Keywords

12.1 Introduction

Quantum mechanics has been a source of astonishment and puzzlement since its very early inception around 1927, when the corpus of the actual theory began to build up from the now legendary fifth Solvay conference (Bacciagaluppi and Valentini 2009). Niels Bohr's philosophical insights at the time were embraced, due to his profound charisma and influence in the physics community. These insights developed after Heisenberg’s realization that the uncertainty relations, in the undulatory formalism, would have an epistemological impact on the classical causal description, involving position and momentum (Bohr 1985). From this fact Bohr rationalized quantum mechanics as a theory that would include the ultimate limits to the human cognition of reality, grounded on his Principle of Complementarity. Although not present even in the late ideas of Bohr, certainly holding a much more cautious stand, a stronger idealistic understanding of quantum phenomena sparked from the concept of limited cognition of reality. It kept feeding the imagination of physicists, mathematicians, and philosophers of Science, over decades. The cultural academic environment has progressively grown used to accept the causal influence of a kind of platonic potential realm, where a wave-like structure metaphysically exists to rule the “creation into the world” of potential atomic corpuscles, brought into existence by acts of conscious observation.

This highly idealistic way of thinking has seemingly shed, upon contemporary reasoning about physics, the idea that mathematical formalism is to be thought as more important than the reality it describes and, surprisingly, even more important than the intelligibility it was supposed to be grounded on. It is rather remarkable that already at the Solvay conference of 1927, Louis de Broglie was trying to counteract this same opacity brought about by the apparent lack of intelligibility quantum wave-corpuscle contradictory experimental results present. At the same event, standing on previous work on the double-solution theory, de Broglie suggested what is now called the Pilot wave model (Bacciagaluppi and Valentini 2009), where wave and corpuscle would both exist at all times, interacting trough what he thought to be a nonlinear guidage effect. The wave would guide the corpuscle, and this would generate the former, behaving as a moving oscillator. De Broglie, deeply influenced by the ideas of Albert Einstein, was aiming at a deterministic theory of quantum mechanics, which of course, might not actually be the case. He was also unaware of the subtill difficulties that the entanglement phenomena and the tunnel effect pose to our understanding of spacetime at the quantum scale. However, he was surprisingly prescient into the fact that a strong relation does exist in Nature between corpuscle behavior and wavelike properties. It so happens that while Bohr explored this fact pursuing an epistemological train of thoughts, de Broglie tried to make sense of it from a more realistic predisposition, aiming at an ontological proposal about how the world might be. It is my humble understanding that quantum mechanics, although fitting phenomena description in a remarkable way, still seems a theory overweighted by epistemological considerations about the limitations of human knowledge, or else about the overexaggerated demand that all presently unobservable aspects of reality are to be simply dismissed as non-existent. As such, perhaps it is time to repursue de Broglie’s ontological way and try to imagine what is implying such possible epistemological limitations, if really there are such. It is surely a very difficult task, but such an enterprise my hopefully bring a theory that will be epistemologically and ontologically weighted in a more balanced sense. It will most probably be a theory that while preserving some of the previous conundrums and perplexities of former attempts, allows framing them in a more intelligible way, pointing to new solutions and unexpected experimental and technological applications. It is a very disturbing sensation to address theoretically an issue without properly understanding it or being taught that one’s understanding capability about it is to be somehow limited, as an a priori assumption held by the theory itself. I propose that such philosophical predisposition is inappropriate when it comes to scientific progress and that one must always do the upmost effort to extend human intelligibility about reality. Even when the generalized present conviction seems to dismiss you from such effort.

In what follows I will try to present a possible intelligible framework for quantum phenomena. Of course, it is just a very limited and initial attempt, only devised to suggest a possible alternative way of thinking. I have hence called it an interpretation because it is just that. This attempt will also feed from the spirit of the present time, while concerning the use of an alternative concept of information, which I have qualified as nomological. What will be proposed will, expectedly and rather unavoidably, have embedded its own failures and perplexities in a mere attempt to understand reality. Perhaps too ambitiously, the present paper will have as its ultimate goal to serve a possible theoretical unification between two apparently irreconcilable descriptions: the pilot wave description and the description using what I have called the Bohrian formalism, to be meant here as the one currently serving the Copenhagen Scholl interpretation. This unification attempt will be done by means of a speculation about quantum reality, involving the idea that there is a kind of natural information encoded in quantum waves affecting the corpuscles behaviors. I have called this kind of natural encoded knowledge the “nomological information”, identifying their carriers as natural memory storing structures. This is the reason why I have called my proposal the “wave-memory interpretation of quantum mechanics”. I will use the words “information”, “encoded” and “memory” in a non-rigorous way, albeit adopting enough semantic charge to expose the present ideas. For instants, although the word “encoded” presupposes the existence of a code, I cannot precisely say what that code precisely is in the actual state of my reasoning. The sense hence taken is only that there is a natural structure able to contain information about other physical entities.

