamics: above a critical value, theparticle self-propelled [43]. The pilot wave form was marked by radial waves with theCompton wavelength radiating energy outwards from the particle path, plus a plane wave
with the de Broglie wavelength moving in synchrony with the particle. The free particle
was found to follow an irregular, quasi-random walk, with a mean momentum ⟨ p2 ⟩ = h̄2 k2
prescribed by the coupling constant. Moreover, the free particle motion was marked by asmaller-scale, erratic component, reminiscent of the Zitterbewegung posited for the Dirac
electron [46].
The question naturally arises: how might Louis de Broglie’s double-solution theory
have evolved if he had the computational facilities available to us today, and the physicalpicture furnished by pilot-wave hydrodynamics? We take another step toward answering
this question by presenting a new class of dynamical systems that represent a modern
version of de Broglie’s double-solution theory, offering a Lorentz-covariant, Lagrangian
formulation of classical pilot-wave dynamics. We emphasize that our work is not an attempt
to refute or modify modern quantum mechanics. Rather, we aim to reformulate and studyde Broglie’s double-solution program in the modern language of classical field theory.
In Section 2, we present the mathematical model under consideration in its mostgeneral form. Specifically, we develop a relativistic, variational theory for a point-particle
and a scalar (Klein–Gordon) field, described in terms of Euler–Lagrange equations. In
Section 3, we introduce a specific, amplitude modulated (AM) limit of our system, wherein
the particle moves in response to the local gradient of the pilot wave. After developing a
version of Noether’s theorem (and expressions for the particle-wave exchange of Noether
currents) in the general case, we study the AM system analytically in Sections 3.1 and 3.2,
characterizing the system’s energetics and showing that a trajectory-invariant, Comptonscale wavepacket is maintained about the particle at all times. In Sections 3.3 and 3.4,
we present our numerical study of the system, which reveals a new, dynamical version ofde Broglie’s harmony of phases. In Section 3.5, we characterize the energy partitioning
between the particle and its immediate surroundings, as well as the manner in which thepilot wave alters the particle’s inertial response. Finally, we demonstrate in Section 3.6 that,in wall-bounded geometries, the particle oscillation and effective mass combine to give rise
to a classical version of Heisenberg’s uncertainty principle for position and momentum
measurements made in any direction.
2. Mathematical ModelIn the following section, we introduce our proposed pilot-wave model in its mostgeneral form, beginning with a Lagrangian framework. Except where stated otherwise,
we apply natural units h̄ = c = 1. For all tensor indices, we use Greek indices (µ, ν,etc.) to run over spacetime indices (0, 1, 2, 3), and Roman indices (j, k, etc.) to run overspace indices (1, 2, 3). We apply the “mostly minuses” convention for the metric tensor:
η µν ∼ (+, −, −, −), so the d’Alembert operator is ∂µ ∂µ = ∂2t − ∇2 . Finally, we use thevector notation ⃗q for three-vectors; if one exists, the corresponding four-vector is simply
denoted without the overline, as q.
Our system is made up of a complex, free Klein–Gordon field ϕ of mass density m̃ anda relativistic particle of mass m at the point q p ∈ R3 , coupled to ϕ through a real-valued
coupling function σ (ϕ(q p ), ϕ∗ (q p )) =: σ (q p ). The total action S may be expressed as thesum of the actions of a free field and of a particle of inertial mass σ:Symmetry 2024, 16, 149

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Sfield =Z

S = Sfield + Spart. ,
d4 q m2 ∂µ ϕ∗ ∂µ ϕ − m̃2 |ϕ|2 ,Spart. = −

Z t′
0

(1)

dt γ−1 mσ(ϕ, ϕ∗ ),where γ = (1 − ( ⃗uc )2 )−1/2 is the Lorentz factor of the particle, and ⃗u its velocity. The
normalization m2 of Sfield is chosen to ensure that ϕ is dimensionless.
Stepping from the action (1) to appropriate dynamical equations requires a formof the Euler–Lagrange equations for pilot-wave systems, which we establish in Lemma 1
below. While similar derivations have likely been performed in the context of classicalelectromagnetism, they are not known to the authors. The proof of the following result isgiven in Appendix A.1.
Lemma 1 (Euler–Lagrange equations for a pilot-wave). Imagine that our action takes thegeneral form (1) with

Sfield =Spart. =ZZ

d4 q Lfield (ϕ, ∂µ ϕ, q),
dt Lpart. (q p , u, τ, ϕ, ∂µ ϕ),

where u = q̇ p and τ is the proper time along the particle’s trajectory. This action is extremized
when ϕ and q p satisfy the following equations of motion:dt δ⃗u Lpart. = δ⃗q p Lpart. ,(∂µ δϕµ − δϕ )Lfield = −δ3 (⃗q − ⃗q p )(∂µ δϕµ − δϕ )Lpart. + ∇δ3 (⃗q − ⃗q p ) · (⃗uδϕ0 − δ∇ϕ )Lpart. ,(2)

where we interpret derivatives of δ3 in a distributional sense (i.e., given a distribution φ : C0∞ → R,
the derivative ∂φ is defined by (∂φ)( f ) = − φ(∂ f )), and where δ∇ϕ represents the three-vectorof variational derivatives δ∂k ϕ . Note that when δ∇ϕ Lpart. = ⃗uδϕ0 Lpart. , the term proportional to∇δ3 vanishes.
Remark 1. At first glance, it appears that the term ∂µ δϕµ Lpart. is not well-defined; namely, as it
is always multiplied by the delta function, we can interpret Lpart. equally well as either a functionof space coordinates ⃗q or of particle position ⃗q p = ⃗q p (t). However, we see that this distinction isinconsequential: by grouping it with the term ∇δ3 (⃗q − ⃗q p ) · (⃗uδϕ0 − δ∇ϕ )Lpart. and integrating
against a test function ψ, we see thatZd3⃗q ψ(⃗q) δ3 (⃗q − ⃗q p )∂µ δϕµ Lpart. + ∇δ3 (⃗q − ⃗q p ) · (δ∇ϕ − ⃗uδϕ0 )Lpart. = − ∂µ ψ(⃗q) δϕµ Lpart. ,which does not depend on spacetime derivatives of Lpart. .
Returning to the action (1) and extremizing against ϕ∗ and ⃗u, as in Lemma 1, gives
rise to the following Lorentz-covariant equations governing the evolution of the wave andparticle, respectively:(∂µ ∂µ + m̃2 )ϕ = m−1 γ−1 δ3 (⃗q − ⃗q p )

∂σ
∂ϕ∗

(3)

dt (mσγ⃗u) = γ−1 m∇σ (q p )There are a few key differences between our formalism and the HQFT program ofDagan and Bush [43]. First, and most notably, HQFT’s guiding equation is only firstorder in the particle position: particle inertia was neglected, precluding its reduction toclassical mechanics in the limit of h̄ → 0. Our Lagrangian formalism forces the guiding
equation to be second order, giving rise to particle inertia and the appropriate classical limit.

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Second, the system (3) is exactly Lorentz-covariant, where HQFT depends on the current
reference frame. As we shall see, this difference has a profound influence on the form of thepilot wave.
Finally, in HQFT, a forcing of the form cos(2ωc t) is imposed over a finite region (of theCompton scale) by the particle on the field. In HQFT, this coupling gives rise to a particle
vibration reminiscent of the Zitterbewegung of the Dirac theory [46]. In Section 3.3, we will
demonstrate that a similar Zitterbewegung emerges naturally (without being imposed)from the Equation (3); moreover, the oscillation frequency adjusts dynamically to alignwith ωc in the rest frame of the particle.
Another nuance of (3) is that the coupling function σ contributes directly to theparticle’s inertial mass. A necessary artifact of Lorentz covariance, this mass-coupling
ensures that the particle remains on-shell (satisfying E2 = m2 + p2 ) at all times; moreover,it enables the particle and field to exchange energy at the point ⃗q p .
2.1. Defining the Field at the Particle Position
We now address the outstanding problem with the system (3); specifically, while thefield ϕ is generally singular at the point ⃗q p , our action depends on the value ϕ(⃗q p ). In theparticular system considered in Section 3, for instance, the field locally takes the form of a
Yukawa potential about the particle: ϕ ∝ e−mr /r. To place our theory on firm mathematicalfooting and sidestep this singularity, we project the field onto its continuous component at the
point ⃗q p :Definition 1. Given a function ϕ : R3 × R → C, define the continuous component ϕ of ϕ at apoint (t, q) asZ1dξ ∂r (rϕ(t, ⃗q + r⃗ξ )),(4)
ϕ(t, q) = lim
r →0 4π S2where ξ parametrizes S2 ⊂ R3 . When ϕ is continuous, we have ϕ = ϕ; however, if
ϕ(t, ⃗q) = ϕ1 (t, ⃗q) + q

a(⃗q − ⃗q p )T A(⃗q − ⃗q p )(5)near a continuous trajectory q p = q p (t), for a positive-definite, symmetric matrix A and a continuous ϕ1 , then we instead find ϕ(q p ) = ϕ1 (q p ). This decomposition demonstrates that theconstruction (4) is Lorentz-invariant when restricted to functions of this form; indeed, any Lorentz
transformation takes the singular term of (5) to another of the same form.
Notably, the Yukawa potential is of the form (5) in all reference frames, so the continuous component is Lorentz-invariant in the setting of Section 3. With this construction in
hand, we clarify that the coupling σ in the action (1) must be of the form∗

σ (ϕ, ϕ∗ ) = σ (ϕ, ϕ ).
Because the projection ϕ 7→ ϕ is linear and preserves continuous perturbations, this redefinition does not affect the variational result (3). It also allows the self-consistent definition∇σ(ϕ, ϕ∗ ) = ∇σ(ϕ1 , ϕ1∗ ),where ϕ1 is defined in a neighborhood of q p by (5). Equivalently, for functions of that form,
this expression is the mean gradient over a ball of radius r → 0.
We note that the construction (4) does not solve the long-standing problem of defining
the self-energy of a classical point-particle, e.g., as discussed by Hammond [47]. In particular, it does not give a consistent, finite integral of energy over the full field. Nevertheless,it does provide a rigorous mathematical footing for the derivation of the Euler–Lagrange
Equation (3).

