A Time-(Anti)symmetric Approach to the Double Solution Theory

Abstract

In this work, we present a new theoretical approach to interpreting and reproducing quantum mechanics using trajectory-guided wavelets. Inspired by the 1925 work of Louis de Broglie, we demonstrate that pulses composed of a difference between a delayed wave and an advanced wave (known as antisymmetric waves) are capable of following quantum trajectories predicted by the de Broglie–Bohm theory (also known as Bohmian mechanics). Our theory reproduces the main results of orthodox quantum mechanics and unlike Bohmian theory, is local in the Bell sense. We show that this is linked to the superdeterminism and past–future (anti)symmetry of our theory.

1. Introduction

Following the success of his idea of wave–particle duality at the core of his quantum-wave mechanics [1,2,3,4,5], Louis de Broglie suggested that this notion should be understood, not as separate waves and particles connected through some constraint, but rather as a particle being itself a subcomponent of a more fundamental undulatory phenomenon. He presented this new approach, which he called the “double solution theory” research program, in a publication in 1927 [6]. Soon, however, due to the mathematical complexity of the double solution, he presented at the 1927 Solvay conference an alternative approach, namely, the pilot-wave theory, receiving mixed reactions overall but still claiming the interest of some of the attendees [7]. History, however, showed that the Copenhagen interpretation of quantum mechanics would become the standard theory of the quantum world, and most of these “classical” ideas were abandoned. Even those of these ideas which survived or were later revived mostly revolved around pilot-wave theories, such as the best-known hidden-variable Bohmian mechanics (also named de Broglie–Bohm theory [8,9,10], hereafter, dBB theory), meaning that waves and particles would long remain distinct objects, even in the eyes of the most critical physicists.

However, de Broglie left us with some pioneering works on the subject, which he first studied around the end of the 1920s [6] and then revived some decades later [11]. In particular, in 1925, he briefly considered, as we do here, the possibility of time-symmetric waves mixing delayed and advanced waves [12,13], but quickly favored some nonlinear dynamics as a model of particles within waves [11]. In a recent work, one of us developed a version of the double solution theory using the time-symmetric description and was able to reproduce the main properties of quantum mechanics [14,15]. This approach, which has some similarities with the theory of action at a distance proposed by Wheeler and Feynman [16,17], implies the existence of time-symmetric divergent fields in the vicinity of the particles. These are treated as singularities of the field, consisting of a half-sum of the delayed field emitted by the particle and the advanced field absorbed by the particle. Note that the field singularity is then eliminated from that theory by introducing a nonlinearity into the field equations that allows the field value to be fixed at a high but finite value in the central region of the particle. This type of object constitutes what is known as a soliton modeling a quantum particle.

However, it is possible to develop another approach, also inspired by de Broglie’s work of 1925 [12,13], which instead of using a half-sum of delayed and advanced waves $\frac{u_{ret.}+u_{adv.}}{2}$ uses a half-difference $\frac{u_{ret.}-u_{adv.}}{2}$. The advantage of using such a difference is that it allows us to remove the singular properties of the field without requiring nonlinear wave equations. This is the subject of the present work.

We thus begin Section 2 with a short historical review of the main ideas behind the double solution theory, followed by the general equations of our approach. In Section 3, we solve these equations in several special cases to illustrate the types of solutions we can expect from this theory. This remains quite qualitative, focusing more on the general aspects of solutions rather than their rigorous mathematical expressions. Section 4 presents a further generalization of our equations recovering the dBB theory. More precisely, we show that our $u$—wavelet associated with particles can be easily guided by a dBB quantum trajectory. Therefore, since the dBB theory is empirically equivalent to the usual quantum approach (as shown, for example, in [10]) the present framework allows us to recover many predictions of the standard quantum theory. In particular, we consider the impact that our fundamentally local theory of a $u\left(x\right)$ wave moving through spacetime has on causality in relation to Bell’s theorem. We show that our approach can be seen in some sense as “superdeterministic” (for discussions of superdeterminism in relation with Bell’s theorem see [14,15,18,19,20,21,22,23]) and that it allows dBB nonlocality to emerge as a valid result at the level of Bohmian trajectories, i.e., when we disregard the fundamental u field bathing spacetime, and which is fundamentally local. We end this article in Section 5 with a broad discussion and conclusion on some important or remarkable aspects of our work, its implications, and a proposal that it could prove very useful even outside quantum mechanics in order to develop a new generation of optical or acoustical beams and pulses guided by trajectories.

We refer reader to previous work and the literature [14,15] for the case of the symmetric solutions $\frac{u_{ret.}+u_{adv.}}{2}$, which share some similarities in the methods and formalism with the one presented here, but which also differ in several key aspects that are mentioned throughout the discussion.

2. General Framework

2.1. The Main Ideas of the Double Solution Theory

It is useful to recall where the idea of wave–particle duality comes from, and how it evolved into both pilot-wave theories as well as the less-known double solution program. In his 1924 Ph.D. thesis [5], French physicist Louis de Broglie famously derived equations for what is referred to as “wave–particle duality”: the idea that to each particle (electrons, photons, …) is associated a wave, so that Nature displays both corpuscular and undulatory phenomena in the microscopic world. Although his eponymous wavelength is now one of the starting points of most quantum mechanics courses and thus remains quite famous, the more general picture that de Broglie developed is often forgotten. In fact, even the expression for this wavelength is only part of a more general (relativistic) identity between waves and particles.

The reason we put this much emphasis on wave–particle duality, is because one can use this single fact to derive intuitively most of the basic quantum phenomena and even some more complex ones. We can then wonder if history would have taken us on a much different course, if the theory of the quantum had been based on these notions rather than wave-functions, matrices, and probabilities. The goal is of course not to throw away the modern formalism of quantum mechanics: it works quite well as it is, and another theory would not add very much. We hope, however, that by studying various approaches, we can pinpoint some of the more essential aspects of quantum mechanics and gain a better understanding of this famously strange theory.

In 1924, Louis de Broglie asked how a microscopic particle like an electron “perceived” time, and more importantly, how it could synchronize itself with the surrounding world. This was obviously very close in spirit to Einstein’s explanation of special relativity, and in fact, the starting point was to assign to the particle—for example an electron—an “internal clock” whose frequency ${\omega }_0$ was given by the Planck–Einstein quantum formula $\hslash {\omega }_0=E_0$, with $E_0$ the rest energy of the particle, so that we obtain

$${\omega }_0=\frac{m_0{c}^2}{\hslash },$$

(1)

with $m_0$ the proper (or rest) mass of the particle. By studying how this frequency was modified in various moving frames, de Broglie showed that it must remain at all times synchronized with a superluminal phase wave of frequency $\omega ={\omega }_0/\sqrt{1-v^2/c^2}$, with v the relative velocity between the observer and the particle. This then implied a beautiful relation between these two objects, a generalization of the Planck–Einstein relation,

$$\left(\begin{array}{c}E\\ \mathbf{p}\end{array}\right)=\hslash \left(\begin{array}{c}\omega \\ \mathbf{k}\end{array}\right),$$

(2)

with p and k the linear momentum of the particle and the wave vector of the corresponding wave, respectively. This new relation $\mathbf{p}=\hslash \mathbf{k}$ is well known as de Broglie’s relation, but should in fact not be separated from its 4-vector counterpart.

Importantly, this new wave is indeed a phase wave, meaning that it propagates faster than light at the velocity $v_{\phi }=c^2/v$ [1,2,3,4,5]. If we want to interpret it as being a real physical object, meaning for instance that it could interact with other objects through some forces, it must obey the usual constraints of our physical world (mainly relativity), so that it is necessary to view it as only a subcomponent of a “real” wave in the following way:

$$\mathrm{physical}\mathrm{wave}=\mathrm{phase}\mathrm{wave}\times \mathrm{group}\mathrm{wave}.$$

(3)

Indeed, in any physical wave, a phase wave must always be accompanied by a group wave. What is remarkable is that one can show in that case that such a group wave

(i)

Has the same velocity as the particle $v_g=v$.

(ii)

Localizes—or more correctly, represents the spatial distribution of—the total energy of the real wave and thus the associated particle.

It is then only natural to suggest that one could describe a quantum object as being this total wave we denote as u (or as being represented by it), with its wave-like properties being determined by the phase wave, while its corpuscular aspects are themselves given by the group wave. This is the essence of the double solution theory, where a single object exists as the combination of two solutions, one for the wave and the other for the particle, but with each of them always inseparable from the other. It is important to note that this is not necessarily the historical motivation behind de Broglie’s theory, but this way of explaining it has the advantage of being very intuitive and tying in nicely with our subsequent derivation.