In Sect. 12.2, I will try to provide motivation for the nomological information hypothesis, starting from the Pilot-wave theory framework. In Sect. 12.3, I will take a more philosophical stand, dealing with some “law-like” or necessitarian aspects of the interpretation, introducing an ontology that tries to capture its main features. Finally, in Sect. 12.4, I will take a rather small part of the Bohrian formalism approach to offer motivation for a possible encoding schema for the nomological information. To accomplish this, I will apply the canonical procedure of wave packaging, first to infinitely spread plane waves and, secondly to space limited Morlet wavelet solitons, both considered as information wave carriers. From the wavelet packaging process, following a previous author (Croca 2003), I will derive a new generalized position-momentum uncertainty relation, which can be interpreted as a measure of nomological order in any physical situation.

12.2 Encoding the Behavior of the Corpuscle in Pilot Wave Phenomena

In recent years a new domain of experimental work developed in fluid dynamics physics: the so-called Hydrodynamic Quantum Analogs (HQA) research domain, with a small world spread community of experimentalists and theoretical physicists, presently highly motivated by John Bush at the MIT in the United States (Bush and Oza 2021). The movement started in 2005 with the pivotal work by Yves Couder, S. Protière, Emmanuel Fort and A. Boudaoud at Paris, France (Couder et al. 2005; Couder and Fort 2006). The French team found out that a millimetric oil droplet could self-propel, bouncing on the surface of a vibrating liquid bath. The sequence of bouncing positions would form a trajectory, each bounce resulting from the resonant interaction between the droplet and the quasi-monochromatic wave field produced by the droplet itself, that have propagated on the oil surface. This physical scenario hence constituted a pilot wave phenomenon, where a generated wave, taking the droplet as the oscillatory corpuscle, would in turn guide the droplet’s trajectory. It showed, once and for all, that pilot wave phenomena and its dynamics do exist, even if at a macroscopic level and with strong deterministic characteristics. The oil bath would have to be acted on from below by a vibrating force, producing an amplitude acceleration below the Faraday threshold. That is, the acceleration value above which pilot wave phenomena will no longer exist, giving way to standing Faraday waves on the oil surface (Bush and Oza 2021). Over the years the HQA community has put the analog approach to the test, trying to reproduce and study, either experimentally or with the use of computational models, several situations resembling the quantum reality cases. From the tunneling effect (Eddi et al. 2009; Tadrist et al. 2020) to the quantum corral effect (Harris et al. 2013; Cristea-Platon et al. 2018), to Landau levels (Fort et al. 2010; Oza et al. 2014), Friedel oscillations (Sáenz et al. 2020), diffraction from slits (Couder and Fort 2006; Pucci et al. 2018; Ellegaard and Levinsen 2020), spin states (Labousse et al. 2016; Oza et al. 2018), Zeeman splitting (Eddi et al. 2012), the quantum harmonic oscillator (Kurianski et al. 2017; Perrard and Labousse 2018), or the Hong-Ou-Mandel effect (Valani et al. 2018). Besides the strong interest that such analogies invoke, the most striking revelation coming from the HQA field is that there seems to be a relation between the resulting trajectories stability and what one can call the path memory encoded in the wavefield. This dependence had already been noted by Antonin Eddi and co-workers in 2011 (Eddi et al. 2011), with Stéphane Perrard and et al. even comparing the droplet- wavefield system to a natural Turing machine (Perrard et al. 2016). The overall pilot wavefield, guiding the corpuscle, is composed of several prior waves, generated by the corpuscle prior bounces. This superposition of waves produces a complicated field that in turn affects the corpuscle next bouncing, acting as a “natural path memory storage system”, and holding kinematic information about the droplet’s behavior. It should be mentioned that the storage process is correlated with the dynamic evolution of the field. Another very revealing fact about this pilot-wave phenomenon is that the decaying time of the prior waves define the wavefield memory storage capacity, experimentally controlled by the acceleration impressed on the bath. This in so much that if, on one hand, the decaying time happens to be very rapid (low acceleration), the pilot wave effect will be less stringent. If, on the other hand, the decaying time is extended (using high acceleration forcing on the oil bath, either near or above the Faraday threshold), the pilot wave phenomena will give way to other kind of effects, including some analogous to quantum chaos, where a droplet can bounce unpredictably over the bath (Bush and Oza 2021). This suggests that the memory encoding physical structure has a storing capability limit with a strong effect on the droplet’s behavior. HQA experimental and research domain is a very rich one, full of insights and surprises that deserve a much wider description than the one presented here. However, the embedded “nomological” memory (nomological, from the Greek: nomos, law, and logos, reason, order) seemingly points to a new way of thinking about the ontological status of a quantum wave, one that I will use in the interpretation I intend to propose further on.