Symmetry 2024, 16, 149

6 of 312.2. Noether’s Theorem for the Pilot-Wave System
With the application to future pilot-wave theories in mind, we leverage Lemma 1 todeduce a version of Noether’s theorem for a general pilot-wave setting, in Lemma 2 below.
Through Corollary 1, we also quantify the exchange of a general Noether current between
the particle and wave. Note that the following results apply equally well to any Lagrangian
pilot-wave theory, including those proposed by Sutherland [10] and Holland [11].
First, we introduce the following non-conservative version of the Euler–Lagrange
Equation (2):dt δ⃗u Lpart. − δ⃗q Lpart. = ⃗F,(6)where ⃗F = ⃗F (q p , u, t, ϕ, ∂µ ϕ) is a given vector field. As in traditional Lagrangian mechanics,
we can think of ⃗F as a non-conservative force applied to the particle directly.
Lemma 2 (Noether’s theorem for a pilot-wave). Let Lfield and Lpart. be as in Lemma 1. Suppose
we have a transformation(qt )k 7→ (qt )k + εQk ,ϕ 7→ ϕ + εΨ,written to the first order in ε, and suppose that the Lagrangian densities transform as follows:′
Lfield 7→ Lfield:= Lfield + ε∂µ Λfield ,µ′
Lpart. 7→ Lpart.
:= Lpart. + εdt Λpart. ,whereΛfield = Λfield (ψ, ∂µ ψ, q),µµΛpart. = Λpart. (q p , u, t, ϕ, ∂µ ϕ).
In this setting, the quantity
jµ := Λfield − (δϕµ Ltotal )Ψ + uµ δ3 (⃗q − ⃗q p )(Λpart. − (δuk Lpart. ) Qk )µis conserved in the Euler–Lagrange evolution of Lemma 1. That is,∂µ jµ = 0,
with derivatives interpreted in a distributional sense (see the note in Lemma 1). If the particle
evolution is instead replaced with the non-conservative version (6) of Lagrange’s equations, wefind that∂µ jµ = −δ3 (⃗q − ⃗q p ) Fk Qk .
Remark 2. We note that this framework includes passive transformations of the coordinate system, simply by writing them with respect to the original coordinates. In general, a spacetimetransformation qµ 7→ qµ + εQµ (q) is given by( q t ) k 7 → ( q t ) k + ε ( Q k − u k Q0 ),ϕ 7→ ϕ − εQµ ∂µ ϕ.

Following directly from the proof of this lemma (given in Appendix A.1), we canquantify the exchange of any Noether current from particle to field and vice versa:µµCorollary 1. In the setting of Lemma 2, define jsing and jfield asjpart. = uµ δ3 (⃗q − ⃗q p )(Λpart. − (δuk Lpart. ) Qk − (δϕµ Lpart. )Ψ),µjfield = Λfield − (δϕµ Lfield )Ψ.
µµSymmetry 2024, 16, 149

7 of 31µµThen jµ = jpart. + jfield , and we have the following balance laws:∂µ jpart. = δ3 (⃗q − ⃗q p )((δϕ Lpart. − ∂µ δϕµ Lpart. )Ψ − Fk Qk ) + Ψ∇δ3 (⃗q − ⃗q p ) · (⃗uδϕ0 − δ∇ϕ )(Lpart. ),µ∂µ jfield = −δ3 (⃗q − ⃗q p )(δϕ Lpart. − ∂µ δϕµ Lpart. )Ψ − Ψ∇δ3 (⃗q − ⃗q p ) · (⃗uδϕ0 − δ∇ϕ )(Lpart. ).
µWe will return to these results in Section 3.1, where we derive a conservation ofstress-energy and of angular momentum for the system of interest.
3. Amplitude-Modulated Dynamics
With these general results in hand, we restrict attention now to the amplitude-modulated
limit of our system (3), defined by
σ = 1 + b2 /4π + b Re ϕ,m̃ = m,for a chosen coupling constant b. The b2 /4π term serves to offset other, constant terms
from the continuous-component construction (4), as is further discussed in Appendix B.1.
This choice of σ decouples the imaginary component of ϕ from the particle, so we cansimply identify ϕ = Re ϕ. This field thus satisfies the modified equation(∂µ ∂µ + m2 )ϕ = bm−1 γ−1 δ3 (⃗q − ⃗q p )(7)

and we demonstrate in Appendix B.1 that the trajectory equation approximately takes
the form
dt (mγ⃗u) = bmγ−1 ∇ϕ(8)This approximation removes the mass coupling term present in the general Equation (3),obviating the need to compute the continuous component ϕ 7→ ϕ. In Appendix B.2, we
present an alternative derivation, using an exact (albeit non-conservative) version of theEuler–Lagrange Equation (2). Notably, the non-conservative component vanishes in the
non-relativistic limit of the theory, and only affects the total energy budget of the joint
particle-wave system.
The form of the pilot-wave accompanying the free particle is depicted in Figure 1,
albeit in two, rather than three, dimensions. Here, the particle was accelerated from rest toa speed u0 = 0.35c. We highlight four key features of the subsequent dynamics:
1.

2.

3.

4.

The particle carries with it a high-amplitude, constant wavepacket of radius ∼ λc ,corresponding to the bright region adjoining the particle in Figure 1. We derive theform of this wavepacket in 3D in Section 3.2, and explore how it affects the particle’s
inertial mass in Section 3.5;
Beyond the energy imparted to the local wavepacket, energy is radiated outward fromthe point of acceleration in a long wavetrain. In the region around the particle, thiswave always satisfies p = h̄k, where p is the instantaneous particle four-momentum.
We derive this result in Section 3.4;
Surfing over the underlying quasi-monochromatic wave causes the particle to vibrate
in-line at the frequency γ−1 ωc , in a dynamical version of the Zitterbewegung seen in
the Dirac theory of the electron [46]. We examine these oscillations in Section 3.3;
All of these dynamics are governed by universal balance laws for stress-energy andangular momentum, which in turn arise from our Lagrangian framework (1). We
investigate the conservation of these Noether currents in Section 3.1, both in a generalpilot-wave system and in the particular limit of interest.

In the remainder of the paper, we rationalize each of these four features in turn, using
both analytical and numerical tools. Finally, we combine them all in Section 3.6 to derive aclassical Heisenberg uncertainty principle:
σx σp = Kh̄/2,

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where σx and σp are variations in position and momentum in any one direction, and
K = K (b). We demonstrate that, for sufficiently large b, this reduces to the uncertainty
principle of quantum mechanics.

(a)

(b)

(c)

Figure 1. (a) A depiction of the free particle in our system, in two dimensions and with the coupling
constant b = 53.3. In this simulation, we accelerate the particle from rest to a velocity u0 = 0.35c.
Radiation is emitted from the point of acceleration, corresponding to the form predicted in Section 3.4.
Axes are given in units of λc . (b) The wave form adjoining the particle, as predicted in Section 3.2,
is visible as a high-amplitude region around the particle, of characteristic radius corresponding to
the Compton scale λc = 2π/m. As predicted in Section 3.4, the local wave field has a characteristic
wavelength λdB = 2π/(γmv), where v is the instantaneous (rather than initial) speed of the particle.
(c) The same simulation at a later time. Because the wave travels out from the particle’s point oforigin, the local curvature of the wavefront decreases as the particle moves forward. Because thedomain is periodic in both directions, the wave field grows complex, and the particle experiences theradiation from its periodic images.

3.1. Conserved Currents in the AM System
We now consider the non-conservative derivation of the AM system detailed inAppendix B.1. Although it does not fit into our general framework (1), it allows us torecover two key conservation laws. The first and most important of these conservation lawsis that of the stress-energy tensor Tβα , which encodes system energy and linear momentum ina Lorentz-covariant 2-tensor. In a given reference frame, E := T00 can be identified with thesystem energy density, and similarly pk := Tk0 with the momentum density. The remaining
terms represent fluxes of these quantities, such that, in the absence of sources or sinks,
we have∂t E + ∂k T0k = 0,∂t pi + ∂k Tik = 0.
We are able to recover a stress-energy conservation law by applying Lemma 2 to spacetime translations. The full derivation is in Appendix A.2. We define the stress-energy tensor
Tα = − 21 δα (∂ν ϕ∂ν ϕ − m2 ϕ2 ) + ∂µ ϕ∂α ϕ + δ3 (⃗q − ⃗q p )mγuµ uα ,
µ

µ

(9)

which is exactly the sum of a relativistic free particle and a free scalar field. Then we recover
the balance equations∂µ T0 = −δ3 (⃗q − ⃗q p )mbγ−1 uµ ∂µ ϕµµ∂µ Tk = 0.

(10)This demonstrates the key benefit of our non-conservative derivation: the momentum
conservation is exact, and the energy balance is neatly encoded by the material derivativedt ϕ(q p ) := uµ ∂µ ϕ(q p ) of the field along the particle trajectory.

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We can find a similar conservation law for the relativistic angular momentum Mσν byexamining spatial rotations and Lorentz boosts. Here, the spatial components Mkℓ form theclassical angular momentum bivector:
Mkℓ = qk pℓ − qℓ pk = ⃗q ∧ ⃗p,
where pk = T̃ k0 is the linear momentum of (9). The temporal components M0k = − Mk0 are
somewhat less useful; we have
M0k = qk p0 − q0 pk = qk E − tpk ,which represents a scaled centre-of-mass value.
Instead of applying Noether’s theorem, we deduce this conservation law more easily
by leveraging (10). We define the angular momentum currents
Mσνµ = qσ T νµ − qν T σµ ,which gives the angular momentum Mσν = Mσν0 . Plugging in the balance Equation (10)and using the symmetry of T µν , we recover∂µ Mσνµ = δ3 (⃗q − ⃗q p )(qσ ∂ν τ − qν ∂σ τ )mbuµ ∂µ ϕ(q p ),where ∂0 τ = γ−1 and ∂k τ = 0 for k ̸= 0. For the spatial components Mkℓµ , which represent
the classical angular momentum current, we deduce a true conservation law∂µ M kℓµ = 0
We expect angular momentum conservation to play an important role in bound or orbiting
states, which are outside the scope of this work. In the context of the linear acceleration of
the free particle, this conservation law requires that the outgoing radiation have vanishing
total angular momentum.
3.2. Steady States of the Free Particle, and the Local Wavepacket
As a first step towards identifying the local form of the wave field, note that one
steady-state solution of (7) and (8) is⃗u = 0,ϕ=be−m|⃗q| .
4πm|⃗q|(11)This corresponds to a Yukawa potential of range 1/m or, in dimensional form, the Compton
wavelength λc .
Now consider a general trajectory ⃗q p (t), and recall the (position-space) Green’s function for the forced Klein–Gordon equation [48]:p
J1 (m t2 − |⃗q|2 )
θ (t)
m2
p
G (⃗q, t) =δ(t − |⃗q|) −θ (t − |⃗q|),(12)
4π |⃗q|2πm t2 − |⃗q|2
with θ the Heaviside step function and J1 a Bessel function of the first kind. We integrate
this expression over the particle trajectory one term at a time, first definingϕ̃(t0 , ⃗q) :=