In previous articles [24,25,26] (written in the framework of hydrodynamic quantum analogs [27,28,29] and “hydrodynamic quantum field” models [30,31,32]), we showed that it was possible to create a basic model of a double solution using a very simple field (scalar- and complex-valued) $u\left(x\right)$ consisting of two counterpropagating plane waves $u_{\pm }$. For a 1D problem, we can write the field qualitatively as

$$u(t,x)=u_{+}+u_{-}=e^{i({\omega }_{+}t-k_{+}x)}+e^{i({\omega }_{-}t+k_{-}x)}=2e^{i(\omega t-kx)}{e}^{i({\omega }_{g}t-k_{g}x)}.$$

(4)

This works on the basis that such a wave appears as a standing wave in the proper frame of the particle, which was also one of de Broglie’s starting points [12,13]. This last expression (see [24,25,26] for explanations) corresponds to a product between a phase wave and a group wave in agreement with Equation (3). The fact that this solution, extracted from a purely classical model, manages to account for many of the phenomena that are usually seen today as manifestations of the quantum nature of microscopic physics rather than something more general, was remarkable and very promising. There were, of course, some issues that were discussed concerning the incompleteness of the solutions ($l=0$ orbitals in the atomic model for instance), as well as a requirement that these physical waves must display some form of superdeterminism. This last point is particularly important as it can be used to motivate the new approach we present here. Indeed, although we do not discuss it in detail before Section 4, we should already mention that Bell’s theorem requires that some trade-off be made in any theory attempting to reproduce quantum mechanics.

2.2. An “Acausal” Approach to Wave–Particle Duality

As mentioned above, two simple counterpropagating waves are sufficient as a basic model of a double solution theory. Instead of writing them as propagating in opposite spatial directions, it is completely equivalent to use two waves in opposite temporal directions. The cost of doing so is obviously to remove the hypothesis of past-to-future causality in our theory, but we should remember that this is not really a drawback but more of a trade-off, since in doing so, we are also removing any form of “magical” and conspiratorial superdeterminism. In other words, as we see in Section 4, the superdeterminism present in our theory is more an illusion linked to an inadequate description of the problem (i.e., a traditional description going from the past to the future). Of course, any Cauchy problem can be translated into such a causal description, as we will see, but in the context of our theory, it is much more natural to consider the u field as a whole unit in spacetime, with some signals corresponding to delayed waves and others to advanced waves.

We stress that the idea of half-advanced–half-delayed waves has been used in the past, most notably by Wheeler and Feynman in their absorber theory [16,17], but also earlier than that with the complete Liénard–Wiechert potentials.

The macroscopic world as we experience it is, of course, causal, and we often impose this constraint on our theories. Another way of thinking is to argue that some thermodynamical phenomenon or other suppresses retrocausal solutions. The emergence of irreversibility from time-symmetric physics is itself a very crucial question, so that we would do it a disservice by adding a causality hypothesis by hand at any point in our work. Let us then set aside this problem, at least for this work, and focus on the very interesting dynamics that can arise from time-symmetric equations and solutions.

We take a d’Alembert equation with a source term $\mathcal{J}\left(x\right)$ using a covariant notation in a flat spacetime

$${\partial }^{2}u\left(x\right)=\mathcal{J}\left(x\right)$$

(5)

with x the position 4-vector (in the following, we use the convention $c=\hslash =1$ and use the Minkowski metric with signature $(+,-,-,-)$). We can write general solutions of this equation as

$$u\left(x\right)=u^{\left(0\right)}\left(x\right)+\int {\mathrm{d}}^{4}x{G}^{\left(0\right)}(x;x_0)\mathcal{J}\left(x_0\right),$$

(6)

where $u^{\left(0\right)}\left(x\right)$ is a solution of the homogeneous equation without source, and $G^{\left(0\right)}(x;x_0)$ is a Green function solution of

$${\partial }^2{G}^{\left(0\right)}(x;x_0)={\delta }^4(x-x_0).$$

(7)

Solutions of this problem are well known and easy to obtain, and in particular, we obtain in momentum space k

$${\tilde{G}}^{\left(0\right)}\left(k\right)=-\frac{1}{k^2}$$

(8)

which gives rise to two completely equivalent solutions indexed $ret./adv.$ (or sometimes ±) in real space

$$G_{ret./adv.}^{\left(0\right)}(t,t_0;\mathbf{r},{\mathbf{r}}_0)=\frac{\delta (t-t_0\mp |\mathbf{r}-{\mathbf{r}}_0\left|\right)}{4\pi |\mathbf{r}-{\mathbf{r}}_0|}.$$

(9)

These two solutions characterize two different propagation directions in time. Interestingly, if we look at the corresponding field solutions $u_{ret./adv.}$, we see that their half-sum is also a solution, but their half-difference is instead a solution of the homogeneous equation

$${\partial }^2\frac{u_{ret.}\left(x\right)-u_{adv.}\left(x\right)}{2}=0.$$

(10)

This second approach has only rarely been considered, but we feel it is worth investigating. We take a more in-depth look at an example in the next section.

Free solutions of the d’Alembert equations are interesting enough, but we should also consider that there could be some external potential acting upon the field. In that case, we can use a covariant derivative D instead of ∂. For example, in the case of an electromagnetic 4-potential, we have $D=\partial +ieA\left(x\right)$ (with e the electric particle charge), and thus

$$D^{2}u\left(x\right)={(\partial +ieA\left(x\right))}^{2}u\left(x\right)=\mathcal{J}\left(x\right).$$

(11)

for the inhomogeneous wave equation and

$$D^{2}u\left(x\right)={(\partial +ieA\left(x\right))}^{2}u\left(x\right)=0.$$

(12)

for the homogeneous wave equation We can use the same procedure and study the Green function

$$D^{2}G(x;x_0)={\delta }^4(x-x_0).$$

(13)

If we expand the covariant derivative, we obtain

$${\partial }^{2}G(x;x_0)={\delta }^4(x-x_0)+{\mathcal{F}}_{x}G(x;x_0)$$

(14)

with ${\mathcal{F}}_x$ a new operator on G

$${\mathcal{F}}_x=-e^2{A}^2\left(x\right)-2ieA\left(x\right)\partial -ie(\partial A\left(x\right)).$$

(15)

Regardless of the value of ${\mathcal{F}}_x$, we see that if we use the previous solution $G^{\left(0\right)}(x;x_0)$ obtained with $A\left(x\right)=0$, we obtain for the difference

$${\partial }^2\left[G(x;x_0)-G^{\left(0\right)}(x;x_0)\right]={\mathcal{F}}_{x}G(x;x_0).$$

(16)

The function $G-G^{\left(0\right)}$ can itself be solved with Green’s method with ${\mathcal{F}}_{x}G(x;x_0)$ acting this time as the source term, so that

$$G(x;x_0)=G^{\left(0\right)}(x;x_0)+\int {\mathrm{d}}^4{x}_1{G}^{\left(0\right)}(x;x_1){\mathcal{F}}_{x_1}G(x_1;x_0).$$

(17)

This equation is fundamental and tells us that the total field G can be seen as the sum of a direct propagation $G^{\left(0\right)}$ and an indirect one $G^{(\mathrm{refl}.)}$ corresponding to one or more (and potentially infinitely many) reflections of $G^{\left(0\right)}$ on the potential $A\left(x\right)$. This is illustrated in Figure 1. This is of course not something we can compute easily but, provided that $A\left(x\right)$ has a small enough amplitude, a perturbative treatment can be performed. In that case, we obtain G as the sum of $n+1$ terms, each representing a different number of punctual interactions with A, i.e., one direct propagation plus a sum of n terms associated with $i=1,2,\dots ,n$ intermediary interactions between $i+1$ propagations. This is written as

$$\begin{array}{cc}\hfill G(x;x_0)& =G^{\left(0\right)}(x;x_0)\hfill \\ \hfill & +\int {\mathrm{d}}^4{x}_1{G}^{\left(0\right)}(x;x_1){\mathcal{F}}_{x_1}{G}^{\left(0\right)}(x_1;x_0)\hfill \\ \hfill & +\int \int {\mathrm{d}}^4{x}_1{\mathrm{d}}^4{x}_2{G}^{\left(0\right)}(x;x_2){\mathcal{F}}_{x_2}{G}^{\left(0\right)}(x_2;x_1){\mathcal{F}}_{x_1}{G}^{\left(0\right)}(x_1;x_0)\hfill \\ \hfill & +\dots \hfill \end{array}$$

(18)

It is important to note that these solutions, both $G^{\left(0\right)}$ and G, are very general and can be applied to a wide range of situations, as we see in the next section. To give back some physical meaning to these equations, let us summarize the above in the following way: a field $u\left(x\right)$ is generated by a (possibly point-like) source at some position $x_0$. If we choose this source to generate a total field expressed as the half-difference of two waves, one delayed and one advanced, the total field then behaves as if no source was present at all and we had a free field. In general, we have an inhomogeneous wave-field:

$$u\left(x\right)=u^{\left(0\right)}\left(x\right)+\int {\mathrm{d}}^{4}xG(x;x_0)\mathcal{J}\left(x_0\right),$$

(19)

from which we can define a new homogeneous field as

$$u\left(x\right)=\int {\mathrm{d}}^{4}x\frac{G_{ret.}(x;x_0)-G_{adv.}(x;x_0)}{2}\mathcal{J}\left(x_0\right),$$

(20)

Because of the spherical symmetry of a point-like source, we can expect the field to display the same symmetry and to decrease in amplitude as we get further from the source, similar to the electrical field radiated from a point charge, for instance. If this is indeed the case, which we verify next, this would constitute a good candidate for a double solution. Indeed, the field can be decomposed into a product of a phase and group wave so that the increase in the amplitude of the group wave (i.e., in the energy of the field) around $x_0$ could account for the corpuscular aspect of the field, while the phase wave could describe its undulatory nature. In this situation, the point-like particle becomes unnecessary, or rather, a useful but virtual intermediary that helps us correctly derive the field.