Hydrodynamic pilot wave phenomena appear to vindicate Louis de Broglie’s original proposals (de Broglie 1960; de Broglie 1969), where the quantum corpuscle acts as a moving oscillator in phase with the quantum wave, guided in turn by what de Broglie called the phase wave of the corpuscle. De Broglie arrived at such a unifying picture of waves and quanta, by means of a law-like equivalence between two principles involving stationary values. In his own words, and from his 1924 Doctoral Thesis: “The Fermat principle applied to the phase wave is identical to the Maupertuis principle applied to the moving entity; the dynamically possible trajectories of the moving entity are identical to the possible rays of the waves” (de Broglie 2021, p. 38). In the course of reasoning following this theoretical equivalence, de Broglie identified a formal connection between the corpuscle’s momentum and his “phase wave”, as he put it, hence performing a major unification in physics between moving atomic entities and quantum waves. Already in the Doctoral Thesis he proposed the following relation between the quantum wave phase ϕ and the relativistic momentum-energy vector J (de Broglie 2021, p. 38):

$$ d\phi =\frac{2\pi }{h}{J}_id{x}^i $$
(12.1)

Which for the simplest case, gives what de Broglie later called the “guidage” relation:

$$ p=\hslash \nabla \phi $$
(12.2)

For this relation between the kinematic behavior of corpuscles and the undulatory properties of matter, Louis de Broglie received the Nobel prize in 1929.

After the 1927 Solvay conference, Louis de Broglie proposals were generally dismissed by the scientific community. This was to go on until American physicist David Bohm appeared with his early 1952 papers about the hidden variable theoretical formulation of quantum mechanics (Bohm 1952a, b), which endured until 1993, when Bohm forwarded his “implicate order” philosophical interpretation of pilot wave theory (Bohm and Hiley 1995, p. 30), still praised today in the Physics community. In Bohm’s approach, the action S would still be related with the wave’s phase, but the guiding process would be primarily due to the action of a quantum force originating from a quantum potential. To express the wave-corpuscle guiding dependence, Bohm considered the classical Hamilton-Jacobi equation with an extra term, corresponding to the quantum energy potential Q. This equation would represent the dynamical behavior of a corpuscle with its momentum depending on the action S, related to the wave phase, hence the connection to de Broglie’s guidage. The corpuscle’s acceleration would then also depend on the quantum potential, which on Bohm’s own account would mean “…that the forces acting on it [the corpuscle] are not only the classical force − ∇V, but also the quantum force − ∇Q” (Bohm and Hiley 1995, p. 30).

This same quantum potential concept, once applied to the n-body case, would be shown as early as 1974, by David Bohm and Basil Hiley (1975) to imply non-locality, in the sense that a “quantum interconnectedness of distant systems” would be valid leading to a “…notion of unbroken wholeness which denies the classical idea of analyzability of the world into separately and independently existent parts” (Bohm and Hiley 1975, p. 96).

As the authors put it, this interconnectedness feature of the world was due to the following main aspects of the quantum potential:

  1. (a)

    The quantum potential Q(X1, …, Xn) does not in general produce a vanishing interaction between two particles i and j as |Xi − Xj| → ∞. In other words, distant systems may still have a strong and direct interconnection. This is, of course, contrary to the general requirement, implicit in classical physics, that when two particles are sufficiently far apart, they will behave independently. Such behavior is evidently necessary if the notion of analysis of a system into separately and independently existent constituent parts, which can conceptually be put together again to explain the whole is to have any real meaning.

  2. (b)

    What is even more strikingly novel is that the quantum potential cannot be expressed as a universally determined function of all the coordinates X1, …, Xn. Rather, it depends on Ψ(X1, …, Xn) and therefore on the “quantum state” of the system as a whole. In other words, even apart from the point made in (a), we now find that the relationships between any two particles depend on something going beyond what can be described in terms of these particles alone. Indeed, more generally, this relationship may depend on the quantum states of even larger systems, within which the system in question is contained, ultimately going on to the universe as a whole. (Bohm and Hiley 1975, p. 99)

This, of course, implies a substantive difference between de Broglie and Bohm. While the former would consider quantum waves to exist as real four-dimensional spacetime physical entities, the later would not. In Bohm’s ontology the quantum wave is only a law-like entity and the guidage effect of the wave over the corpuscle is to be though as a pure idealistic and non-material process, converting implicate order into explicate order. This can cause conceptual tension with the idea of a “quantum force” caused by the quantum potential but, interestingly, is something that already resembles some kind of information transfer happening in Nature. A process that, on the other hand, still dismisses the possibility for a four-dimensional physical structure where nomological information can be encoded. Another of Bohm’s insight is non-locality, equated with the concept of wholeness. This can be understood as a law-like (nomological) feature of the universe, once one accepts the possibility that “interaction ruling” between several systems might also be encoded in a physical underlying structure. Non-locality would follow from the necessity that each and every corpuscle, involved in an interaction, must be “informed” about all its possible behaviors in relation to all remaining possible behaviors of all other corpuscles. I will address again this topic in Sect. 12.3.