Z t0
−∞

dt

bγ−1
δ(t − |⃗q − ⃗q p |).
4πm|⃗q − ⃗q p (t)|This term reduces toϕ̃(t = 0, ⃗q) = ∑tbγ−1 (t),4πm|⃗q − ⃗q p (t)|Symmetry 2024, 16, 149

10 of 31

where the sum is taken over times t < 0 such that |⃗q − ⃗q p (t)| = |t|. Since the particle istraveling strictly slower than c ≡ 1, however, it can only cross this locus once—say, att(⃗q)—and the sum reduces further toϕ̃(t = 0, ⃗q) =bγ−1 (t(⃗q)).
4πm|⃗q − ⃗q p (t(⃗q))|(13)

Suppose we are in the instantaneous rest frame of the particle, so that |⃗q p (t)| = O( at2 )for some acceleration amplitude a > 0. For ⃗q within the ball Bε (0), ε > 0, we find that|⃗q p (t(⃗q))| = O( aε2 ), and thus |t(⃗q)| = |⃗q| + O( aε2 ), yieldingϕ̃(t = 0, ⃗q) =1 + O( aε2 )1=+ O( aε).
4πm|⃗q|4πm|⃗q|With this result in hand, we reduce the total field to the local expression (11); using theexpression (12) and subtracting the components giving (11), we find
q


p
Z 0
J1 (m t2 − |⃗q − ⃗q p |2 )2 − |⃗
2)
J
(
m
t
q
|
b
mb
 + O( aε)qϕ(t = 0, ⃗q) =
dt 
− 1 p
e−m|⃗q| −2 − |⃗
24πm|⃗q|
2π −ε
2
2
m
t
q
|
m t − |⃗q − ⃗q |
p


 q
p
Z 0
m t2 − |⃗q − ⃗q p |2
2 − |⃗
2
m
t
q
|
b
mb
 + O( aε)=− pe−m|⃗q| −dt  q2 − |⃗
24πm|⃗q|
4π −ε
2
2
m
t
q
|
m t − |⃗q − ⃗q p |

=

(14)

b
e−m|⃗q| + O( aε),4πm|⃗q|using the expansion J1 ( x ) = 21 x + O( x2 ). In particular, the expression (11) holds in aneighborhood of the particle in the particle’s instantaneous frame of reference, up to a finite
contribution of amplitude O( aε). By performing a Lorentz boost in the ê1 direction, we
recover the more general formpb exp −m γ2 (q1 − tu1 )2 + (q2 )2 + (q3 )2
p
ϕ=
(15)
4πm γ2 (q1 − tu1 )2 + (q2 )2 + (q3 )2
for the wavepacket adjoining a particle of velocity ⃗u = u1 ê1 . As we discuss further inAppendix B.3, this extension follows exactly from the approximate Lorentz-covariance of
the AM system.
In short, the analysis above demonstrates that the particle is dressed with a trajectoryindependent Yukawa potential, constant up to a length contraction. In Section 3.5, we will
demonstrate that this “wavepacket” modifies the particle’s effective inertial mass and momentum. We see a numerical depiction of this wavepacket in Figure 1, albeit in two rather
than three dimensions. There, the local wavepacket corresponds to the high-amplitude
region of radius ∼ λc centered on the particle.
Another important inference may be made by re-examining (14). Consider again therest frame of the particle at t = 0, and suppose as before that the particle is accelerating as|⃗q p (t)| = O(at2 ). Then the second term in (14) corresponds precisely to the radiation createdby the particle’s motion between time t = −ε and t = 0; that is, the change in the scalar valueof the field ϕ away from its steady state (15). Our calculation shows that this value is at mostO(aε) = O(∆v), where ∆v = O(aε) is the magnitude of the velocity change in this interval. If
the particle is not accelerating at all (and so a = 0), it does not radiate any waves outward:
at a fixed velocity ⃗u, the particle carries only the wavepacket (15). If the particle acceleration
has a magnitude O(a), as in the above analysis, the value of ϕ at a later time is changed atmost at the rate O(a). The resulting radiation becomes significant only if aε = O(1), as willarise if the particle rapidly changes from one velocity state to another. This calculation will be
substantiated in our numerical study of wave radiation in Section 3.4.

Symmetry 2024, 16, 149

11 of 313.3. Zitterbewegung: Particle Oscillation at the Compton Frequency
Spontaneous particle oscillations have been shown to arise in several models ofclassical pilot-wave systems. In-line speed oscillations have been reported to arise in
several settings in the walking droplet systems, including the hydrodynamic analogue ofFriedel oscillations [32]. In-line oscillations with amplitude comparable to the wavelengthof the pilot wave have been shown to be a robust feature of the generalized pilot-wave
framework (GPWF) [37], a parametric generalization of the walking droplet system [20].
Moreover, one-dimensional motion of the free particle in HQFT [43,44] is marked by erratic
in-line oscillations at the Compton frequency. Notably, in all of these examples, particle
oscillations are restricted to the in-line direction, and the coupling strength between particle
and wave is a periodic function of time.
We proceed by demonstrating that both features of de Broglie’s harmony of phases,
an internal particle oscillation at frequency γ−1 ωc and an accompanying wave of wavelength λdB , emerge naturally from the time-invariant dynamics (7) and (8). Moreover, theoscillation frequency and wavelength update dynamically as the particle’s momentum
changes, in order to preserve the de Broglie relation p = h̄k. Finally, the particle vibrates inall directions when interacting with a wall-bounded geometry.
In our first series of tests, we start a particle at rest and accelerate it quickly to a speed
u0 , from which it settles quickly into a steady speed u = u(u0 , b) < u0 . We discuss the formof u(u0 ) further in Section 3.5, and in particular, we derive a nonzero virtual mass imparted
to the particle’s rest mass by the surrounding wavefield.
For u0 = 0.5c and b = 53.3, a spectrogram of the resulting in-line position oscillations
is shown in Figure 2. We highlight two noteworthy effects:
Initially, the particle undergoes in-line oscillations with frequency γ−1 ωc in response
to outgoing radiation from the point of acceleration, effectively surfing over its own
radiative wave field. The form of this radiation is discussed in Section 3.4;
1 − ebAfter ∼ 50 Compton periods, the particle oscillates with amplitude n−
λc ∼ λc
2.
b γ
−
1
2
at frequencies between γ ωc and γ(1 + u )ωc . This is an artifact of our periodic
domain, and specifically the particle interacting with the wave form generated by its
periodic images. However, we expect the same effect to occur any time the particle
interacts with a wall-bounded geometry, and its wave form reflects off the boundaries.
We quantify the first effect in Figure 3a, where we repeat this experiment across a wide
range of velocities and b values. We see that the oscillation frequency conforms closely to
γ−1 ωc in all cases. In all simulations, the dominant frequency is constant until the particle
encounters radiation from the opposite side of the domain.

1.

Figure 2. A spectrogram of in-line oscillations for our two-dimensional system, with coupling
constant b = 53.3. The shown color values are normalized by ∆x 7→ arctan(50 · ∆x/λc ), where ∆x is
the oscillation magnitude. Here, we give the particle an initial velocity u0 /c = 0.5c, which quickly
relaxes to a mean velocity u/c = 0.455c. For t < 50 Compton periods, the particle undergoes anoscillation at the frequency γ−1 ωc . Thereafter, waves cover the entire periodic domain, and theparticle vibrates at frequencies between γ−1 ωc and γ(1 + v2 )ωc . Note, the diminishing intensity of
the yellow line at γ−1 ωc reflects the temporal decay of the in-line Zitter.

Symmetry 2024, 16, 149

12 of 31

(a)

b

eb

nb

13.3
26.7
40.0
53.3
66.7
80.0

3.88
3.52
2.87
2.38
2.10
1.97

238.
69.
39.
28.
23.
21.

(b)

(c)

Figure 3. (a) Dominant oscillation frequencies at the beginning of each trajectory, for particles acrossthe range of initial velocities u0 = 0.2 to u0 = 0.65. We observe that the particle oscillates at thefrequency γ−1 ωc independent of the coupling constant and velocity. (b) Amplitudes of in-line
oscillations in the long-time limit of Figure 2, i.e., after waves have covered the entire periodic
1 − ebdomain. Curves of the form n−
λc are shown for reference, where nb and eb are least-squares fitsb γ
(reported in the Table (c)).

Numerical Simulation of the AM System.
Our numerical code is based on that developed by Faria [49] for walking droplets. We
model the space as a two-dimensional periodic domain, allowing us to use high-accuracy
pseudo-spectral methods to resolve the wave field. In turn, we evolve the wave fieldusing a fourth-order Runge–Kutta method, separately tracking ϕ and η := ∂t ϕ in order tobreak (3) into two first-order equations.
Our algorithm takes the following form. At each timestep, we add an approximatedelta function—taken to be a Gaussian of variance 2m−2 = 2λ2c —to the field η at the current
particle position ⃗q p , scaled by bγ−1 . Suppose ϕ⃗ℓ and η⃗ℓ are the discrete Fourier transforms⃗of ϕ and η, respectively, so that, for instance, ϕ = ∑ ϕ⃗ℓ eiℓ·⃗q . We evolve the fields ϕ and η
for one time-step according to the equations∂t ϕ⃗ℓ = η⃗ℓ ,∂t η⃗ℓ = (−|ℓ|2 + m2 )ϕ⃗ℓ ,and calculate the gradient ∇ϕ(⃗q p ) using the field’s Fourier expansion:⃗∇ϕ(⃗q p ) = ∑ i⃗ℓϕ⃗ℓ eiℓ·⃗q p .

We use this to evolve γ⃗u for one time-step, and we move the particle accordingly.
As we argue in Appendix B.3, we can deduce the same behavior after a Lorentz
transformation. Namely, oscillations at γ−1 ωc arise regardless of initial velocity. If theparticle is moving at a velocity ⃗u′ and accelerates to a steady velocity ⃗u, it begins to vibrateat the frequency γu−1 ωc . This demonstrates a relativistically-correct internal clock at theCompton frequency, as in de Broglie’s original model, but with two distinctions. First, itemerges naturally from a time-invariant dynamics, without reference to intrinsic particle
oscillation. Second, the emergent particle clock updates dynamically to adjust to theparticle’s current velocity, preserving the synchrony of de Broglie’s clock even under the
influence of applied forces. We detail the wave form complementing this clock in thesubsequent section.
3.4. A Dynamical Harmony of Phases
We proceed to examine the wave form generated following a particle acceleration, firstfor the case of acceleration from rest (see Figure 1), then for a more general acceleration(see Figure 4). We shall demonstrate that the resulting wave form naturally gives riseSymmetry 2024, 16, 149

13 of 31

to the periodic forcing needed for the Zitterbewegung reported in the previous section.
Specifically, we will show that the oscillation frequencies reported in Section 3.3 arise fromthe particle being repeatedly washed over by quasi-monochromatic waves of wavelength
λdB and phase velocity c2 /u. This analysis allows for a detailed comparison of the wave
forms arising in pilot-wave hydrodynamics, in HQFT, and in our new model.
u0

u1

F
usource ∼ u0

Figure 4. A diagrammatic sketch of the radiation pattern following a general acceleration of theparticle in our system. Suppose the particle (shown in black) initially moves at a steady velocity
u0 , carrying with it the Yukawa wavepacket (15), and is then accelerated by an external force F toa new velocity u1 . This acceleration spawns a new, continuous source of waves that drifts along
the extrapolated original trajectory of the particle (shown in gray). The wave source spreads out inthe ellipsoid (18) at a speed uexpansion ∼ |⃗u1 − ⃗u0 | given by (20), all the while drifting at the velocity⃗usource ∼ ⃗u0 defined by (19). We stress that the gray curves are approximate level sets of the wave
amplitude, but not generally of the wave phase or wavenumber.