3. Special Cases and Examples

There are many special cases we can study. We chose two of them that are of particular interest in order to better understand and visualize these solutions.

3.1. A Particle at Rest

Let us consider a point-like particle of mass m which is located at $\left(x_0^{\mu }\right)=(t,{\mathbf{r}}_0)$. We can choose ${\mathbf{r}}_0=0$ without loss of generality. This particle is also characterized by an internal periodic phenomenon $e^{-i{\omega }_0\tau }$ with proper frequency ${\omega }_0$, which we set as the source of the field u. In that case, the source term can be written as

$$\mathcal{J}\left(x\right)=g{e}^{-i{\omega }_{0}t}{\delta }^3\left(\mathbf{r}\right),$$

(21)

with g some coupling constant. According to the equations we derived in the previous section, the total field radiated from this source particle is given in its antisymmetric form by

$$u(t,\mathbf{r})=\frac{g}{4\pi }\int {t^{\prime }}^3{\mathbf{r}}^{\prime }{e}^{-i{\omega }^{\prime }{t}^{\prime }}\frac{\delta (t-t^{\prime }-|\mathbf{r}-{\mathbf{r}}^{\prime }\left|\right)-\delta (t-t^{\prime }+|\mathbf{r}-{\mathbf{r}}^{\prime }\left|\right)}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}{\delta }^3\left({\mathbf{r}}^{\prime }\right).$$

(22)

Using the properties of the Dirac deltas, we easily see [33] that this reduces to the simple expression

$$u(t,\mathbf{r})=ig\frac{\sin \left({\omega }_{0}r\right)}{4\pi r}{e}^{-i{\omega }_{0}t},$$

(23)

where $r=\left|\mathbf{r}\right|$. Some interesting comments can be made on this solution, which is drawn qualitatively on Figure 2.

First of all, we observe that the amplitude of the field decreases as $1/r$, which was expected. It remains finite everywhere, even at the origin, because of the fact that we chose the half-difference of our advanced and delayed propagators. If we had chosen a half-sum instead, the field would have been $\cos \left(kr\right)/r$ and have diverged at zero. It is also remarkable that a solution to a linear and homogeneous wave equation can be this reminiscent of solitonic physics. Indeed, because of the equivalence of inertial frames, a source moving at a constant velocity produces the same field, possibly contracted or stretched in some directions because of relativistic effects. In other words, a small region of space contains most of the total field energy, which can move without dissipation. This means that, according to the definition we gave in the previous section, this solution behaves in some sense both as a free wave and as a free particle. Going outside of the sole topic of quantum mechanics, this also has many implications and applications for experimental physics, optics in particular, where one could want to trap a small particle. We discuss more of this later. An important historical point is that this particular wavelet was obtained by de Broglie as early as 1925. It describes the case of a corpuscle at rest or in uniform motion and clearly shows the possibility of seeing the quantum object described by the u field as a kind of standing wave in the proper frame of reference where the particle is at rest, though more fundamentally it really is a half-difference between a delayed wave and an advanced wave.

3.2. Qualitative Discussion of an Atomic Model

We discussed in Section 2.2 how one would find the field solutions for a relativistic particle like an electron immersed in an electromagnetic four-potential. This is a difficult thing to do properly, but we can make a few assumptions in order to obtain a qualitative discussion of the solutions.

First of all, if the electron is sufficiently slow, we expect relativistic effects to disappear. In the case of hydrogen, the first energy level corresponds to a velocity $v/c=\alpha \approx 0.007$, with $\alpha$ the fine structure constant. This, in turn, gives a Lorentz factor $\gamma =1/\sqrt{1-v^2/c^2}\approx 1+2\times {10}^{-5}$. We recall that this factor is used to quantify how much characteristic durations (frequencies) and distances (wavelengths) need to be corrected because of relativistic effects. Here, we obtain a correction of the order of 0.002%, which is sufficiently small for us to neglect it most of the time, and most importantly, in our current discussion.

The second difficulty concerns the presence of a Coulomb potential $V\left(r\right)$. We showed that we could use a kind of perturbative treatment to obtain the complete solution, where each additional term corresponded to an increasing number of punctual interactions. This should work in principle because of the fact that each new interaction adds a factor $\alpha$ which is of the order of 0.7% of the previous term. Once again, neglecting corrections due to the presence of this potential seems like a reasonable approximation, though it is more delicate in this case because of the additional $1/r$ factor in V, which suggests that we are also required to consider only points which are at large enough distances from the origin. In more general terms, we have to make the assumption of a weak field regime.

With these two elements in mind, we make the suggestion that the field generated by a point-like harmonic source orbiting in an electrostatic Coulomb potential should be, as a first approximation, that of a similar source at rest (this agrees with more general results derived in Section 4.1). This means that we expect the total field to be written as

$$u(t,\mathbf{r})\approx ig\frac{\sin \left({\omega }_0\right|\mathbf{r}-{\mathbf{r}}_0\left(t\right)\left|\right)}{4\pi |\mathbf{r}-{\mathbf{r}}_0\left(t\right)|}{e}^{-i{\omega }_{0}t}$$

(24)

with ${\mathbf{r}}_0\left(t\right)$ describing a circular orbit.

A more complete and rigorous derivation of these solutions in spherical coordinates can be found in [33], where a decomposition on a basis of Bessel functions $j_l$ and spherical harmonics $Y_{lm}$ was used with the following result for the field u:

$$u(t,r,\theta ,\phi )=ig\sqrt{1-v_n^2}{e}^{i{L}_{n}t}\sum _{l,m}{Y}_{lm}(\theta ,\phi )Y_{lm}^{*}({\theta }_n,{\phi }_n\left(t\right)){\chi }_n{j}_l\left({\chi }_{n}r\right)j_l\left({\chi }_n{r}_n\right).$$

(25)

The various quantities are indexed n when corresponding to the circular motion of the particle on a specific quantized orbit (the model used in [33] used a semiclassical approach for the trajectory corresponding to the Bohr–Sommerfeld quantization procedure applied to the hydrogen atom). For instance, $r_n,{\theta }_n$, and ${\phi }_n\left(t\right)$ are the spatial spherical coordinates describing the uniform circular motion of the source, $v_n$ is the instantaneous velocity, and $L_n=\gamma m{v}_n-E_n$ the conserved relativistic Lagrangian of the system. $L_n$ is also generally equal to the frequency of the field, but we introduce another quantity, ${\chi }_n=\left|L_n-m{v}_n/r_n\right|$, which acts as an effective frequency (or rather wave vector for the spatial component of the field) corrected for the presence of the central Coulomb potential.

Showing this requires lengthy calculations, and the final mathematical expression is not particularly easy to interpret, which is why we do not get into much more of the details here, but interested readers are welcome to look at the full derivation (see Chapter 6, Sections 6.2.1 and 6.3.1 of [33]). This general field was then evaluated numerically, as seen in Figure 3, and showed, remarkably, that the qualitative treatment in the present article was, in fact, a very good approximation, though there remained the assumptions that the electrostatic potential should not be too strong and the velocities non-relativistic.

Note that the semiclassical model is only considered here as an illustration. In a purely quantum framework, it makes more sense to use dBB dynamics, which are consistent with the results of standard quantum mechanics as far as statistical predictions are concerned. This is discussed in more detail in Section 4.2.

It is worth noting that from a historical point of view, this type of object with a local structure in $\sin \left(x\right)/x$ for a field u was previously studied by Mackinnon [34,35] and Barut [36] and even more recently by Borghesi [37] in cases of simple motion. However, the novelty of our approach lies in the possibility of generalizing in principle to any type of wavelet motion (i.e., to cases where the equality $u=\sin \left(x\right)/x$ is only approximate). The key point is the use of Green’s function and the half-difference between delayed and advanced solutions.

4. Generalization

4.1. General u Field

The preceding analysis, in particular, that given in Section 2, is in fact of a very general scope that goes far beyond the examples schematized in Section 3. Starting from Equation (20), we see that if we choose a source term $\mathcal{J}\left(x\right)$ centered on a mean trajectory $z\left(\tau \right)$ in spacetime (i.e., in order to model a moving source), then, by subtracting an advanced field from a delayed field, we can define an arbitrary solution of the general homogeneous Equation (12) imposed to follow the overall motion of the source term $\mathcal{J}\left(x\right)$. To be more specific, we consider a particular source term associated with the motion of a fictitious material point moving along the trajectory $z\left(\lambda \right)$ parameterized by the affine variable $\lambda$ during motion. We have, for the source term,

$$\begin{array}{c}\hfill \mathcal{J}\left(x\right)={\int }_{\left(C\right)}g\left(z\left(\lambda \right)\right)e^{iS\left(z\right(\lambda \left)\right)}{\delta }^4(x-z\left(\lambda \right))\sqrt{\left(\dot{z}\left(\lambda \right)\dot{z}\left(\lambda \right)\right)}d\lambda \\ \hfill ={\int }_{\left(C\right)}g\left(z\left(\tau \right)\right)e^{iS\left(z\right(\tau \left)\right)}{\delta }^4(x-z\left(\tau \right))d\tau \\ \hfill =g(t,\mathbf{z}\left(t\right)){\delta }^3(\mathbf{x}-\mathbf{z}\left(t\right))e^{iS(t,\mathbf{z}(t\left)\right)}\sqrt{1-{\mathbf{v}}^2\left(t\right)}.\end{array}$$

(26)

The first line corresponds to the most general parametrization and involves the velocity $\dot{z}\left(\lambda \right):=\frac{d}{d\lambda }z\left(\lambda \right)$, the second line uses parametrization by the proper time along the path $\left(C\right)$, and the last line corresponds to the choice of parameter $\lambda =t$, i.e., the laboratory time. We have also introduced a complex source term evolving along the trajectory $\left(C\right)$ with amplitude $g\left(z\right)$ and phase $S\left(z\right)$. Note that nothing limits a priori the velocity of this fictitious point particle. The trajectory may well contain time-like segments such that $dzdz>0$, corresponding to subluminal motion or space-like $dzdz<0$, i.e., tachyonic and corresponding to superluminal motion. In the following, we limit our analysis to the more natural subluminal regime and go back to the superluminal case briefly later.