Portuguese physicist José Croca (born 1944), started from de Broglie’s so called causal tradition in Quantum Mechanics developing mainly from 2003 on, yet another view about Pilot-wave theory with several original and insightful contributions. He started by reasoning that the prohibition posited by the Heisenberg uncertainty relations to well-defined trajectories for quantum corpuscles was, in fact, due to the adoption of what he called a Fourier ontology in our modes of thought about quantum phenomena. In so much that the quantum formalism uses the composition of infinite plane waves to form a corpuscle wave package, which in turn leads unavoidably to Heisenberg’s uncertainty relations and to Bohr’s Complementarity Principle, grounding the Copenhagen School interpretation. The Fourier treatment approach is, of course, only an ideal mathematical one, since in Nature one does not expect to have infinite waves of any sort. Croca hence proposed that a much more reasonable mathematical tool to apply to quantum mechanics would be wavelet local analysis, using Morlet solitons (Croca 2003). These are mathematically equivalent to plane waves modulated by amplitude gaussian functions. Croca’s approach would, in fact, introduce a third parameter to describe quantum waves, besides wavenumber and frequency, namely the spatial spreading of the wavelet soliton, represented by the Greek letter σ (sigma) in the gaussian modulation. A quantum Morlet soliton would immediately exhibit a most favorable property, that of having well-defined wavelength and frequency, while being spatially localized. This somehow surprising feature would, consequently, avoid the problem of having to build a wave package from infinite plane waves in order to get a localized corpuscle, while allowing to generalize Heisenberg’s uncertainty relations, as it will be seen in Sect. 12.4. Next, Croca went on to derive a one body wavefield non-linear differential equation, having as its natural solution precisely the Morlet soliton. He did this by starting from the two classical mechanic equations already considered by Bohm, the continuity equation and the Hamilton-Jacobi equation, although, unlike Bohm, without introducing an ad hoc quantum potential in the Hamilton-Jacobi equation. Croca hence obtained what he called a non-linear Master Schrödinger equation, with the nonlinear quantum potential term appearing formally in it, as a formal consequence from using the two classical equations at the start (Croca 2003, p. 72). Croca thought this to be a signature of a possible unification between the classical and quantum worlds. Significantly, one can also invert his derivation, giving thus strength to the interpretation that those two fundamental equations of classical mechanics will, in turn, result as formal consequence of a quantum Master equation, to be considered epistemologically more general. Croca also noted, as it is clear from (12.3) and (12.4) below, that if the Morlet soliton’s intensity is constant, corresponding to the special case where its spatial spreading σ is exceptionally large, one gets both the linear Schrodinger equation and its usual plane wave natural solution. This strongly suggests that wavelet local analysis can be a more general approach to quantum mechanics, with the Fourier situation appearing as a particular limiting case of the former treatment. Although I have recently come to the idea, to be developed in a future work, that both dynamics, the one given by the Schrodinger linear equation, and the one given by the non-linear master equation are to be though as complementary in quantum phenomena description. That is, both are necessary to completely describe quantum events and while the later relates directly to the four-dimensional physical wavefield and the corpuscle kinematic behavior, the former describes the dynamics of the nomological information encoded in the four-dimensional wavefield.

José Croca’s own thoughts about the two kinds of equations and natural solutions are the ones that follows:

$$\begin{aligned} &-\frac{{\hslash}^2}{2m}{\nabla}^2\theta +\frac{{\hslash}^2}{2m}\frac{\nabla^2{\left(\theta {\theta}^{\ast}\right)^{1/2}}}{{\left(\theta {\theta}^{\ast}\right)^{1/2}}}\theta + V\theta =i{\hslash}{\theta}_t\\&\kern0.5em\text{ If}\quad {{(\theta {\theta}^{\ast})^{1/2}}= const.}\quad \text{then{:}}\\&-\frac{{\hslash}^2}{2m}{\nabla}^2\theta + V\theta =i{\hslash}{\theta}_t \end{aligned}$$
(12.3)
$$ \begin{aligned}\theta &=A{e}^{-\frac{{\left(x- vt\right)}^2}{2{\sigma}_x^2}+i\left( kx-\omega t\right)}\kern1em \underset{\sigma_x\to \infty }{\to}\kern1em A{e}^{i\left( kx-\omega t\right)}\kern1.25em \\&\text{Implying}\kern1em {\left(\theta {\theta}^{\ast}\right)^{1/2}}\underset{\sigma_x\to \infty }{\to }A\ (const.) \end{aligned}$$
(12.4)