We first review some elementary properties of Klein–Gordon waves before returning
to our particular system. Consider the dispersion relation of Klein–Gordon waves:
ω 2 = m2 + |⃗k|2 ,or in dimensional form, ω 2 = ωc2 + c2 |⃗k|2 . Here, ω is the local oscillation frequency and ⃗k
the local wavevector. The group velocity of the wave is⃗v g = ∇⃗k ω = q⃗km2 + |⃗k |2

,

(16)

which corresponds precisely to the velocity of a point-particle of mass m and momentum ⃗k,
as can be seen by inverting the above relationship: ⃗k = mγv g ⃗v g . The system is dispersive:the group velocity depends on wavelength. Thus, following a wave disturbance from apoint source, different
component wavelengths λ := 2π/k travel outwards at different√
speeds v g = k/ m2 + k2 . The result is a wave train centered on the original source, witheach excited wavenumber k spreading outward at its corresponding group speed v g . If
a particle of mass m is traveling away from this point-source at a velocity ⃗u, the region
immediately adjoining the particle must then carry a wavelength λdB = 2πh̄/|⃗p|, where⃗p = γm⃗u is the particle’s momentum. This region grows in size over time—as wave
crests of wavenumber ⃗k and (⃗k + ε) drift apart with velocity O(ε/k)—giving rise to aquasi-monochromatic wave form in the vicinity of the particle.
The above behavior occurs in our system when a particle is accelerated from rest (seeFigure 1). At the particle’s point of acceleration—the origin, in the case of Figure 1—theparticle excites a range of wavevectors ⃗k defined byh̄|⃗k | ≲ mγu,(17)

Symmetry 2024, 16, 149

14 of 31

where u is the new particle speed. The different components spread out from the origin in
a spherical wave train according to (16), and the particle surfs along the outgoing, origincentered, spherical region of local wavelength λdB . The particle is repeatedly washed overby these waves, which possess a phase speed
ω/k =

q

k2dB + m2 /k dB = 1/u = c2 /u.

The relative speed between the wave phase and the particle is then c2 u−1 − u. We may
thus deduce the forcing frequency of the wave on the particle by multiplying the relative
velocity by the local wavenumber:ωparticle = u−1 − u k dB = u−1 − u γmu = γ−1 ωc .
This result is in accord with the estimate ωparticle = ωc γ−1 evident in Figure 3 for t < 50Compton periods, that is, before the particle encounters radiation from its periodic image
sources. We note that the response to the quasi-monochromatic waves of wavelength λdB in
the particle’s immediate vicinity is the dominant effect on the subsequent particle motion.
In Section 3.2, we have shown that the particle emits radiation at a rate proportional to its
instantaneous acceleration. As such, waves generated by the particle’s resulting in-line
Zitter over a single oscillation are at most of amplitude O(ε), where ελc is the amplitude
1 − ebof particle vibration. We plot values of ε = n−
in Figure 3b, but note that ε ≪ 1 inb γ
all cases. Finally, we note that in an unbounded domain, the resulting in-line Zitter dies
down over time, since the wave amplitude necessarily decays as the wave disperses. Thisbehavior may be seen by comparing Figure 1a–c.
Upon accelerating, the particle excites a range of wavenumbers including the set (17).
We proceed to demonstrate that the dominant wavenumbers of the outgoing wavefront are,in fact, bounded above in norm by the value ∼ k dB . Consider the behavior of the particle
for t > 50 Compton periods in Figure 2, when the particle has circled the far side of its
periodic domain and encounters its own waves head-on. The relative speed between theparticle and oncoming de Broglie waves is now c2 u−1 + u, so, following a similar derivation
as above, the excited vibration frequency in the particle isωmax = γ(1 + u2 )ωc .
In Figure 2, this value appears as an approximate upper-bound for the particle oscillations;higher oscillation frequencies, which would correspond to waves of smaller wavelength,
are not evident. Although we see in Figure 1a that some waves of momentum k > mγu
are excited—i.e., those moving faster than the particle itself—this spectrogram demonstrates that they are negligible compared to their lower-momentum counterparts. Thus, inaccelerating from rest, the particle effectively excites only the wavevectors (17).
We emphasize that when the particle starts from rest, waves are radiated outward
from the point of acceleration. This marks a point of contrast with the behavior in the walking
droplet system [18] and HQFT [43,44], where waves are radiated continuously along theparticle’s trajectory. In our system, the free particle travels alongside a nearly-planar
wavefront of the de Broglie wavelength, as shown in Figure 1. Finally, our results support
the prediction of Section 3.2, that a nearly constant-velocity state of the free particle is alsonearly non-radiating; specifically, an O(ελc )-amplitude vibration about a constant velocity
induces only an O(εωc ) rate of radiation.
We now consider a more general particle acceleration: a particle moving steadily withvelocity ⃗u0 is accelerated quickly to a new mean velocity ⃗u1 . As we show rigorously inAppendix B.2, we can apply a Lorentz transformation to reduce this case to that treatedpreviously, and thus deduce the form of the resulting pilot wave. Figure 4 illustrates theform of radiation arising in a Lorentz-boosted coordinate system. The wave can no longer
simply radiate from the point of acceleration, which is not a well-defined notion underSymmetry 2024, 16, 149

15 of 31

Lorentz symmetry. Instead, a new, continuous wave source is spawned at the point ofacceleration, and travels along the extrapolated, original trajectory of the particle at a velocity⃗usource comparable to ⃗u0 , to be derived shortly. This “virtual” source continues to radiatewaves in all directions, their form depending on the velocity difference between the particle
and wave source.
To quantify this effect, suppose that ⃗v ∼ ⃗u1 − ⃗u0 is the velocity of the outgoing particle
in the rest frame of the incoming particle. Without loss of generality, we suppose that⃗u0 = u0 ê1 lies in the x direction. In the rest frame of the incoming particle, the outgoing
wavefront takes the form described above: a spherically symmetric wavefront expanding(approximately) at the speed v = |⃗v|. In particular, the wavefront satisfies the equations( x ′ )2 + ( y ′ )2 + ( z ′ )2 = v2 ( t ′ )2 .
Transforming back to the laboratory frame, the wavefront must satisfy the transformed equations
1 − u20 v2
1 − u20

1 − v2x−u0 t
1 − u20 v2!2+ y2 + z2 =1 − u20 2 2v t .
1 − u20 v2(18)Specifically, this is the Lorentz transformation of the fastest-moving wavefront in theparticle’s rest frame (wherein λ = λdB ), which approximately defines the outer edge of thefull wave form.
One must be careful in interpreting the wave geometry in our new frame of reference.
In a frame where the particle accelerates from rest, the outer edge of the wave form isa level set simultaneously of wavenumber, phase, and amplitude, but such is not thecase in general. First, note that in transforming from the particle’s original rest frame
to our new, laboratory frame, the wavenumber becomes higher in front of the particle
and lower behind it; thus, the locus (18) is no longer a level set of the wavenumber. It
is also no longer a level set of the wave phase; indeed, the front (18) corresponds tovalues of ϕ from a finite time-interval in the particle’s original rest frame (specifically, aninterval ∼ (vu0 /c2 )t′ , following the equations of a Lorentz transform), over which the highfrequency ∼ ωc = mc2 /h̄ of the wave would spread the wave front over a range of phases∆θ ∼ ωc (vu0 /c2 )t′ = (mu0 v/h̄)t′ . However, it is approximately a level set of the wave
amplitude, which is a Lorentz scalar (avoiding the first problem) and modulates far slower
than the phase (avoiding the second). We proceed by thinking of it in these terms. Other
amplitude level sets, corresponding to longer wavelengths in the particle’s rest frame, canbe found by replacing v with the corresponding group speed (16).
Theqrelation (18) reveals the wave form to be ellipsoidal, dilated in the direction ⃗u0 by
a factor(1 − u20 v2 )/(1 − u20 ). The center of the ellipsoid is traveling with the velocity⃗usource =1 − v2⃗u0 ,1 − u20 v2(19)which reduces to ⃗u0 if either v → 0 or u0 → 0. The ellipsoid is expanding at the speedsuexpansion = v

1 − u20,
1 − u20 v2(20)which reduces to v in the same limits. The excited wavevectors in this wave form arenecessarily given by the Lorentz-transformed version of the ball (17) in reciprocal space.
Figure 4 sketches level sets of the wave amplitude following a general acceleration,
as discussed above, but we stress again that these are not level sets of the phase or of thewavenumber; in fact, the phase generally oscillates rapidly around the outer edge of the
wavefront, and up to a rescaling, the wavevector aligns exactly with the group velocity of
the immediate wave field, including drift. A particular outcome of this is that even though
the particle appears to be traveling askew (not perpendicular) to the plane of the wavefront