In this formalism, the homogeneous antisymmetric field associated with the point source is written as follows:

$$\begin{array}{c}\hfill u\left(x\right)={\int }_{\left(C\right)}\frac{G_{ret.}(x;z\left(\lambda \right))-G_{adv.}(x;z\left(\lambda \right))}{2}g\left(z\left(\lambda \right)\right)e^{iS\left(z\right(\lambda \left)\right)}\sqrt{\left(\dot{z}\left(\lambda \right)\dot{z}\left(\lambda \right)\right)}d\lambda \\ \hfill ={\int }_{\left(C\right)}\frac{G_{ret.}(x;z\left(\tau \right))-G_{adv.}(x;z\left(\tau \right))}{2}g\left(z\left(\tau \right)\right)e^{iS\left(z\right(\tau \left)\right)}d\tau \end{array}$$

(27)

The crux of the problem is that here, the particle with trajectory $z\left(\tau \right)$ is purely virtual in the sense that it is only a mathematical tool for finding a solution to the homogeneous equation guided by this central trajectory $z\left(\tau \right)$.

We emphasize that the formalism used here was partly developed within the framework of the model involving time-symmetric fields (see refs. [14,15]) for solitons. Here, however, we consider fields that are solutions of linear equations, due to the temporal antisymmetry of the propagators.

An interesting particular solution corresponds to the material point moving in vacuum in the absence of an external field $A\left(x\right)$. We then have the Green’s functions in Equation (9), which, after insertion in Equation (27), gives:

$$\begin{array}{c}\hfill u\left(x\right)=\frac{1}{2}\left({\left[\frac{g\left(\tau \right)e^{iS\left(z\right(\tau \left)\right)}}{4\pi \rho \left(\tau \right)}\right]}_{{\tau }_{ret.}}-{\left[\frac{g\left(\tau \right)e^{iS\left(z\right(\tau \left)\right)}}{4\pi \rho \left(\tau \right)}\right]}_{{\tau }_{adv.}}\right)\end{array}$$

(28)

with $\rho \left(\tau \right)=|(x-z\left(\tau \right))·\dot{z}\left(\tau \right)|$, and where the delayed proper time ${\tau }_{ret.}$ (respectively, advanced proper time ${\tau }_{adv.}$) corresponds to the point $z\left({\tau }_{ret.}\right)$ (respectively, $z\left({\tau }_{adv.}\right)$) belonging to the trajectory $\left(C\right)$, in which the u radiation propagating along the forward light cone (respectively, backward light cone) reaches the point x. This u field is clearly reminiscent of the delayed and advanced Lienard–Wiechert potentials in classical electrodynamics and has several remarkable properties. Most importantly, near the point-particle trajectory $x\sim z\left(\tau \right)$, i.e., for points located at a distance $r=\sqrt{-{\xi }^2}$ from the singularity in the space-like (rest frame) hyperplane $\Sigma \left(\tau \right)$ defined by $\xi ·\dot{z}\left(\tau \right)=0$ (with $\xi :=x-z\left(\tau \right)$), we have approximately [14]:

$$\begin{array}{c}\hfill u\left(x\right)=\frac{ig\left(\tau \right)e^{iS\left(z\right(\tau \left)\right)}}{4\pi }[-(\dot{S}-i\frac{\dot{g}}{g})(1+\xi ·\ddot{z})+\frac{i}{3}\xi ·z^{\left(3\right)}+O\left(r^2\right)].\end{array}$$

(29)

where $\dot{g}:=\frac{dg\left(\tau \right)}{d\tau }$, $\ddot{z}:=\frac{d^{2}z\left(\tau \right)}{d^2\tau }$, and $z^{\left(3\right)}:=\frac{d^{3}z\left(\tau \right)}{d^3\tau }$. In particular, for a uniform motion, if $g\left(\tau \right)=const.$ and $S\left(\tau \right)=-{\omega }_0\tau$, we recover the near-field of the stationary wavelet Equation (23) in the rest frame (i.e., we have $\frac{\sin \left({\omega }_{0}r\right)}{r}\simeq {\omega }_0=-\dot{S}$). This allows us to justify the local approximation of Equation (24). More generally, from Equation (29), we deduce the field value on the trajectory:

$$\begin{array}{c}\hfill u\left(z\left(\tau \right)\right)=\frac{-ig\left(\tau \right)e^{iS\left(z\right(\tau \left)\right)}}{4\pi }(\dot{S}-i\frac{\dot{g}}{g})\end{array}$$

(30)

More generally, Equation (29) separates into two contributions: (i) a term $\frac{-ig\left(\tau \right)e^{iS\left(z\right(\tau \left)\right)}}{4\pi }(\dot{S}-i\frac{\dot{g}}{g})(1+\xi ·\ddot{z})$, which reduces in the case of uniform motion to the monopole field Equation (24), (ii) a contribution $\frac{ig\left(\tau \right)e^{iS\left(z\right(\tau \left)\right)}}{4\pi }\frac{i}{3}\xi ·z^{\left(3\right)}$, which depends on the derivative $z^{\left(3\right)}$ and defines deviations from this field Equation (24) in accelerated motion. Term (ii) is generally of smaller amplitude than (i) and is neglected as a first approximation (strictly speaking, (ii) defines deviations from the dBB guiding law in our model).

A second interesting property of the wavelet concerns the local phase gradient. In the particular case where $g\left(z\left(\tau \right)\right)=g_0=const.$ and where the acceleration of the particle is not too large (the general case is obtained in Appendix A), we have approximately

$$\begin{array}{c}\hfill {\partial }^{\mu }\phi (x=z\left(\tau \right))\simeq \dot{S}{\dot{z}}^{\mu }\end{array}$$

(31)

In other words, the local four-vector gradient of the phase is parallel to the four-vector velocity of the particle. This is actually reminiscent of the so-called de Broglie guidance formula introduced by de Broglie in his double solution theory [6,11]. We can rewrite this formula as

$$\begin{array}{c}\hfill {\dot{z}}^{\mu }\simeq \frac{\pm {\partial }^{\mu }\phi (x=z\left(\tau \right))}{\sqrt{[{\partial }^{\mu }\phi (x=z\left(\tau \right){\partial }_{\mu }\phi (x=z\left(\tau \right)]}}\end{array}$$

(32)

where the sign ± depends on the sign of $\dot{S}$. If we impose $\dot{S}<0$ (in order to recover the limit of the uniform motion where $\dot{S}=-{\omega }_0<0$), we have

$$\begin{array}{c}\hfill {\dot{z}}^{\mu }\simeq \frac{-{\partial }^{\mu }\phi (x=z\left(\tau \right))}{\sqrt{[{\partial }^{\mu }\phi (x=z\left(\tau \right){\partial }_{\mu }\phi (x=z\left(\tau \right)]}}\end{array}$$

(33)

or equivalently

$$\begin{array}{c}\hfill \frac{d}{dt}\mathbf{z}\left(t\right)=-\frac{\nabla \phi (x=z)}{{\partial }_t\phi (x=z)}\end{array}$$

(34)

which is the formula used by de Broglie in his pilot-wave theory (i.e., Bohmian mechanics) [7,10]. The present analysis can be generalized to describe the motion of the wavelet in the presence of an external electromagnetic field $A\left(x\right)$. Using results developed in [14,15] for a time-symmetric soliton, we can generalize the guidance formula to

$$\begin{array}{c}\hfill {\dot{z}}^{\mu }\simeq \frac{-({\partial }^{\mu }\phi (x=z\left(\tau \right))+e{A}^{\mu }\left(z\left(\tau \right)\right))}{\sqrt{\left[({\partial }^{\mu }\phi (x=z\left(\tau \right))+e{A}^{\mu }\left(z\left(\tau \right)\right))({\partial }_{\mu }\phi (x=z\left(\tau \right))+e{A}_{\mu }\left(z\left(\tau \right)\right))\right]}}\end{array}$$

(35)

which leads to

$$\begin{array}{c}\hfill \mathbf{v}\left(t\right)=\frac{d}{dt}\mathbf{z}\left(t\right)=-\frac{\nabla \phi (x=z)-e\mathbf{A}(x=z)}{{\partial }_t\phi (x=z)+eV(x=z)}\end{array}$$

(36)

again in agreement with de Broglie’s double solution.