Using Morlet wavelet solitons as quantum waves, Croca went on to make a wave package composition, following the same procedure as Bohr did when he used plane waves to get to the Heisenberg uncertainty relations (Bohr 1985). Croca was then able to derive the set of generalized uncertainty relations (12.5a) and (12.5b) below (Croca 2003, p. 94), containing in a consistent manner with (12.3) and (12.4) the Heisenberg’s ones as a particular limiting case. As I will refer later in Sect. 12.4, one can also reason that Croca’s relations also contain the Newtonian case (no correlation between position and momentum) as the other limiting case (Castro et al. 2017). The uncertainty brought about by the correlation between position and momentum (or similarly between time and energy) would now also depend on the wavelet’s spatial spreading σ, as can be seen directly from (12.5a) and (12.5b).

$$ \Delta {p}_x=\frac{\hslash}{\Delta x}\sqrt{1-{\Delta x}^2/{\sigma}_x^2} $$
(12.5a)
$$ \Delta E=\frac{\hslash}{\Delta t}\sqrt{1-{\Delta t}^2/{\sigma}_t^2} $$
(12.5b)

Some years later, in 2015, Croca developed his quantum mechanics ontology. He introduced what he called philosophical Principle of Eurythmy (Croca 2015) as a foundational basis for the extremal principles and as a foundation principle for the pilot wave guidage effect over the corpuscle. In a certain sense, one might say that he balanced the phenomenal roles of both entities, wave and corpuscle, highlighting the ontological role of the corpuscle, as a complex active entity of its own. In his words:

The Principle of Eurhythmy comes from the Greek word, ενριτμια (euritmia) which is the composition of the root eu plus rhythmy. Here, eu stands for the right, the good, the adequate; and rhythmy for the way, the path, the harmonic motion. The composite word means: the adequate path, the good path, the good way, the right way, the golden path, and so on (Croca 2015, p. 15) (…) The Principle of Eurhythmy states that complex systems in nature, in order to continue to exist, must “choose” and follow optimal dynamic pathways. If a given system does not – on average – follow the optimal, most efficient pathway of development, it will not survive for long. All systems, which are able to exist long enough to be observable, even at a very small scale of observation, necessarily follow the Principle of Eurhythmy. (Croca 2015, p. 38)

According to the Principle of Eurhythmy, in order to keep its structural integrity, the corpuscle would move into the regions where the intensity of the quantum wave was higher, hence explaining the origin of Born’s probability rule. On the other hand, the Fermat and Maupertuis Principles would be instantiations of the Principle of Eurhythmy.

Using his Principle and following a stochastic reasoning, Croca was finally able to derive an additional second guidage formula, different from the one used by de Broglie’s and Bohm, reading:

$$ p={\hslash}\frac{\nabla \left(\theta {\theta}^{\ast}\right)}{\left(\theta {\theta}^{\ast}\right)} $$
(12.6)

It is a remarkable thing that using (12.6) one can write the following heuristic relations, where in (12.7a) the second order differentiation term represents the kinetic energy associated to the quantum wave phase, while in (12.7b) the quantum potential can now be interpretated as the guidage kinetic energy of the corpuscle:

$$ {p}_{\phi }={\hslash}\nabla \phi \kern3em \to \kern1.5em {E}_{\phi }=\frac{{\hslash}^2}{2m}{\nabla}^2\phi\kern26pt $$
(12.7a)
$$ {p}_g={\hslash}\frac{\nabla \left(\theta {\theta}^{\ast}\right)}{\left(\theta {\theta}^{\ast}\right)}\kern2em \to \kern1.5em {E}_g=\frac{{\hslash}^2}{2m}\frac{\nabla^2{\left(\theta {\theta}^{\ast}\right)^{1/2}}}{{\left(\theta {\theta}^{\ast}\right)^{1/2}}} $$
(12.7b)

In both expressions presented in (12.7b) the term operated by the gradient in the numerator is also present in the denominator, thus providing a kind of nonlinear normalization of that term. This means that only the topological curvature of the quantum field, and not its intensity, is relevant for the pilot-wave effect. I find this to be a signature for the presence of a sort of nomological information encoded in the field, being informationally read by the corpuscle as I will propose later on.

The kinetic energy of the corpuscle will be the one associated to the wave phase in the particular case where the wave’s intensity is constant, corresponding to the situation where the wave’s extension is exceptionally large, compared to the corpuscle’s dimension. In this case, the corpuscle moves randomly within the wave, inheriting its average kinematic characteristics from the wave’s phase. In this sense, de Broglie’s and Bohm’s original phase guidage can be thought of as a kind of inertial momentum formula, mainly attributable to the four-dimensional physical wave’s dynamic evolution. When, on the other hand, the wave’s topology is asymmetric, then the kinetic energy of the corpuscle will be a composition of both energies depicted in (12.7a) and (12.7b). In this case, the corpuscle will still move within the wave but will also be highly dependent on the local topology of the wave, that is, on the topology at the particular position where the corpuscle happens to be, in a given instant, within the wave.