Symmetry 2024, 16, 149

16 of 31

in Figure 4, the wavevector at the particle’s position must (by Lorentz symmetry) match the
new momentum of the particle, and thus the phase increases most strongly in the particle’s
direction of motion. By Lorentz symmetry, we also see that this wave form continues to drive
particle oscillations at the Compton frequency, although now, these generally have bothin-line and lateral components. The wave form thus continues to surround the particle
with a quasi-monochromatic region of wavelength λdB , or wavenumber k dB , even though
the wavevector is not perpendicular to the amplitude level sets depicted in Figure 4.
Taken together with the particle oscillation of the previous section, we can understandthe radiative behavior in our system as a dynamical version of de Broglie’s harmony of phases,
wherein the particle’s internal clock and the underlying guiding wave remain locked inphase throughout its motion. Just as the particle’s vibration updates dynamically to remain
at the characteristic frequency γ−1 ωc , the wavelength of the field (at the particle position)
updates to preserve the relation p = h̄k. Together, these effects lock the particle andwave in phase—necessarily, as the wavelength update creates the corresponding frequency
update—and preserves the harmony of phases throughout the particle motion.
3.5. Virtual Mass in the Local Wavepacket
We proceed by highlighting a critical feature of the free pilot-wave system, madeevident by the local wavepacket (15): in steady state, the particle shares a certain amount ofenergy with the field around it, spread over a radius ∼ λc . We refer to this as a virtual mass.
While it is not readily apparent in the Equation (3), it impacts the particle’s inertial response
through a constant augmentation m 7→ m + δm of the particle mass. Quantitatively, thisvirtual mass is exactly the energy carried by the wavepacket (15); because the wavepacket
is independent of particle dynamics, so too is the virtual mass δm.
Now, we note that the virtual mass fraction δm/m likely differs between our twodimensional simulations and the three-dimensional model developed previously. With thatin mind, we derive this effect in a three-dimensional system, but provide numerical results
only for the two-dimensional AM system.
Recall the momentum continuity Equation (10):µµµ∂µ Tk =: ∂µ (( Tfield )k + ( Tpart. )k ) = 0,with

( Tpart. )k = δ3 (⃗q − ⃗q p )γmuµ uk ,µ

( Tfield )k = − 21 m2 δk (∂ν ϕ∂ν ϕ − m2 ϕ2 ) + m2 ∂µ ϕ∂k ϕ.
µµNote that these are simply the momentum and momentum flux densities of a free particle
and field, respectively. In a fixed reference frame, we define the corresponding threemomenta as the space integrals( Ppart. )k =( Pfield )k =ZZd3⃗q ( Tpart. )0k = γmuk ,d3⃗q ( Tfield )0k = m2Z

d3⃗q ∂t ϕ∂k ϕ,which satisfy the conservation law
dt ( Ppart. + Pfield ) = 0,
obtained by integrating (10) over space. Now, decompose the wave field as ϕ = ϕwav + ϕrad ,where ϕwav is simply the wavepacket (15). This gives us another natural field momentum,( Pwav )k =Z

d3⃗q ( Twav )0k := m2Zd3⃗q ∂t (ϕwav )∂k (ϕwav ),which dominates the total field momentum in the case that radiation rates are small. Suchis true in all of our numerical experiments, as we quantify in Appendix B.1.

Symmetry 2024, 16, 149

17 of 31Since the particle must bring along a wavepacket of the form ϕwav , it is the combined
momentumPeff. = Ppart. + Pwav
that determines the particle’s inertial response. The remainder Pfield − Pwav determines
only higher-order couplings to the underlying wave field.
Since ϕwav = ϕwav (⃗q − ⃗ut), we find that ∂t ϕwav = uk ∂k ϕwav = ⃗u · ∇ϕwav . Critically,
we note that the integral for Pwav only involves derivatives of ϕwav in the direction of ⃗u.
Indeed, the coefficient ⃗u · ∇ϕwav only depends on this directional derivative by construction,and likewise for the vector ∇ϕwav by symmetry. We can then write the integrand as
Pwav = b2 γ2⃗u ·Z⃗ ⊗ U,⃗
d3⃗q U⃗ := mb−1 (∇ϕ)(⃗q) to emphasize that the latter is now a ⃗u-independent
using the notation U
function of space. Finally, we switch coordinates of integration to those of the rest frame ofthe particle, ⃗q′ , and thus pick up a factor of γ−1 :Pwav = b2 γ⃗u ·Z⃗ ⊗U⃗ =: b2 γ⃗u · A.
d3⃗q′ U

Here, A ∈ R3 ⊗ R3 depends only on the mass density m of the field. Recall from symmetrythat Pwav must point parallel to ⃗u, restricting A = (δm)1 for some δm ∈ R3 independent ofvelocity. In fact, we know that δm must be proportional to m, as the only remaining massscale in the system. In total, we find thatPwav = b2 (δm)γ⃗u,

Peff. = (m + b2 δm)γ⃗u = γmeff⃗u(21)This effective mass, rather than the “decoupled mass” m, thus determines the inertial
response of the particle.
We can confirm the influence of the wave-induced virtual mass by looking again at thenumerical experiments of Section 3.3. Recall that, in those experiments, we start a particle
at rest, impart a momentum ⃗ptot = γm⃗u0 to the particle, and observe its evolution. Though
we focused on velocity oscillations before, we now look at how the particle approaches asteady-state speed us < u0 .
Figure 5a shows horizontal particle position after imparting a velocity u0 = 0.35c. First,
note that the relaxation to the steady-state speed us < u0 occurs over the Compton timescale(roughly the period of several oscillations, as seen in the cutout). This short-time dynamics
is characterized by a transfer of momentum from the particle to its adjoining wavepacket,
as predicted by Corollary 1. We can think of this exchange as a reflection of the particle’s
delocalised nature: since the local wavepacket requires momentum everywhere over aradius ∼ λc , it requires a time ∼ 1/ωc to distribute momentum appropriately. This also
explains why radiation closely follows a single-source approximation, as we encounteredin the preceding section; radiation occurs upon particle-to-wave energy transfer, which
occurs over a Compton timescale.

Symmetry 2024, 16, 149

18 of 31(a)(b)

Figure 5. (a) Six particle trajectories in our two-dimensional system, all given an initial velocity
u0 = 0.35c (or p0 ∼ 0.37mc). Within a Compton timescale, the particle exchanges a certain amount of
momentum with the underlying wave field and settles into a lower, steady-state velocity. (b) Fractionof total momentum retained by the particle after an initial acceleration from rest, with zero initialwave field. The resulting momentum transfer depends on both p0 and b, and can be broken into twostates. The first is the “high-radiation regime” below the black curve, where the particle first loses
much of its momentum to the outgoing wavefront (18). The second is the “low-radiation regime”
above the black curve, where momentum is transferred almost exclusively to the local wavepacket(15)—i.e., only the momentum b2 (δm)γ⃗u is lost by the point particle.

Figure 6 shows the exchange of (horizontal) momentum between particle and wave
for the initial time period in Figure 5a (for b = 80.0). Here, we see that the magnitude ofoscillations dies down significantly after t ∼ 8 Compton periods—i.e., as the particle approaches its steady-state velocity. After this point, the particle velocity continues oscillating
at a lower amplitude, corresponding to the in-line Zitterbewegung of Section 3.3.

Figure 6. Exchange of horizontal (i.e., in-line) momentum between the particle and field after theparticle is accelerated to an initial velocity u0 = 0.35. The coupling constant is b = 80.0, corresponding
to the lowermost curve in Figure 5a. Note that the “wave” component of momentum incorporates
both the local wavepacket of Section 3.2, which gives rise to the virtual mass δm, and the radiating
component detailed in Section 3.4. The momentum exchange is most pronounced for t ≲ 8 Compton
oscillations, after which the system approaches a quasi-steady, periodic state. The former regime
reflects the establishment of the robust wavepacket (15), and the latter the Zitterbewegung detailed in
Section 3.3.

Figure 5b shows the quantity | Ppart. |/| P(t = 0)| over a range of initial velocities u0 .
This measures the fraction of momentum retained by the particle, which we use as a roughSymmetry 2024, 16, 149

19 of 31

measure of the fraction of momentum transferred to the wavepacket. The limitation of thismetric is that momentum can either be transferred to the wavepacket or radiated away.
Here, we see both effects. First, since the momentum fraction carried by the steady-state
wavepacket is velocity-independent—as quantified in (21)—each curve in this figure isbounded above by 1 − | Pwav |/| P(t = 0)|. To continue, we assume that at the maximum
point in each curve in Figure 5b, the particle does not radiate any momentum away. That is,for each b,
Pwav = Pfieldat the maximum point on each curve. With this approximation, the best fit to | Pwav |/| Ptotal |is given by2bδm/m =,(22)163.8
with a maximum relative error of 0.5%. This represents a very close fit to our predictionδm ∝ b2 .
Finally, looking beyond the maximum point of each curve, note that a significant
amount of momentum is lost beyond the (δm)γus transferred to the wavepacket. As b
increases, the wavepacket itself grows, and more momentum is taken up by the virtual
mass. Conversely, as p0 decreases, more is radiated away on top of the wavepacket. We canunderstand this through Corollary 1; the momentum transfer between particle and field is∂µ [ Tfield ]k = −δ3 (⃗q − ⃗q p )γ−1 mb∂k ϕ ∝ γ−1 b2 ,µso as γ−1 b2 → 0 with increasing particle velocity or decreasing coupling constant, lessmomentum is available to radiate.
As a point of note, we see that the curve γ−1 roughly cuts the system into two states:when accelerated (from rest) above a critical momentum p∗ = p∗ (b), the particle settles
quickly into a steady state with virtual momentum fraction
1∗ 2 −1/2
pvirtual /ptot ∼ γ−.
p ∗ = (1 + ( p ) )This “low-radiation regime” characterizes the experiments above the black curve in Figure 5b.
When accelerated below this critical momentum, however, the particle initially loses a
substantial fraction of its momentum to radiation.
3.6. Heisenberg Uncertainty, and the λc Particle Cloud
Recall from Section 3.4 that, in the case of a free particle, an in-line Zitter is excited by thephase waves washing over the particle. Now suppose the particle is not free, but confined
to a finite geometry—we assume only that its waves wash over it from all directions, either
reflected off of walls or generated by its periodic images. The resulting wave interference
characterizes our periodic system in the long-time limit, as in Figure 1c, but it is also expected
to arise when the particle interacts with a variety of common quantum apparatuses (e.g.,slits, corrals, or interferometers) or indeed during any position measurement. In suchconfined geometries, Zitter occurs in all directions, owing to the complex geometry of theincoming waves. The resulting motion is characterized by a region of scale λc around themean trajectory, about which the particle vibrates at a characteristic frequency γωc . We callthis region the particle cloud, whose form is shown in Figure 7b.
We investigate this vibration in 1D by returning to the numerical experiments of
Figure 2, and focusing on their long-time limit, wherein previously-radiated waves areincoming from all directions. Amplitudes of these oscillations are given in Figure 3 over arange of velocities and coupling constants. Namely, we calculate each amplitude as theL1 norm (over frequency space) of the short-time Fourier transform discussed in Figure 2,
averaged over the last 1000 time-steps. Note that higher L p norms are bounded by thisvalue up to a constant multiple, as the frequency spectra have (approximately) uniformlycompact support.