4.2. Connection with Bohmian Mechanics: A Wavelet Guided by a Quantum Path

To bring the analysis of our wavelet’s motion closer to Bohmian theory, we now impose that the trajectory $z\left(\tau \right)$ be a trajectory given by Bohmian mechanics for the Klein–Gordon equation. More precisely, we recall that according to quantum theory, for a scalar particle (i.e., without spin) the relativistic wave function $\Psi \left(x\right)$ (not to be confused with our field $u\left(x\right)$) obeys the Klein–Gordon equation

$$D^2\Psi \left(x\right)={(\partial +ieA\left(x\right))}^2\Psi \left(x\right)=-m^2\Psi \left(x\right)$$

(37)

where $m:={\omega }_0$ is the mass of the particle. Within the framework of the Bohmian theory, we postulate that a particle with trajectory $z\left(\tau \right)$ is guided by the wave function $\Psi$, and we demonstrate that we must have the so-called de Broglie–Bohm guiding law:

$$\begin{array}{c}\hfill {\dot{z}}^{\mu }=\frac{-({\partial }^{\mu }S(x=z\left(\tau \right))+e{A}^{\mu }\left(z\left(\tau \right)\right))}{\sqrt{\left[({\partial }^{\mu }S(x=z\left(\tau \right))+e{A}^{\mu }\left(z\left(\tau \right)\right))({\partial }_{\mu }S(x=z\left(\tau \right))+e{A}_{\mu }\left(z\left(\tau \right)\right))\right]}}\end{array}$$

(38)

where we use the polar expression $\Psi \left(x\right)=R\left(x\right)e^{iS\left(x\right)}$. In particular, the $S\left(x\right)$ phase is the generalization of the relativistic Hamilton–Jacobi action and obeys the equation:

$$\begin{array}{c}\hfill ({\partial }^{\mu }S\left(x\right)+e{A}^{\mu }\left(x\right))({\partial }_{\mu }S\left(x\right)+e{A}_{\mu }\left(x\right))=m^2+Q\left(x\right)\end{array}$$

(39)

in which the quantum potential $Q\left(x\right)=\frac{{\partial }^{\mu }{\partial }_{\mu }R\left(x\right)}{R\left(x\right)}$ introduced by de Broglie and Bohm is involved. We also deduce the conservation law:

$$\begin{array}{c}\hfill \partial \left[R^2\left(x\right)(\partial S\left(x\right)+eA\left(x\right))\right]=0,\end{array}$$

(40)

linked to the local probability current conservation (see below). In this theory, the particle has a varying mass $\mathcal{M}\left(z\left(\tau \right)\right)=\sqrt{[m^2+Q\left(z\left(\tau \right)\right)]}$ along its trajectory (in the following, we limit our analysis to the cases where $m^2+Q\left(x\right)>0$ corresponding to subluminal particle motions). The Hamilton–Jacobi action defines an integral along the trajectory:

$$\begin{array}{c}\hfill S\left(z\left(\tau \right)\right)=-{\int }^{z\left(\tau \right)}\mathcal{M}\left(z\left({\tau }^{\prime }\right)\right)d{\tau }^{\prime }-e{\int }^{z\left(\tau \right)}{A}^{\mu }\left(z\left({\tau }^{\prime }\right)\right){\dot{z}}_{\mu }\left({\tau }^{\prime }\right)d{\tau }^{\prime }\end{array}$$

(41)

and we have $\mathcal{M}\left(z\left(\tau \right)\right){\dot{z}}^{\mu }\left(\tau \right)=-({\partial }^{\mu }S(x=z\left(\tau \right))+e{A}^{\mu }\left(z\left(\tau \right)\right))$.

We stress that from Equations (38)–(40), we obtain the second-order relativistic “Newton” law already found by de Broglie in 1927 [38]

$$\begin{array}{c}\hfill \frac{d}{d\tau }\left[\mathcal{M}\left(z\left(\tau \right)\right){\dot{z}}^{\mu }\left(\tau \right)\right]={\partial }^{\mu }\left[\mathcal{M}\left(z\left(\tau \right)\right)\right]+e{F}^{\mu \nu }\left(z\left(\tau \right)\right){\dot{z}}_{\nu }\left(\tau \right)\end{array}$$

(42)

with $F^{\mu \nu }\left(x\right)={\partial }^{\mu }{A}^{\nu }\left(x\right)-{\partial }^{\nu }{A}^{\mu }\left(x\right)$ the Maxwell tensor field at point $x:=z$. The varying de Broglie mass $\mathcal{M}\left(z\right(\tau \left)\right)$ (i.e., varying quantum potential $Q\left(z\right(\tau \left)\right)$) is central in order to recover the non-classical features of quantum mechanics specific of the dBB pilot-wave theory.

Returning to Equation (40), we remind the reader that the Klein–Gordon equation is associated with the local conserved current

$$\begin{array}{c}\hfill J_{\mu }\left(x\right):=\frac{i}{2m}({\Psi }^{*}\left(x\right)D_{\mu }\Psi \left(x\right)-\Psi \left(x\right)D_{\mu }^{*}{\Psi }^{*}\left(x\right))=\frac{-R^2\left(x\right)}{m}({\partial }_{\mu }S\left(x\right)+e{A}_{\mu }\left(x\right))\end{array}$$

(43)

which reads, in the dBB framework, $J_{\mu }\left(z\left(\tau \right)\right)=\frac{\mathcal{M}\left(z\right(\tau \left)\right)}{m}{R}^2\left(z\left(\tau \right)\right){\dot{z}}_{\mu }\left(\tau \right)$. In particular, in the non-relativistic regime where $\mathcal{M}\sim m$, we have

$$\begin{array}{c}\hfill J_{\mu }\left(z\right)\simeq R^2\left(z\right)[1,\mathbf{v}\left(t\right)]\end{array}$$

(44)

in agreement with the Schrödinger equation. Indeed, we recall that in the non-relativistic limit, the Klein–Gordon equation reduces to the Schrödinger equation. In that regime, the wave function $\Psi (t,\mathbf{x})$ obeys

$$\begin{array}{c}\hfill i{\partial }_t\Psi (t,\mathbf{x})=\frac{-1}{2m}{(\nabla -ie\mathbf{A}(t,\mathbf{x}))}^2\Psi (t,\mathbf{x})+(m+eV(t,\mathbf{x}))\Psi (t,\mathbf{x})\end{array}$$

(45)

and the guiding velocity dBB becomes

$$\begin{array}{c}\hfill \mathbf{v}\left(t\right)=\frac{d}{dt}\mathbf{z}\left(t\right)=\frac{\nabla S(x=z)-e\mathbf{A}(x=z)}{m}\end{array}$$

(46)

$R^2\left(x\right)={|\Psi |}^2\left(x\right)$ can be interpreted as a probability density in agreement with Born’s rule. In other words, if Bohmian particles are ${|\Psi |}^2$-distributed according to Born’s law, we can recover all the statistical consequences and predictions of standard quantum mechanics with the dBB theory (at least in the non-relativistic domain). The $u(t,\mathbf{x})$ field associated with the wavelet defined in spacetime is as follows:

$$\begin{array}{c}\hfill u(t,\mathbf{x})\simeq g_0{\int }_{\left(C\right)}\frac{G_{ret.}(t,\mathbf{x};t^{\prime },\mathbf{z}\left(t^{\prime }\right))-G_{adv.}(t,\mathbf{x};t^{\prime },\mathbf{z}\left(t^{\prime }\right))}{2}{e}^{iS(t^{\prime },\mathbf{z}\left(t^{\prime }\right))}d{t}^{\prime }\end{array}$$

(47)

This theory makes it possible to define wavelets following dBB trajectories, for example, in the case of motion inside an atom, during a scattering process by a potential, or during a diffraction experience by a double slit giving rise to interference.

It is important to note that Barut’s previous approaches to diffractive phenomena could not account for the localization of the corpuscle along the trajectory (see [39] and a similar problem in the context of hydrodynamic analogies [40]) due to the dispersion in the considered wavelets. It is worth noting that back in the 1950s, Fer [41,42], in collaboration with de Broglie, proposed models of particles guided by delayed waves emitted by the same particle. However, as explained in [14,15], these waves cannot account for the guiding formula Equation (35) and are incapable of explaining interference phenomena precisely. However, in our approach involving half-delayed and advanced fields, all these problems disappear completely. On the other hand, as we shall see, our model involving delayed and advanced waves can account for the violation of Bell’s inequalities without departing from Einstein’s locality. This is in fact impossible in a model involving only delayed waves.

As an illustration of the new theory, we consider the case of a particle described in the non-relativistic limit by a quantum superposition $\Psi (t=0,x,y,z)$ of two 1D Gaussian wave packets (along the x axis) separated by a distance $d=8$ with unit variance, zero mean velocity, and initial mean position equal to −4 and +4, respectively (see [43]). The $\Psi (t,x,y,z)$ field is distorted over time, with each Gaussian dilating towards the future and towards the past from $t=0$. The superposition creates Young-type interferences, and the associated calculated Bohmian trajectories are shown in Figure 4 in the x–t plane.