There is now an important remark to make. The nonlinear interaction between the quantum guiding wave and the guided corpuscle was always a motive of concern, even for de Broglie himself. A quantum real wave after traversing a beam splitter, in the realistic approach, splits in two branches: one carrying the corpuscle, and the other corresponding the so-called empty wave. For the majority of experimental cases, where the quantum wave is very large, compared to the corpuscle dimension, one can assume according to (12.7a) that the wave’s energy is practically equal the energy of the corpuscle. This poses a striking objection to the existence of an empty wave in the other branch, since this empty wave seems to be a physical entity devoid of energy, as all energy seems to be in the occupied branch. In fact, it renders rather strange the pilot wave hypothesis entirely, because then one would have in general a non-energetic entity, a quantum wave, piloting a comparably high energetic entity, the corpuscle.

I wish to suggest that such a conundrum can be removed if one thinks in terms of information transfer, rather than in terms of energy balance checks. Although this is a possible way of reasoning out the of the problem, it must nevertheless be emphasized that a quantum empty wave can still have an energy. Croca has estimated that the energy of the wave would be of about 1054 orders less than the energy of the corpuscle (Croca 2004). This is about the same as saying that it has no energy, and of course means that whatever guiding interaction is happening between the wave and the corpuscle it must be of a nonlinear kind, in the sense that a natural amplification process must be at work. This amplification process must be able to gather energy from the vacuum or, otherwise, steer the comparably formidable energy of the corpuscle, in order to preserve energy conservation.

In a special issue about Pilot wave theory (Drezet 2023), an article by Bloch and Cohen was published relating mean quantum potential with the Heisenberg uncertainty relations, via the Fisher information (Bloch and Cohen 2022). The significant aspect I wish to note here is that the authors have shown that the mean quantum potential is proportional to the Fisher information applied to the corpuscle position measurement. In that particular context, the Fisher information represents the amount of empirical information that the measurement of the corpuscle’s position provides, conditioned to the possible positions it can occupy. The original definition, as the authors have referred to, comes from Marcel Reginato, as stated in 1998:

Consider the problem of estimating a parameter X in the presence of unknown added noise x. A measurement Y of the parameter will be related to X and x by:

$$ Y=X+x $$
(12.8)

For example, Y might be a measurement of the position of a corpuscle, while X is the actual position of the corpuscle. The probability distribution P(x) for the noise x will be related to P(Y| X) by

$$ P\left(Y|X\right)=P\left(Y-X\right)=P(x), $$
(12.9)

(…) In this case, the Fisher information I, is given by

$$ I\equiv \int \frac{1}{P\left(Y|X\right)}{\left(\frac{\partial P\left(Y|X\right)}{\partial X}\right)}^2 dX=\int \frac{1}{P(x)}{\left(\frac{\partial P(x)}{\partial x}\right)}^2 dx $$
(12.10)

Reginatto (1998, p. 1775)

Bloch and Cohen made the following remark about the formulation and identities assumed by Reginato:

(…) it is clear that for the identification of Fisher information with the quantum potential, one must have,

$$ {\left(\frac{\partial P(x)}{\partial X}\right)}^2={\left(\frac{\partial P(x)}{\partial x}\right)}^2. $$
(12.11)

While the previous assumptions are partially justified in the literature, this assumption goes unnoticed. This statement is equivalent to,

$$ {\left(\frac{\partial x(X)}{\partial X}\right)}^2=1, $$
(12.12)

which means that an increment of the “hidden” position results in an equal increment in the noise. In other words, the noise is proportional to the actual position of the corpuscle. What is the justification for such an assumption?” (Bloch and Cohen 2022, p. 117)

Providing a possible answer to this problem, I will also suggest that what is called the corpuscle position “noise” is, in fact, a kind of nomological information about a diversity of possible positions available to the corpuscle. In other words, the mathematical formalism, used to treat the uncertainty about the corpuscle position in a measurement, masks the existence of two different representations of the corpuscle position: its actual position X, the one where the corpuscle in fact is, and the overall information about all other possible positions x, that the corpuscle can potentially have. This information is encoded in the wave, being accessed by the corpuscle, via the wave’s topology curvature. That is, via the encoded value of momentum that the corpuscle can assume to reach a new position from its actual one. The proportionality between the noise and the corpuscle actual position, only means that wherever the corpuscle is, there will also be the nomological information about other possible positions to be reached.