Symmetry 2024, 16, 149

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Figure 7. (a) A trajectory of the same form as Figure 1, but with b = 66.7 and initial velocity⃗u = (0.15c, 0.25c). By breaking the up-down symmetry of previous simulations, we can see particle
vibrations in both in-line and transverse directions. Both axes are given in units of λc . (b) Particle
vibrations about the mean trajectory in the x and y directions, scaled by λc . To resolve this motion,
we performed a high-pass filter on the full trajectory. These vibrations form a moving particle cloud,which satisfies the Heisenberg uncertainty relation (24). Created with the MATLAB package [50].
1 − ebIn Figure 3b, we show best-fit curves of the form n−
λc for each value of b. Here,b γ
nb and eb both decrease with b for all values within the tested range. Critically, note thateb < 2 for sufficiently large b. This gives an oscillation amplitude
1 − ebδx ∼ m−1 n−b γ

in any direction, where eb and nb are as in Figure 3. Using the characteristic oscillation
frequency γωc , based on our discussion in Section 3.4, we find1
δv ∼ γωc δx ∼ γ1−eb n−b(23)

To estimate the corresponding momentum amplitude δp, suppose that the particle oscillates
from a minimum momentum p0 = meff γ0 v0 to a maximum p1 = meff γ1 v1 in the directionof δx. Here, meff is the effective mass of both the particle and its steady-state wavepacket,
as discussed in Section 3.5. Furthermore, note that γ0 ≥ γv0 := (1 − v20 )−1/2 and similarly
γ1 ≥ γv1 , with equality only if the particle is confined to move in one direction. In either
case, the equality dp = γ3 dv holds, where γ is the total Lorentz factor of the particle. Then,supposing transverse oscillations are small, we haveδp = p1 − p0 = m′

Z v
v0

1

dv γ3 ≥ meff (v1 − v0 )γv3 = meff γv3 δvfrom Jensen’s inequality, using v = (v0 + v1 )/2 + (orthogonal) to denote the particle’s
mean velocity. In turn, our estimate (23) gives1
δp ∼ meff γ4−eb n−b .

√√Modeling x and p as monochromatic oscillations, we have σx ∼ δx/ 2 and σp ∼ δp/ 2, or
2
σx σp ≥ 12 (meff /m)γ4−2eb n−b .

Finally, applying eb < 2 gives
σx σp ≥

1
2
(m /m)n−
b ,
2 eff

(24)

Symmetry 2024, 16, 149

21 of 31or in dimensional form,12h̄(meff /m)n−b .
2
Given (22), this inequality reduces to the traditional uncertainty principle for sufficiently
large b. Note that a linear regression suggests that we should find eb < 1 at b ∼ 105; in thisstronger coupling regime, these oscillations would obey the stronger relativistic uncertainty
principle of Putra and Alrizal [51]: σx σp ≥ 21 h̄γ2 .
We emphasize that the uncertainty relation (24) is somewhat different in nature toits counterpart in quantum mechanics. Instead of an underlying property of a wave-likesystem, as in quantum theory, our uncertainty relation characterizes a classical uncertaintyof the Compton-scale particle dynamics, brought about by its waves interacting with awall-bounded geometry. Averaging over the Compton timescale of the particle, the particle
appears to take up a Compton-scale volume in phase space, reminiscent of quantum
mechanics. However, in quantum mechanics, these scales can be squeezed in either
position or momentum coordinates, giving rise to (a) highly localized states of indefinite
momentum and (b) delocalised states of definite momentum. This type of squeezing does
not appear to have an analogue in our uncertainty relation.
σx σp ≥

4. Discussion
The physical picture furnished by pilot-wave hydrodynamics has motivated a revisitation of de Broglie’s mechanics. With a view to extending his double-solution theory [2,5],
we have presented a general Lagrangian framework for relativistic pilot-wave dynamics,
and studied a particular limit of this framework in depth. Our framework has several
structural advantages over the HQFT program of Dagan, Durey, and Bush [43–45]. First,
our Lagrangian framework is Lorentz-covariant, and so in accord with the principles of
special relativity. Second, it gives rise to a second-order equation for the particle trajectory,
allowing classical dynamics to naturally emerge from our model in the h̄ → 0 limit.
The dynamics of our system recovers a variety of behaviors familiar from de Broglie’s
theory, but with several key differences and advantages. In both de Broglie’s doublesolution program and in HQFT [43–45], the internal particle vibration takes the form of an
enforced particle clock at the Compton frequency. In our model, the particle clock emerges
naturally from a time-invariant dynamical system, in the form of regular particle vibrations,or Zitterbewegung. These vibrations take the form of in-line oscillations for a free particle,
but may occur in all directions when the guiding wave is altered by bounding walls.
The Zitter of de Broglie’s theory is subject to his harmony of phases, according to which
the internal particle oscillation is locked in phase with the oscillation of the pilot wave. In
de Broglie’s work, this phase-locking gave rise to the key relation p = h̄k, the identificationof the de Broglie wavelength λdB = h̄/(mγu), and the adjustment of the particle oscillation
to the Doppler-shifted frequency γ−1 ωc . The de Broglie relation also emerges from HQFT;
however, there the particle momentum ⟨ p2 ⟩ = h̄2 k2 is fixed by the wave-particle coupling
constant. Our system recovers a dynamical version of de Broglie’s harmony of phases: notonly does the pilot wave radiate at the de Broglie wavelength and the particle oscillate at
the Doppler-shifted Compton frequency, but both effects update dynamically as the particle
accelerates, preserving the de Broglie relation p = h̄k even when the momentum p changes.
The distinction between the wave form of the free particle deduced here and that of
HQFT is noteworthy. While both are characterized by the particle moving in conjunction
with a pilot wave with wavelength λdB , the manner in which the Compton wavelength appears is markedly different. In HQFT, waves of the Compton wavelength are continuously
generated along the particle path. In our system, the Compton wavelength appears only
in setting the scale of the Yukawa potential adjoining the particle. Thus, the pilot-wave
form of the free particle is effectively monochromatic in the vicinity of the particle, as is
the case in pilot-wave hydrodynamics. Nevertheless, wave interference can give rise toparticle oscillations with a characteristic scale prescribed by the Compton wavelength.

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Furthermore, the de Broglie waves in our system are not generated continuously by theparticle, as they are in HQFT, but arise only as a result of particle acceleration. When the particle
accelerates from one velocity ⃗u0 to another velocity ⃗u1, it spawns a new, continuous source ofwaves, which continues at a velocity ⃗usource ∼ ⃗u0 along the extrapolated, original trajectory of theparticle. We note that a comparable physical picture was explored by Fort [52] in his investigationof orbiting ‘inertial walkers’, a theoretical abstraction of the walking-droplet system in whichwave sources do not strictly follow the particle, but instead extend in rays along the instantaneousparticle paths. In the first simple example considered here, when the particle accelerates fromrest, waves are generated continuously from the point of acceleration. After traveling an appreciable
distance, the free particle thus surfs on an approximately planar wavefront of the de Broglie
wavelength at nearly constant speed, as its in-line Zitter diminishes with time.
Another important feature of the present system is that our particle always travels
with a Yukawa wavepacket, which stores energy in a region of radius ∼ λc adjoining theparticle. We have shown that this energy takes the form of a wave-induced virtual massthat augments the particle’s inertial mass to m 7→ m + δm. A similar wave-induced mass
has been reported and rationalized in pilot-wave hydrodynamics and accounts for the
anomalously large radii of inertial orbits in that system [53]. A critical difference is thatin the hydrodynamic system, the wave-induced mass is velocity-dependent, while ours is
not. This difference simply reflects the Lorentz-covariance of the present model: the virtual
mass of our system, as a Lorentz scalar, cannot depend upon the particle velocity.
Finally, by combining the virtual mass and the multi-directional Zitterbewegung of oursystem, we have recovered a dynamical version of the Heisenberg uncertainty principle:
σx σp ≥ Kh̄/2,
where K depends only on the coupling constant b, and σx and σp are standard deviationsof the position and momentum in any one direction. For sufficiently large particle-field
coupling constants, b, this inequality reduces to the standard uncertainty principle ofquantum mechanics.
While we have focused here on a particular limit of our Lagrangian framework, ourmodel allows for the consideration of other wave-particle couplings that will be exploredelsewhere. In future studies, we shall adopt the present model to perform simulationsof the classic single- and double-slit diffraction experiments, and compare the emergent
statistics with those predicted by standard quantum mechanics. We shall also examine
the case in which the particle moves in response to the gradient of the wave’s phase, anddemonstrate that this particular variant allows for a recovery of Bohmian mechanics and acompelling model of wavefunction collapse following a position measurement.
Finally, it is perhaps worth reminding the reader that the mathematical framework
developed here is not an attempt to replace modern quantum theory; moreover, we do notclaim that it will reproduce the results of quantum field theory. However, it does represent
a rigorous, modern version of the mechanics envisaged by de Broglie in his double-solution
program. As such, we hope that it may prove fruitful in exploring the boundary between
classical and quantum behavior.
Author Contributions: Conceptualization, D.D. and J.W.M.B.; methodology, D.D.; software, D.D.;
validation, D.D.; formal analysis, D.D.; investigation, D.D.; resources, J.W.M.B.; data curation, D.D.;
writing—original draft preparation, D.D.; writing—review and editing, D.D. and J.W.M.B.;
visualization, D.D.; supervision, J.W.M.B.; project administration, J.W.M.B.; funding acquisition,
J.W.M.B. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by the National Science Foundation through grant CMMI-2154151.
Data Availability Statement: All data for the numerical experiments of the present work is available
in our GitHub repository, at https://github.com/ddarrow90/hqftii (accessed on 18 December 2023).
Acknowledgments: The first author would like to thank MathWorks, whose generous support
through the MathWorks Fellowship has made this research possible.

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Conflicts of Interest: The authors declare no conflict of interest.