The double solution theory allows us to calculate numerically the field $u(t,x,y,z)$ in the x–t plane for a given Bohmian trajectory (the one shown in red in Figure 4). To do this, we use Equation (47) with an action $S\simeq -m\tau$ (here, $m=1$, and $\tau$ is the proper time). The antisymmetric field, calculated in Figure 5 in the x–t plane, demonstrates the superdeterministic character associated with delayed and advanced wave superposition. Note that the wavelet follows (i.e., is guided by) the Bohmian trajectory. In turn, this theory eliminates the paradoxes and limitations of previous models that do not consider advanced wavelets and allows us to fully justify the interference fringes (analyzed in a Bohmian framework in Figure 4) using our time-antisymmetric double solution theory.

A central point of the theory proposed here concerns the distinction between the field $u\left(x\right)$, which is of a more fundamental or “ontological” nature, and the quantum wavefunction $\Psi \left(X\right)$, which is defined in configuration space. In line with de Broglie’s vision, the $\Psi$ wavefunction is more epistemic in nature and is more similar to the Hamilton–Jacobi action function $S\left(X\right)$, which, in classical physics, is also defined in configuration space (for a single particle, the configuration space is a three-dimensional-like physical space but it has $3N$ dimensions in the case of an N-particle system). The $\Psi$ wave function and the S action are therefore seen as tools for calculating particle trajectories. More precisely, these tools are used for calculating a whole family of trajectories defined using the guidance formula Equation (38). In contrast, the u field is more akin to a classical field such as the electromagnetic or gravitational field.

In the examples shown in Figure 4 and Figure 5, the $\Psi$ wave function defines a set of possible trajectories for the particle. This information is epistemic in nature, in the sense that only one trajectory occurs at a time. It is only by repeating the experiment several times (with the same $\Psi$ wave function, i.e., the same set of possible trajectories) that different trajectories in the set are realized experimentally. This involves a statistical element: as in classical physics, we need to know the distribution of initial conditions $\rho \left(z\right(0\left)\right)$ defining a statistical or probabilistic set. According to the dBB theory, we must assume Born’s rule, i.e., $\rho \left(z\left(0\right)\right):={|\Psi \left(z\left(0\right)\right)|}^2$ (this rule is well defined in the dBB theory, at least in the non-relativistic limit). In our example of Figure 5, we only show one realization of the $z\left(t\right)$ trajectory together with the associated physical u field. To reproduce the interference fringes obtained with the $\Psi$ wave function, we need to consider a large set of particles statistically distributed according to Born’s law. This statistical distribution enables us to reproduce the experimental results observed in the laboratory concerning the statistical arrival of particles in an interference pattern. Our theory is therefore empirically equivalent to traditional quantum theory as far as statistical predictions are concerned.

Let us therefore summarize the double solution theory developed here:

(1)

We introduced a Bohmian trajectory $z\left(\tau \right)$ guided by a wave function $\Psi \left(z\right)$ solution of the Klein–Gordon Equation (37).

(2)

We associated a $u$—field solution of Equations (11) and (26), which was expressed in the form

$$\begin{array}{c}\hfill u\left(x\right)=g_0{\int }_{\left(C\right)}\frac{G_{ret.}(x;z\left(\tau \right))-G_{adv.}(x;z\left(\tau \right))}{2}{e}^{iS\left(z\right(\tau \left)\right)}d\tau \end{array}$$

(48)

and was guided by the Bohmian trajectory $z\left(\tau \right)$.

(3)

The guiding velocity of the Bohmian particle given by Equation (38) was equal to the velocity of the center of the wavelet $u\left(x\right)$ moving through spacetime (see Equation (35)). We therefore had the local equality:

$$\begin{array}{c}\hfill {\partial }^{\mu }\phi (x=z\left(\tau \right))\simeq {\partial }^{\mu }S(x=z\left(\tau \right))\end{array}$$

(49)

where $S\left(z\right)$ is the Hamilton–Jacobi action for the Klein–Gordon equation and $\phi \left(z\right)$ the local phase of the wave field $u\left(x\right)$ at the point $x=z\left(\tau \right)$.

(4)

Finally, each wavelet guided by a trajectory $z\left(\lambda \right)$ corresponded to a specific realization of dBB mechanics for a set of initial conditions $z(\lambda =0)$ and a predefined wave function $\Psi$. The statistical predictions of quantum mechanics were recovered by varying the initial conditions $z(\lambda =0)$ assumed to be distributed in accordance with Born’s ${|\Psi |}^2$ rule (i.e., admitting the local Bohmian conservation law).

This model and the consequences drawn from it can be generalized to systems of entangled Bohmian particles, as shown in Section 4.3.

4.3. Causality, Time-(Anti)symmetry, and Bell’s Theorem

The theory developed here brings together two bodies of work that originated both in de Broglie’s quest for an understanding of quantum and wave mechanics, i.e., (i) pilot-wave theory also known as dBB dynamics, and (ii) the double solution theory. The great specificity of our result is that it succeeds in uniting two apparently contradictory physical and ontological points of view. Indeed, the dBB theory (Bohmian mechanics) is widely known as the nonlocal hidden variable theory par excellence. This means that when several entangled quantum particles are considered, the dBB theory predicts instantaneous action at a distance, which violates the spirit of the relativity theory, but which, in turn, recovers quantum results and agrees with Bell’s theorem on nonlocality (or rather nonlocal causality). On the other hand, de Broglie developed the double solution project in order to keep within a local framework in line with Einstein’s relativity theory.

In our approach, this duality between the dBB theory and the double solution is resolved, but this can only be achieved at a price, that of using a causality involving delayed and advanced u waves (more precisely, their half-difference $\frac{u_{ret.}-u_{adv.}}{2}$).

In our opinion, the consequence is remarkable because while the theory of the double solution presented here is clearly local, by applying it to an entangled particle system of the EPR (Einstein–Podolsky–Rosen) particle pair type, we can explain the results observed in experiments of the Bell inequality violation type [44] (like the famous Aspect et al.’s experiment [45]) with a completely local approach!

To fully understand how this is possible, we need to extend our discussion of the dBB theory, summarized in Section 4.2, to a set of N entangled (bosonic) particles with trajectories $\left\{z_j\left(\lambda \right)\right\}:=[z_1\left(\lambda \right),z_2\left(\lambda \right)\dots ,z_N\left(\lambda \right)]$ and common rest mass $m={\omega }_0$. We keep this rather schematic, as this theory is discussed in more detail in [14,15] in the context of a different approach. The central point is to use a multi-time formalism adapted to the Klein–Gordon equation. In the context of the dBB pilot-wave theory, it is natural to introduce the wave function ${\Psi }_N\left(\left\{x_j\right\}\right)=a_N\left(\left\{x_j\right\}\right)e^{i{S}_N\left(\left\{x_j\right\}\right)}$ solution of the set of N coupled Klein–Gordon equations:

$$\begin{array}{c}\hfill D_j^2{\Psi }_N\left(\left\{x_j\right\}\right)=-m_0^2{\Psi }_N\left(\left\{x_j\right\}\right)\end{array}$$

(50)

with $D_j:={\partial }_j+ieA\left(x_j\right)$ and ${\partial }_j$ being the four-gradient operator for the jth particle. These equations can be deduced from a second quantization formalism, but this is not shown here. Using the polar representation, we can write

$$\begin{array}{c}\hfill {({\partial }_j{S}_N\left(\left\{x_j\right\}\right)+eA\left(x_j\right))}^2={\omega }_0^2+Q_j\left(\left\{x_j\right\}\right):={\mathcal{M}}_j^2\left(\left\{x_j\right\}\right)\end{array}$$

(51)

$$\begin{array}{c}\hfill {\partial }_j\left[a_N^2\left(\left\{x_j\right\}\right)({\partial }_j{S}_N\left(\left\{x_j\right\}\right)+eA\left(x_j\right))\right]=0,\end{array}$$

(52)

with $Q_j\left(\left\{x_j\right\}\right)=\frac{{\square }_j{a}_N\left(\left\{x_j\right\}\right)}{a_N\left(\left\{x_j\right\}\right)}$ being a quantum potential. In the context of the relativistic dBB theory, the particle velocity for the jth particle is supposed to be

$$\begin{array}{c}\hfill \frac{d{z}_j\left(\lambda \right)}{d\lambda }=-({\partial }_j{S}_N\left(\left\{z_j\left(\lambda \right)\right\}\right)+eA\left(z_j\left(\lambda \right)\right)\sqrt{\left(\frac{{\dot{z}}_j\left(\lambda \right){\dot{z}}_j\left(\lambda \right)}{{\mathcal{M}}_j^2\left(\left\{z_j\left(\lambda \right)\right\}\right)}\right)}.\end{array}$$

(53)

Once the initial conditions $z_1\left(0\right),z_2\left(0\right),\dots ,z_N\left(0\right)$ are given, this set of coupled equations can be integrated to obtain N coupled (i.e., entangled) trajectories for the N particles. It is essential to note that all these trajectories are parameterized by a single $\lambda$ variable, which synchronizes the motion of the N entangled corpuscles. Without this common parameter, it would be impossible to integrate the set of N coupled equations of motion Equation (53). This system of coupled equation generates N-particle motion, which is generally strongly nonlocal due to the synchronization between the $\lambda$ parameter and the presence of nonlocal quantum potentials $Q_j(z_1\left(\lambda \right),\dots ,z_N\left(\lambda \right))$ coupling even very distant particle.