Following Bloch and Cohen about the relation between the mean quantum potential and the Fisher information, for one spatial dimension, and with \( \rho (x)={\left(\theta {\theta}^{\ast}\right)^{1/2}} \), one can thus write:

$$ \overline{Q}=\frac{{\hslash}^2}{2m}\int \rho (x)\frac{\partial^2\rho (x)/\partial {x}^2}{\rho (x)} dx $$
(12.13a)
$$ I(x)=\int \frac{1}{\rho (x)}{\left(\frac{\partial \rho (x)}{\partial x}\right)}^2 dx $$
(12.13b)

Where (12.13a) is the mean quantum potential and (12.13b) is the Fisher information for the measurement of a corpuscle’s position (Reginatto 1998). From these expressions we get, after some calculation:

$$ \overline{Q} \propto \frac{{\hslash}^2}{8m}I $$
(12.14)

This identity is a remarkable one. Using (12.7b) one can now propose that the average guidage kinetic energy is, in fact, closely related to the overall positional nomological information encoded in the quantum wave. Surprisingly, it seems to correlate (in an informational way) energy with a set of possible locations in space, invoking the idea that such positions demand energy and therefore momentum to be reached. This gives strong bearing to the idea that the guidage effect is an information transfer phenomenon, where the corpuscle is using a sort of encoded information to acquire its state of movement, that is, momentum. It also gives credit to the possibility that the guidage effect cannot be strictly thought as an energy transfer process, since information is the actual fundamental concept to be attributed to the wave. The guidage effect seems to be more adequately described as a transduction phenomenon between information and energy. A process that in purely energetic terms will present itself as a nonlinear one. I must emphasize that at this stage it is not at all clear to me if the Landauer type of reasoning about information and energy (Landauer 1993, 1996, 1999) necessarily applies here. Anyhow, the information concept with which I am trying to cope with must be understood as a fundamental and primary phenomenal kind of information in Nature. One that is closer to what can metaphorically be envisaged a “rule or law of Nature” encoded in a quantum wave four-dimensional structure.

Before proceeding to the next section, I want to signal the theoretical work of Lucien Hardy (Hardy 2001), which also seems pertinent to the present proposals. Although the literature has developed since then, I find it quite significant that Hardy was able to show that standard quantum mechanics is a probability theory that only differs from the classical one by a simple axiom. In Hardy’s own words and from his paper:

(…) it is shown that quantum theory can be derived from five very reasonable axioms. The first four of these axioms are obviously consistent with both quantum theory and classical probability theory. Axiom 5 (which requires that there exist continuous reversible transformations between pure states) rules out classical probability theory. If Axiom 5 (or even just the word “continuous” from Axiom 5) is dropped, then we obtain classical probability theory instead. (Hardy 2001, p. 1)

Again, this seems to give strength to the hypothesis that Bohrian quantum formalism could be primarily thought as a theory about information in a very general sense, and not about the phenomena to which such information refers to. It must, nevertheless, be noted that this overall picture, where quantum waves encode information, posits considerable pressure upon quantum realists such as me, since we will have the onus of showing that empty (informational) quantum waves do exist. Quantum wave experimental detection, which also has its own history, therefore becomes another key element in the memory-wave interpretation. To that matter a set of quantum wave detection experiments have been devised and published here (Croca et al. 2023).

12.3 Strong Nomological Physicalism

In what follows I will try to give some philosophical motivation for the possibility of having nomological encoded information in the world. This will of course be a brief account on the topic, since my main concern here is to suggest an alternative way of thinking about quantum phenomena, rather than to solve old mysteries with a few proposals, something that definitely no one should expect here.

The first sub-section will deal with the core ideas that I’ve gathered about a possible nomological approach to pilot wave phenomena. The second sub-section will introduce what I have called the ontology of the wave memory interpretation of quantum mechanics. Both themes define in a very initial approach what I have named “Strong nomological physicalism”, the idea that one can have in Nature physical structures that can encode information about the physical behaviors of other structures.

12.3.1 Nomological Constructivism and Probabilistic Necessitarianism in Natural Laws

One of the major and deepest mysteries of Nature is its innate order and stability. To this, one might say, David Bohm answered with his implicate order concept, while José Croca forwarded the Principle of Eurythmy (Croca 2015), which in a broad sense establishes the relation between existence or being, on one hand, and order, harmony, and persistence, on the other. Around the same subject, I wish to contribute to a related topic, which can naively be formulated by asking where are the so-called “laws of Nature” written. That is, where does such natural order and harmony comes from and not so much why does it exist, which is the problem Bohm and Croca tried to address. In the prior section I gave experimental and some formal motivation to the idea that pilot-wave fields can encode information about corpuscles kinematic behaviors. Now I will try to make sense of it from a more philosophical point of view in route to what I will introduce as an ontology grounding the wave memory interpretation of quantum mechanics.