Appendix A. Derivation of the Variational Theory
Appendix A.1. Variational Proofs in the General Case
We here present the proofs of our variational results, i.e., the pilot-wave version of theEuler–Lagrange equations and of Noether’s theorem, respectively. See Lemmas 1 and 2 forthe statement of these results.
Proof of Lemma 1. By linearity of the functional derivative, the action is extremized overvariations in both ϕ and q p when it is extremized over variations in each individually. As
δq p Sfield = 0, the particle’s trajectory equation follows from the classical Euler–Lagrange
equations.
For the field equation, we make use of the Euler–Lagrange equations for a scalar field:
δϕ L = ∂µ δϕµ L.
Specifically, viewing the particle and interaction Lagrangians as singular Lagrangian fields
over spacetime, we apply the above to findδϕ (Lfield + δ3 (q − q p )Lpart. ) = ∂µ δϕµ (Lfield + δ3 (q − q p )Lpart. ).
On the left-hand side, the variational derivative commutes with the delta function, giving
two terms of the field equation. On the right-hand side, the variational derivative commutes
similarly. Combining with the fact that∂t δ3 (q − q p ) = −⃗u · ∇δ3 (q − q p )gives the desired field equation.
′′
Proof of Lemma 2. Let Lfield= Lfield + ε∂µ Λfield and Lpart.
= Lpart. + εdt Λpart. . We deal
with field and singular components separately, allowing us to quantify the exchange ofconserved quantities between them. Beginning with the field Lagrangian, we find (to firstorder in ε) thatµ′ε−1 (Lfield− Lfield ) = (δϕ Lfield )Ψ + (δϕµ Lfield )∂µ Ψ= (δϕ Lfield − ∂µ δϕµ Lfield )Ψ + ∂µ ((δϕµ Lfield )Ψ)= Ψδ3 (⃗q − ⃗q p )(∂µ δϕµ Lpart. − δϕ Lpart. ) + ∂µ ((δϕµ Lfield )Ψ) − Ψ∇δ3 (⃗q − ⃗q p ) · (⃗uδϕ0 − δ∇ϕ )(Lpart. ),giving∂µ (Λfield − (δϕµ Lfield )Ψ) = Ψδ3 (⃗q − ⃗q p )(∂µ δϕµ Lpart. − δϕ Lpart. ) − Ψ∇δ3 (⃗q − ⃗q p ) · (⃗uδϕ0 − δ∇ϕ )(Lpart. ).
µ(A1)

For the singular component, we similarly find′ε−1 (Lsing− Lsing ) = (δϕ Lpart. )Ψ + (δϕµ Lpart. )∂µ Ψ + (δqk Lpart. ) Qk + (δuk Lpart. )dt Qkp= (δϕ Lpart. − ∂µ δϕµ Lpart. )Ψ + ∂µ ((δϕµ Lpart. )Ψ) + (δqkp Lpart. − dt δuk Lpart. ) Qk + dt ((δuk Lpart. ) Qk ).
Inserting the generalized Euler–Lagrange Equation (6) as well, we finddt (Λpart. − (δuk Lpart. ) Qk ) = ∂µ ((δϕµ Lpart. )Ψ) + (δϕ Lpart. − ∂µ δϕµ Lpart. )Ψ − Fk Qk .
Before combining the particle and field components, we note that
δ3 (⃗q − ⃗q p )∂µ ((δϕµ Lpart. )Ψ) = ∂µ δ3 (⃗q − ⃗q p )(δϕµ Lpart. )Ψ + Ψ∇δ3 (⃗q − ⃗q p ) · (⃗uδϕ0 − δ∇ϕ )(Lpart. ).

Symmetry 2024, 16, 149

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With this in mind, we combine the above with (A1) to recover∂µ (Λfield − (δϕµ Ltotal )Ψ) + δ3 (⃗q − ⃗q p )dt (Λpart. − (δuk Lpart. ) Qk ) = −δ3 (⃗q − ⃗q p ) Fk Qk .
µ

We note that dt δ3 (⃗q − ⃗q p ) = (⃗u · ∇ − ⃗u · ∇)δ3 (⃗q − ⃗q p ) ≡ 0, so we can commute the time
derivative with the delta function. Furthermore, we can substitute dt = uµ ∂µ , as thelatter is the material derivative along the particle trajectory. Note that this substitution
works whether we interpret Qk as a function of q p = q p (t) or as a function of the spatial
coordinates q, as we can see by the chain rule. The theorem follows.
Appendix A.2. Stress-Energy Conservation in the AM System
We now restrict our attention to the amplitude-modulated (AM) system introduced in
Section 3, and derive the balance law (10). In short, this approach guarantees an exact
conservation of momentum and a near conservation of the total energy budget. The latter
evolves over time in a manner proportional to dt ϕ(⃗q p ) = uµ ∂µ ϕ(⃗q p ).
We consider the specific pilot-wave framework introduced in Appendix B.2. That is,
our Lagrangians take the following forms:
Lfield = 12 ∂µ ϕ∂µ ϕ − m2 ϕ2 ,(A2)Lpart. = −γ−1 m − bτuµ ∂µ ϕ,where b is a fixed coupling constant and τ is the proper time along the particle’s trajectory(synchronized at t = 0, say). Recall that, while the above Lagrangians do not obey the generalform (3), they give rise to the AM system when we further apply the non-conservative force⃗F = −mbdt (δ⃗u τ )uµ ∂µ ϕ(qt ) ,as in the Equation (6).
Now, suppose we have a constant spatial translation qk 7→ qk + εQk , so that(q p )k 7→ (q′p )k := (q p )k + εQk ,ϕ 7→ ϕ′ := ϕ − εQk ∂k ϕ.

With this spatial translation, we find that Lpart. remains invariant. Indeed,′
Lpart.
= −γ−1 m − bτuµ ∂µ ϕ′ (q′p ),but ϕ′ (q′p ) = ϕ(q p ) and, since the velocity u is invariant, neither γ nor τ changes. The field
Lagrangian is translated as′
Lfield= Lfield − εQk ∂k Lfield = Lfield − εQk ∂µ δk Lfield ,µ

as it has no explicit spatial dependence. Applying Lemma 2 above, we recover
Tk = − 21 δk (∂α ϕ∂α ϕ − m2 ϕ2 ) + (∂µ ϕ − δ3 (⃗q − ⃗q p )mbτuµ )∂k ϕ + uµ δ3 (⃗q − ⃗q p )(mγuk + mbτ∂k ϕ − Jk )µµ

= − 21 δk (∂α ϕ∂α ϕ − m2 ϕ2 ) + ∂µ ϕ∂k ϕ + uµ δ3 (⃗q − ⃗q p )(mγuk − Jk ),µwhere Jk is given by (A6). Supposing that our particle evolves under the non-conservative
Equation (6) with ⃗F = dt⃗J = uµ ∂µ⃗J, as we do in Appendix B.2, we find that the Jk term
disappears in the balance equation: that is,T̃k = − 21 δk (∂α ϕ∂α ϕ − m2 ϕ2 ) + ∂µ ϕ∂k ϕ + uµ δ3 (⃗q − ⃗q p )mγukµµµ

satisfies ∂µ T̃k = 0 exactly.

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Moving on to time translations, suppose that t 7→ t + εQ0 and spatial coordinates areleft fixed. Then we have instead that(q′p )k = (q p )k − εQ0 uk ,ϕ′ = ϕ − εQ0 ∂0 ϕ.

The field Lagrangian translates as before:′
Lfield= Lfield − εQ0 ∂0 Lfield ,although the singular Lagrangian does not. If Lpart. had no explicit time dependence, wewould find?
′
Lpart.
= Lpart. − εQ0 dt Lpart. .
However, by replacing the passive coordinate transformation with an active transformation of q p and ϕ, we see that we must remove a term corresponding to ∂t τ = γ−1 fromthe above:′
Lpart.
= Lpart. − εQ0 dt Lpart. − εQ0 mbγ−1 uµ ∂µ ϕ.
Rt
Write R(t) = 0 dt′ γ−1 uµ ∂µ ϕ(q p ), so that′
Lpart.
= Lpart. − εQ0 dt Lpart. + mbR(t) .
Then we find
T0 = − 12 δ0 (∂α ϕ∂α ϕ − m2 ϕ2 ) + ∂0 ϕ∂µ ϕ − δ3 (⃗q − ⃗q p )mbτuµ ∂0 ϕµµ

+ uµ δ3 (⃗q − ⃗q p )(mγ−1 + mbτuα ∂α ϕ + mbR(t) − uk mγuk − uk mbτ∂k ϕ(q p ) + uk Jk )= − 12 δ0 (∂α ϕ∂α ϕ − m2 ϕ2 ) + ∂0 ϕ∂µ ϕ − δ3 (⃗q − ⃗q p )mbτuµ ∂0 ϕµ

+ uµ δ3 (⃗q − ⃗q p )(m(γ−2 − uk uk )γ + mbτ∂0 ϕ + mbR(t) + uk Jk )= − 21 δ0 (∂α ϕ∂α ϕ − m2 ϕ2 ) + ∂0 ϕ∂µ ϕ + uµ δ3 (⃗q − ⃗q p )(mγu0 + mbR(t) + uk Jk ).
µ

As before, the Jk term cancels in the balance equation, so the modified vector
T̃0 = − 12 δ0 (∂α ϕ∂α ϕ − m2 ϕ2 ) + ∂0 ϕ∂µ ϕ + uµ δ3 (⃗q − ⃗q p )mγu0µµ

satisfies∂µ T̃0 = −∂µ (uµ δ3 (⃗q − ⃗q p )mbR(t)) = −mbδ3 (⃗q − ⃗q p )γ−1 uµ ∂µ ϕ(q p ),µwhere we differentiated uµ ∂µ R = dt R. We have thus deduced the desired energy balance relation.
Appendix B. Deriving the Approximate Equation (8)We here provide two different derivations of the amplitude-modulated (AM) system
of Section 3. The first employs a scaling argument to demonstrate that the AM systemapproximately fits the form (3) demanded by our general Lagrangian framework. The
second deviates slightly from this framework, employing a non-conservative variational
approach to reach the AM system. The latter derivation has two key advantages. First, it
gives rise to useful stress-energy (and thus angular momentum) conservation relations,derived in Section 3.1. Second, it allows us to obtain bounds on the approximate Lorentz
covariance of the AM system, derived in Appendix B.3.
Appendix B.1. An Approximate Route to the AM System
Below, we use the notation ⟨ A⟩ to denote a scale estimate of a given quantity A, rather
than a mean. If the estimates ⟨γ⟩ and ⟨u⟩ of the Lorentz factor and velocity, respectively,
arise from the same mean state, this means that we can take ⟨γu⟩ = ⟨γ⟩⟨u⟩ to hold exactly.

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To derive the Equation (8) for the AM dynamics, take ϕ := Re ϕ as before andσ (ϕ) = 1 + b2 /4π + bϕ,
so the full particle equation reads
dt mγ(1 + b2 /4π + bϕ)⃗u = bmγ−1 ∇ϕ.
Recall that ϕ denotes the continuous component of ϕ, as laid out in Section 2.1. Splitting
ϕ = ϕrad + ϕwav into radiative and wavepacket terms, where ϕwav is given by (15), we
deduce the continuous component of ϕ:
1
b
· lim
ϕ = ϕrad + ϕwav =4πm r→0 4π

ZS2dξ ∂r e−mr + ϕwav = −b+ ϕwav ,4π

by using the fact that ϕwav is continuous and that ϕ 7→ ϕ is Lorentz-invariant (from Section 2.1)
on our domain. The latter allows us to carry out the above computation in the reference
frame of the particle, without loss of generality. In turn, this gives the modified (exact!)
trajectory equationdt (mγ(1 + bϕrad )⃗u) = bmγ−1 ∇ϕrad .
(A3)
From Section 3.3, we know that the field tends to vary at the length and time scales
L := λdB =

1
,
m⟨γu⟩

T :=

1
= L ⟨ u ⟩.
m⟨γ⟩

We use these scales to define the dimensionless derivatives dt∗ = Tdt , ∂t∗ = T∂t , ∇∗ = L∇,and ∆∗ = L2 ∇2 . To estimate ϕrad , recall from Section 3.3 that momentum fluctuations of
the particle are of the order1δp ∼ γ4−eb n−b mc,
where eb and nb are given in Figure 3. However, from the trajectory Equation (8), we also
know thatδp∼ γ−1 bL−1 ⟨ϕrad ⟩,mT
where ⟨ϕrad ⟩ is our desired ϕrad scale. Equating the two estimates yields⟨ϕrad ⟩ =which we plug into (A3) to find
dt m′ γ⃗u = bmγ−1 ∇ϕ,⟨ γ 5− e b ⟩,⟨u⟩bnbm′ = m + O(mγ5−eb /nb u).