It is this type of nonlocality that, in the dBB theory, justifies the violation of Bell’s inequalities. The relativistic dBB theory presented here in schematic form also makes it possible to recover a large number of quantum results traditionally considered inexplicable by a deterministic approach. An essential point is that, as in the single-particle case presented in Section 4.2, the dBB theory recovers the statistical results of quantum mechanics, especially in the non-relativistic regime where we can define a probability density associated with all N particles. In this non-relativistic regime, we can introduce a single common time $t:=t_1=\dots =t_N$ such that ${\partial }_t={\sum }_j{\partial }_{t_j}$, and we deduce

$${\partial }_t{a}_N^2+\sum _j{\nabla }_j\left(a_N^2{\mathbf{v}}_j\left(t\right)\right)=0$$

(54)

which recovers the standard Bohmian probability law for the many-body Schrödinger equation with the definition $a_N:=a_N(t,{\mathbf{z}}_1\left(t\right),\dots ,{\mathbf{z}}_N\left(t\right))$. Here, $a_N={\left|{\Psi }_N\right|}^2$ defines the density of probability in configuration space in agreement with Born’s rule.

Having summarized the dBB theory adapted to N entangled bosons, we can now construct a completely local double solution theory. To do this, we start with N solutions $u_j\left(x\right)$ of the linear Equation (12) defined in spacetime, which read:

$$\begin{array}{c}\hfill u_j\left(x\right)=g_0{\int }_{\left(C_j\right)}\frac{G_{ret.}(x;z_j\left(\lambda \right))-G_{adv.}(x;z_j\left(\lambda \right))}{2}{e}^{i{S}_N\left(\left\{z_j\left(\lambda \right)\right\}\right)}\sqrt{\left({\dot{z}}_j\left(\lambda \right){\dot{z}}_j\left(\lambda \right)\right)}d\lambda \end{array}$$

(55)

Thus, the full field reads

$$\begin{array}{c}\hfill u\left(x\right)=\sum _{j=1}^{j=N}{u}_j\left(x\right).\end{array}$$

(56)

The central point here, and we emphasize this once again, is that the $u\left(x\right)$ field is the solution of a linear, local equation defined in spacetime, whereas the Bohmian trajectories come from a highly nonlocal description defined in configuration space for the N particles.

What makes all this possible is that the $u\left(x\right)$ field is the difference between a delayed and an advanced contribution. The advanced contribution, in particular, carries information associated with various $C_j$ trajectories in the past. This makes it possible to re-express the problem in terms of a more classical Cauchy description requiring initial conditions for the field $u\left(x\right)$ and its first derivative $\partial u\left(x\right)$ along a space-like hypersurface localized in the distant past. More precisely, applying Green’s theorem to a four-volume $\Omega$ surrounded by a closed hypersurface (boundary $\partial \Omega$), we can express the field $u\left(x\right)$ ($x\in \Omega$) solution of Equation (12) as the integral:

$$\begin{array}{c}\hfill u\left(x\right)={\oint }_{\partial \Omega }{\epsilon }_y{d}^3{S}_y{n}_y·[u\left(y\right){\partial }_{y}G(x;y)-G(x;y){\partial }_{y}u\left(y\right)]\end{array}$$

(57)

where $G(x;y)$ is a Green propagator solution of Equation (13) ($d^3{S}_y$ is a three-dimensional scalar elementary volume belonging to the boundary $\partial V$ at point y, and $n_y$ is the outwardly oriented normal unit four-vector at point y, such that ${\epsilon }_y=\mathrm{sign}\left(n_y^2\right)=\pm 1$). In particular, if we use the delayed Green function $G_{ret.}(x;y)$, we can express the field at point $x:=[t,\mathbf{x}]$ as an integral over the hyperplane $t_y:=t_{in}=const.$ (with $y:=[t_y,\mathbf{y}]$) localized in the remote past of $t\gg t_{in}$ (e.g., $t_{in}\to -\infty$):

$$\begin{array}{c}\hfill u(t,\mathbf{x})=-\int d^3\mathbf{y}·[u(t_{in},\mathbf{y}){\partial }_{t_y}{G}_{ret.}(t,\mathbf{x};t_{in},\mathbf{y})-G_{ret.}(t,\mathbf{x};t_{in},\mathbf{y}){\partial }_{t_y}u(t_{in},\mathbf{y})]\end{array}$$

(58)

where the minus sign comes from the fact that $n_y$ is time-like and past-oriented (i.e., ${\epsilon }_y=1$, $n_y:=[-1,\mathbf{0}]$).

Moreover, from Equations (55) and (56), the u field along the hyperplane $t_{in}$ contains information about the trajectories $C_j$ coming from all times $t>t_{in}$. In other words, the initial conditions required for computing the fields $u\left(x\right)$ at any points x with $t>t_{in}$ are fine-tuned, i.e., superdeterministic!

The implications for our understanding of causality involving quantum systems are far-reaching. Indeed, what we see is that starting from a traditional causal description going from the past to the future (i.e., expressed in Equation (57) with a usual delayed Green’s function), we can reconstruct a set of correlated u wavelets in such a way that the associated Bohmian trajectories $z_j\left(\lambda \right)$ are nonlocally entangled. Thus, from this point of view, due to the fine-tuned initial conditions of the u field, the quantum nonlocality of Bohmian trajectories is justified from a local $u$—field theory. In other words, nonlocality becomes emergent and is no longer fundamental, contrary to what Bohmians usually think!

According to our approach, there are two levels of understanding. At the $u$—wavelet level, the description is local but superdeterministic, whereas it is nonlocal if we focus only on the motion of wavelet centers with correlated $z_j\left(\lambda \right)$ trajectories. These two levels of description are not antagonistic, however, if we recall the particular structure of u fields, implying a difference between delayed (causal) and advanced (anti-causal) waves. In turn, this mixture of delayed and advanced actions explains the emergence of nonlocality at the level of $z_j\left(\lambda \right)$ trajectories.

The most important consequences concern, of course, Bell’s theorem [44,45,46,47] and more specifically, the nonlocality implied in de Broglie–Bohm’s theory to explain the violation of Bell’s inequalities [10]. Let us consider a pair of entangled particles. Although Bell’s theorem is generally discussed for spin or polarization observables, we recall that for spinless particles, it is possible to develop protocols for testing Bell’s inequalities using dichotomous observables such as directions and momenta using interferometers (see for example [48] and discussion [49]). Without going into details, which are unnecessary here, we assume a quantum state ${\Psi }_{1,2}$ associated with two entangled particles 1 and 2, with trajectories $z_1\left(\lambda \right),z_2\left(\lambda \right)$ influencing each other nonlocally at a distance (see Figure 6). The two particles are separated at such distances that any subluminal communication during the experiment can be neglected. In the Bell test Alice observes particle 1 and Bob particle 2, and each particle is acted upon by a field (characterized by “settings” $\mathbf{a}$ and $\mathbf{b}$, respectively) which define the type of measurements made by Alice and Bob. As shown in Figure 6, these settings $\mathbf{a},\mathbf{b}$ can, in principle, be determined at the last moment by Alice and Bob via two independent photons from a very distant cosmological past (e.g., emitted by two stars or quasars [50,51,52,53]). The two Bohmian trajectories are therefore nonlocal functions of the settings and of the Bohmian initial conditions, i.e., the initial positions at a common past time ${\lambda }_{in}$. At a common time ${\lambda }_{out}$ after the measurement, we have:

$$\begin{array}{c}\hfill z_1\left({\lambda }_{out}\right):=F_{1,{\Psi }_{12}}({\lambda }_{out},z_1\left({\lambda }_{in}\right),z_2\left({\lambda }_{in}\right),\mathbf{a},\mathbf{b})\\ \hfill z_2\left({\lambda }_{out}\right):=F_{2,{\Psi }_{12}}({\lambda }_{out},z_1\left({\lambda }_{in}\right),z_2\left({\lambda }_{in}\right),\mathbf{a},\mathbf{b}).\end{array}$$

(59)

In turn, these trajectories precisely determine the dichotomous $\alpha$ and $\beta$ observables measured by Alice and Bob in their Bell test. We therefore have the nonlocal functions:

$$\begin{array}{c}\hfill \alpha =A_{{\Psi }_{12}}(z_1\left({\lambda }_{in}\right),z_2\left({\lambda }_{in}\right),\mathbf{a},\mathbf{b}):=A_{{\Psi }_{12}}(\mathbf{\Lambda },\mathbf{a},\mathbf{b})\\ \hfill \beta =B_{{\Psi }_{12}}(z_1\left({\lambda }_{in}\right),z_2\left({\lambda }_{in}\right),\mathbf{a},\mathbf{b}):=B_{{\Psi }_{12}}(\mathbf{\Lambda },\mathbf{a},\mathbf{b})\end{array}$$

(60)

which depend nonlocally on the settings $\mathbf{a},\mathbf{b}$ and the local observables $\mathbf{\Lambda }\equiv [z_1\left({\lambda }_{in}\right),z_2\left({\lambda }_{in}\right)]$. These are the quantities involved in Alice and Bob’s joint measurements leading to the violation of Bell inequalities. Once more, for a Bohmian, the situation is clearly demonstrating the necessary nonlocal link based on an instantaneous action at a distance.