In the range 13.3 ≤ b ≤ 80.0 used in our simulations, and assuming that ⟨u⟩ is neither
too small nor too large, this corresponds to an error in the mass of at most a few percent.
Dropping this oscillating mass in favor of its mean, we recover the amplitude-modulated
system of Section 3.
Appendix B.2. An Exact, Non-Conservative Route to the AM System
Although we show in Appendix B that the Equations (7) and (8) follow approximatelyfrom the action (3), we can deduce exact conservation laws by deriving the AM model fromSymmetry 2024, 16, 149

27 of 31an alternate, non-conservative set of Euler–Lagrange equations. In the language of Lemma 1,
we temporarily introduce the new Lagrangians
Lfield = 12 m2 ∂µ ϕ∂µ ϕ − m2 ϕ2 ,Lpart. = −γ−1 m − mbτuµ ∂µ ϕ,where τ is the proper time along the particle trajectory. In the language of our generalframework (A2), these Lagrangians roughly identify the coupling σ as
σ = σ(uµ , ∂µ ϕ) = 1 + bτγuµ ∂µ ϕ.
As σ depends on uµ and ∂µ ϕ, rather than just ϕ, this coupling does not fit our proposed
framework. With that said, we can still identify dynamical equations and conservation
laws from Lemmas 1 and 2.
We derive the field and particle equations in turn, as the latter (despite appearancesfrom Lemma 1) is more involved. In applying Lemma 1 to our particular system, note thatδ∇ϕ Lpart. = −mbτ⃗u = ⃗uδϕ0 Lpart. ,so terms involving ∇δ3 vanish from the field equation. We then evaluate(∂µ δϕµ − δϕ )Lfield = m2 ∂µ ∂µ ϕ + m4 ϕ,and(∂µ δϕµ − δϕ )Lpart. = −∂µ (mbτuµ ) = −mb∂t τ = −γ−1 mb,
noting that u0 ≡ 1 and dτ = γ−1 dt. Putting this together, we find the Equation (7). Turning
now to the particle trajectory equation, we findδ⃗q p Lpart. = −mbτuµ ∂µ ∇ϕ(qt ),(A4)

δ⃗u Lpart. = mγ⃗u − mbτ ∇ϕ(qt ) − mb(δ⃗u τ )uµ ∂µ ϕ(qt ).

(A5)

At this stage, we rewrite the final term above usingαδu j τ = ∂γuδ α τ = γ(δjα − γ2 uα u j )δγuα τ,∂u j γuwhich reveals it to be a Lorentz-invariant impulse of interaction:
Jj = −(δjα − γ2 uα u j )(δγuα τ )mbγuµ ∂µ ϕ(qt ).

(A6)

d⃗
Defining ⃗F = dt
J and differentiating (A5) yields

dt δ⃗u Lpart. = dt (mγ⃗u) − γ−1 mb∇ϕ(qt ) + mbτuµ ∂µ ∇ϕ(qt ) + ⃗F.
Subtracting off (A4) and adding ⃗F as a non-conservative force, as in (6), we recover the
trajectory Equation (8).
Although we could have proceeded with a simpler Lagrangian, inspired by (1), there
are two key benefits to our choice. First, the non-conservative component ⃗F vanishes in the
non-relativistic limit of our system, reducing our equations again to exact Euler–Lagrange
equations. Second, we will see that the remaining effect of ⃗F applies exclusively to theenergy budget of the system, leaving both momentum and angular momentum untouched.
We derive these conservation laws in detail in Section 3.1.

Symmetry 2024, 16, 149

28 of 31

Appendix B.3. Approximate Lorentz Covariance
One caveat with the non-conservative derivation of Appendix B.2 is that the vectordFα := dt
Jα does not take the correct form for a relativistically compatible force. While γFα is
Lorentz-covariant, it does not satisfy the requirement γFα uα = 0 for the particle to remain
on-shell. The result is that (8) is not exactly Lorentz-covariant if b ̸= 0, as it cannot be
extended to an equation on four-vectors.
In this appendix, we quantify the error in this approximate Lorentz-covariance, anddetermine where we can still make claims based on Lorentz transformations. Approximate
covariance is used in three claims throughout our paper. First, in Section 3.2, we apply
a Lorentz transformation to the steady-state solution (11) in order to deduce the form of
the traveling wavepacket (15). Second, in Section 3.3, we use this principle to argue that
the Zitterbewegung frequency γ−1 ωc is obtained after an acceleration from an arbitrary
initial walking state. Finally, in Section 3.4, we use this principle to derive the form of theaccompanying waveform, as illustrated in Figure 4.
In our laboratory frame, we have the trajectory equationdt (mγ⃗u) = γ−1 mb∇ϕ(qt ).
First consider a mean velocity ⃗u = uê1 , and suppose we perform a Lorentz boost withvelocity ⃗v = vê1 , resulting in the new four-momentum
γ′ = γv γ(1 − uv),

γ ′ u ′ = γv γ ( u − v ) .

On the one hand,dτ (mγ′ u′ ) = γv (1 − uv)dτ (mγu) = (γ′ /γ)dτ (mγu),or equivalently,

dt′ (mγ′ u′ ) = dt (mγu)

with the new time coordinate t′ . On the other hand, note thatγ−1 = (γv γ′ (1 + u′ v))−1and∂ x ϕ = γv (∂ x′ + v∂t′ )ϕ.
Suppose ϕ has the local phase velocity c2 ê1 /s in the boosted frame, corresponding to a de
Broglie wave with group velocity sê1 . In this setting, s∂t′ ϕ(q p ) = ∂ x′ ϕ(q p ), and we recovera final equation1 − v/sdt′ mγ′ u′ = (γ′ )−1 mb∂ ′ ϕ ( q t ).
1 + u′ v x
With this equation in hand, consider the case of a particle starting from rest (in the original,or laboratory, frame) and accelerating in the ê1 direction. As we demonstrate in Section 3.4,
this acceleration gives rise to quasi-monochromatic radiation of wavelength λdB in a region
around the particle, and thus s = ⟨u′ ⟩. Writing u′ = s + ε(t′ ), with ε encapsulating thesmall-amplitude Zitterbewegung of Section 3.3, we finddt′ mγ′ u′ = (γ′ )−1 mb̃(∂ x′ ϕ(qt ))(1 + O(εv)),
where
b̃ =

1 − v/sb.
1 + vs1
As shown in Figure 3, |ε| ∼ n−b ≤ 1/21.0 in all of our numerical experiments. Even formoderate-to-large initial velocities v, our trajectory thus satisfies the modified equationdt′ mγ′ u′ = (γ′ )−1 mb̃∂ x′ ϕSymmetry 2024, 16, 149

29 of 31in the boosted frame. Since the field Equation (7) is invariant only without a change in b, wenow define the following rescalings of both b and ϕ:qpb′ = b̃b,ϕ′ = b̃/bϕ.
(A7)
Under these rescalings, ϕ′ satisfies the transformed equation(∂µ ∂µ + m2 )ϕ′ = (γ′ )−1 mb′ δ3 (⃗q − ⃗q p ),and u′ satisfies the transformed equationdt′ mγ′ u′ = (γ′ )−1 mb′ ∂ x′ ϕ′ ,up to a relative error O(εv). Making the same argument in the other direction, suppose we
begin with a coupling constant b′ , and accelerate a particle from rest to a velocity ⃗u = w1 ê1 .
Then the dynamics from Sections 3.3 and 3.4 hold. The field radiates continuously fromthe particle’s point of origin with wavenumbers ≲ k dB , and the particle oscillates at thefrequency γ−1 ωc . However, the same transformation shows that, in our original reference⃗ , and theframe, the particle started with a velocity ⃗u and ended with a velocity ⃗u + w
Lorentz covariance of the wave four-vector recovers the appropriate wavelength andoscillation frequency.
We recover the steady-state wavepacket (15) for free in this derivation. Indeed, if theparticle undergoes no Zitterbewegung, the transformed fixed-velocity state ⃗u satisfies the
new trajectory equation exactly. Since the field equation is exactly Lorentz covariant, thetransformed wavepacket remains a solution in the new reference frame.
Before moving onto transverse Lorentz transformations, we note that a core weaknessof this argument is that we cannot boost into the particle’s post-acceleration frame ofreference. Our dynamical harmony of phases does not extend to the case in which theparticle is spontaneously brought to rest. Study of this case is outside the scope of thecurrent work.
Now, transverse Lorentz boosts leave the derivative ∂ x ϕ = ∂ x′ unchanged, so ourinline dynamics continue to satisfy the equationdt′ mγ′ u′ = (γ′ )−1 mb(∂ x′ ϕ(qt )).
The only (indirect) effect is that the ∂t′ ϕ term now introduces a particle oscillation of
the order O(εv) in the transverse direction. In turn, the particle now experiences ∇ϕfrom slightly off the axis of symmetry, reducing the derivative ∂ x′ ϕ(qt ) by a relative errorO(ε2 v2 ). Applying a similar argument as before shows that, if we start a particle at rest in
the transformed frame and accelerate it in the ê1 direction, the particle continues to vibratein the direction ê1 at the frequency γ−1 ωc , and is now forced in the transverse direction at
the same frequency. The continuous source of radiation at the particle’s point of origin is as
before. Unlike the inline direction, however, there is no rescaling of either b or ϕ.
The difference of scaling between transverse and in-line Lorentz boosts means that,
under a general Lorentz transformation, b becomes anisotropic. If we break a generic
Lorentz boost into in-line and transverse directions, the rescaled value b̃ governs dynamics
in the in-line direction and the value b governs dynamics in the orthogonal direction. Thisanisotropy does not affect the form of the wavepacket (15) or the wavelength and frequency
seen in the dynamical harmony of phases. Nevertheless, it does affect the amplitude of theparticle’s vibration in each direction. A more detailed study of this vibration amplitude isoutside the scope of the present study.
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