However, from the point of view of our local wavelet theory, the total u field reads

$$\begin{array}{c}\hfill u\left(x\right)=g_0{\int }_{\left(C_1\right)}\frac{G_{ret.}(x;z_1\left(\lambda \right))-G_{adv.}(x;z_1\left(\lambda \right))}{2}{e}^{i{S}_N\left(\left\{z_j\left(\lambda \right)\right\}\right)}\sqrt{\left({\dot{z}}_1\left(\lambda \right){\dot{z}}_1\left(\lambda \right)\right)}d\lambda \\ \hfill +g_0{\int }_{\left(C_2\right)}\frac{G_{ret.}(x;z_2\left(\lambda \right))-G_{adv.}(x;z_2\left(\lambda \right))}{2}{e}^{i{S}_N\left(\left\{z_j\left(\lambda \right)\right\}\right)}\sqrt{\left({\dot{z}}_2\left(\lambda \right){\dot{z}}_2\left(\lambda \right)\right)}d\lambda \end{array}$$

(61)

which is a function of the two Bohmian trajectories $z_1\left(\lambda \right),z_2\left(\lambda \right)$. This antisymmetrical u field, made up of a delayed and an advanced part, can alternatively be seen (see Equation (58)) as being generated by knowledge of the u field and its ${\partial }_{t}u$ derivative on a $\Sigma \left({\lambda }_{in}\right)$ space-like hyperplane associated with an initial time $t_{in}$ in the distant past (see Figure 6).

This initial field seems conspiratorial or superdeterministic, as it already contains, in advance, information about what will happen to the wave packets $u_1$ and $u_2$ in the future via the interaction with the settings $\mathbf{a},\mathbf{b}$. From this point of view, we have demonstrated that Bohmian quantum nonlocality can be alternatively reproduced by a local but superdeterministic model involving delayed and advanced u fields.

To sum up, the notion of nonlocality is central to the de Broglie–Bohm theory, where it is seen as a necessary and unavoidable tension with the theory of relativity. However, in our approach, we demonstrate that nonlocality is only an effective description that appears when we limit ourselves to describing the motion of particles assumed to be point-like and with $z_j\left(\lambda \right)$ trajectories. In reality, according to the theory developed here, the trajectory is only part of the description associated with the particle. Each particle is described by a field $u\left(x\right)$ propagating in spacetime, and this field can be decomposed into a delayed and an advanced part. In particular, the advanced part corresponds to information moving from the center of the particle towards the past. However, if we look at wave propagation in the usual past–future temporal direction, what we see is a form of superdeterminism, i.e., an apparently conspiratorial u field prepared exactly at a past instant so as to arrive with the right phases and amplitudes on the particle at the present time. This eliminates the tension with relativity. In fact, nonlocality arises from the fact that many particles described as entangled and described by a $\Psi$ wave function (propagating in the configuration space) are actually $u\left(x\right)$ wave systems moving in spacetime and involving numerous advanced fields. It is these advanced fields that, looking in the standard past–future direction, bring conspiratorial information from the distant past to the present time. This information defines—from an effective point of view—at the present time a nonlocal force acting upon the particle trajectories (i.e., in accordance with the de Broglie–Bohm pilot-wave theory). From then on, apparent nonlocality is well and truly explained by a more fundamental and local theory using $u\left(x\right)$ waves propagating in spacetime and not in configuration space. The fundamental structure of our theory is local and respects the theory of relativity. Moreover, the presence of delayed and advanced waves emphasizes the profound past–future temporal symmetry of this theory. These aspects help explain the nonlocality of the pilot-wave theory as an emergent property.

5. Summary and Conclusions

To sum up our work, we presented an alternative theory that was empirically equivalent to traditional quantum mechanics, based on the idea of a double solution dating back to de Broglie. In our approach, systems of N quantum particles (entangled or not) were described by wavelets $u\left(x\right)$ moving in spacetime and solutions of a linear wave equation. The wavelets were also spatially centered on trajectories given by the Bohmian mechanical theory (or the de Broglie–Bohm theory). These guiding trajectories were themselves predicted by the $\Psi$ function propagating in configuration space for a set of N particles. Therefore, from the point of view of our theory, the $\Psi$ wave function was simply a tool for solving our linear wave equation for the u field in spacetime. A philosophical note on vocabulary might be needed: the nature of the $\Psi$ wave function is often discussed by philosophers, and debates concern the ontic, epistemic, or nomological nature of $\Psi$. Bohmians, in fact, are part of this debate, along with many other interpretative currents. In our case, the fundamental object is not the $\Psi$ wavefunction (propagating in the high-dimensional configuration space) but the local $u\left(x\right)$ field evolving in the four-dimensional spacetime. The u field clearly corresponds to an object of a much more ontological nature than $\Psi$ and can be compared to the electromagnetic field $F_{\mu \nu }\left(x\right)$ or even to the gravitational field $g_{\mu \nu }\left(x\right)$ of classical physics. The aim here is to unify contemporary physics with the classical perspective defended by Maxwell, Einstein, and Mie, to describe particles as localized objects in spacetime.

Our approach reproduces the consequences of Bohmian mechanics and hence of quantum physics. Moreover, unlike Bohmian mechanics, it is entirely local in the Bell sense. In fact, Bohmian nonlocality, which justifies experimentally observed violations of Bell’s inequalities, emerges here as a consequence of a local theory involving delayed and advanced waves (more precisely, their half-difference). This mixed delayed–advanced character in turn allows our theory to be interpreted as a superdeterministic approach, without having the miraculous or conspiratorial character usually associated with it. We want to insist, however, on the fact that this superdeterministic interpretation is only here if we insist on imposing a single time direction, in particular by reinterpreting advanced waves as being convergent delayed waves coming from a common past. Should we embrace the existence of advanced solutions with the same credence as delayed ones, then there is absolutely no need to invoke superdeterminism since the advanced waves are generated by the source itself rather than the past. In other words, an acausal (time-(anti)symmetric) phenomenon appears superdeterministic only for causal observers but is fundamentally not.

We have deliberately left vague the origin or nature of the $u\left(x\right)$ field used in our theory. Indeed, the central point here is the locality and linearity of our approach, which requires only standard d’Alembert-type wave equations that can appear in many electromagnetic, gravitational, etc., variants of classical field theory in the linear limit. In works published elsewhere, we presented more complex models involving $u$—wave field nonlinearities (see [14,15]), implying the existence of solitons and a modification of known classical field theories. In the same sense, this raises a fundamental question about our approach to experimental validation or falsification. Indeed, as previously mentioned, the aim here is to recover the pilot-wave theory by means of a more fundamental local theory. However, if the equivalence is correct, then the gain would be purely explanatory: we could rewrite a highly nonlocal theory (i.e., pilot-wave theory) into an equivalent local theory. This is already a huge gain from a logical point of view, as nonlocality has always been perceived as a thorn in the side of quantum theory. However, we believe that we can also go beyond this minimal gain or advantage. Indeed, if the quantum particle is an object with, as we propose, a spatial extension, then the u fields must undoubtedly be perturbed when the wavelets (or solitons in nonlinear cases) approach each other. Similar situations could occur in regimes where the u field strongly interact with an external potential $A^{\mu }$ so as to disturb the soliton or wavelet structure and motion. This could lead to new couplings and other interactions that disrupt the purely Bohmian dynamics of the objects, with potentially detectable empirical consequences. In the same vein, we neglected in our analysis the deviations due to the relativistic acceleration field in Equation (29). These purely relativistic effects could also modify the idealized Bohmian laws of motion. We leave these interesting speculations to future work.

It should be noted, however, that the consequences of the present work go beyond the scope of quantum mechanics and its interpretation. Indeed, from an application point of view, we demonstrated that it was theoretically possible to generate beams guided by arbitrary $z\left(\tau \right)$ trajectories. However, in the optical or acoustic field, it was demonstrated in the past, since the work of Durnin et al. [54], that certain beams known as “focus wave modes”, “Bessel Gauss Pulses”, “X Waves”, etc. [55,56,57], possessed a set of properties associated with a particle or energy packet (some can even move locally faster than light [58,59]). Recent works even showed the possibility to observe some de Broglie–Mackinnon pulses [57] which are particular cases of the waves considered here. In the framework we are developing here, we see that we can use a clever mix of delayed and advanced waves to generate new trajectory-guided beams. Since the trajectory is not necessarily time-like, we can also imagine a new family of superluminal “X-wave”-type beams: this opens up interesting prospects in communication and optical or acoustic trapping. Finally, it also opens up fascinating prospects for developing optical or acoustic quantum analogues using “superdeterministically” pre-prepared waves (somewhat like the time-reversed beam reconstruction methods developed by Fink [60]) so as to mimic quantum predictions exactly.