Let us start by the tension between indeterminism and determinism that divides micro and macro reality in present day Physics. I suggest that it is a dismissible tension, should one assume that the universe has a history. That is, an evolution of regularities and pattern forming that has been slowly and progressively been fixated in natural nomological memory structures during the evolution of the universe. Whatever physical entities have emerged and developed, by whatever physical processes, a probabilistic overall process has occurred, making behaviors and interactions to be describable using probability distributions, and consequently informationally encodable in nomological structures. Of course, in this universal plot there should be a few grounding and primeval “laws”, ruling nomological information storage and retrieval. However, even if the existence of such rules again presupposes an a priori regularity in Nature, as something embedded in existence itself, this evolutionary nomological picture significantly deflates the ontological weight of the metaphysical origin of regularity. It so happens that the majority of the law-like behaviors we actually find in the world can be in principle retraced to a nomological memory, which has been constructed over time and encoded in the proper physical structures. One can even presuppose that a possible overall candidate for nomological memory storing is space itself, given its ubiquity and given that it can be viewed as the set of all possible behaviors that have occurred and that can possibly still occur in Nature by moving corpuscular entities. In this tridimensional realist view of physical phenomena, space would hence be seen as the memory of the world, while time would become the signature of all nomological information storage and retrieval processes. In what can be dubbed as a “nomological constructivist” view of Nature, nomological information retrieval (and storing) would always be a process of memory altering, rather than simply one of information flux. This means that nomological memory is always dynamic, with different timescale regularity stabilities. The main thesis of this article is that quantum waves are the physical carriers of nomological information, that is, of information about the behaviors of corpuscles. Adding to this, one can also suggest that space can be seen as an overall nomological memory structure, therefore implying that quantum waves are a sort of primeval elements that confer spatial and temporal organization to corpuscles behaviors. The corpuscles in turn, retrieving and storing nomological information in quantum waves, through their actualized behaviors and by way of the guidage effect, thus instantiating a pilot wave physical phenomenon. Since in the nomological evolution process there is a kind of a priori indeterminism for the majority of behaviors, one can therefore reason that determinism is but an end case of probabilistic behavior that due to habit repetition and nomological memory fixing, has stabilized, becoming a highly probable type of physical behavior at the macroscopic scale. This almost certain repeatable behavior, nomologically encoded as such, can be recognized to be deterministic, for all practical purposes, hence resolving the tension between indeterminism and determinism. This last assertion allows me to pass on to the next topic I wish to consider.

The necessitarian approach to natural regularity proposes that the laws of Nature are more than just a descriptive summary of past and future events, occurring accordingly to a regulatory scheme or upholding empirically recognized patterns (Cover et al. 2012, p. 813). Therefore, the universality and necessity of natural laws in this view are posed as real objective features of the world. I can very much relate to that belief, in so much that I think that there is a nomic necessitation upon behaviors coming from the nomological memory structure, that is, from the quantum waves upon their corpuscles. In other words, since there is no a priori necessity for the majority of behaviors in the world, there is a “nomological necessitarianism” at play, which results from the fact that there is nomological encoding. It is, in fact, something of a double ontological effect. On one hand, the initial encoded information could have been about a few of many possible behaviors, thus being contingent. On the other hand, once the encoding is done, the propensity for such behavior will now depend on its encoded probability of occurrence. This points to a kind of “probabilistic necessitarianism” that gains or losses its ontological necessitarian strength upon the occurrence of behaviors, depending on the dynamics of the nomological information carrier structure, that alters the encoded probability distributions of those behaviors.

Regarding the causal relations that can be hypothesized to hold between events, one should justly remember Hume when he noted that “The mind can never find the effect in the supposed cause by the most accurate scrutiny and examination. For the effect is totally different from the cause, and consequently can never be discovered in it” (Hume 1999, p. 111). This, I believe, could be due to the fact that a causal relation can now be understood as being of a nomological probability nature, which is, of course, nomologically encoded in the quantum waves and, although affecting the occurrence of behaviors in a necessitarian probabilistic way, being of an a priori contingence nature.

Before moving on to the wave memory ontology, there is something important to be said about interaction schemes that are to be considered, in general, possible between corpuscles or different entities. Once one assumes that strong nomological physicalism is true, thus entailing a form of necessitarianism, one must also accept that a kind of nomological non-locality is also true. The reason for this is that, besides all nomological information about the corpuscles individual behaviors, the nomological information concerning all possible interactions between the corpuscles must also be encoded. This relational encoding being a sort of second order nomological information level, coming next in the encoding hierarchy of the nomological information about each corpuscle possible behaviors. As such, whenever a corpuscle stands in some relation towards other corpuscles, it must be that all relations in which that corpuscle could stand with all other, must already be encoded or can be encoded in the nomological memory. This must surely be true for all corpuscles and, in particular, for those corpuscles that act as signal carriers in spacetime background theories, such as special relativity. This finally means that each and every interacting corpuscle, once it has assumed a certain relata role within a certain relation, must be able to retrieve the nomological information about that particular relation concerning its and all other corpuscles relata roles. This strongly suggests that, at least, in a nomological sense, all corpuscles are informationally connected, which points to a sort of nomologic informational holism and therefore, to a sort of nomological non-locality. It also seems highly resonant with the non-locality aspect of the quantum potential, as identified by David Bohm in his wholeness conceptualization of reality. It is equally consistent with the fact, presented in Sect. 12.2, that the quantum potential is related with the Fisher information about the corpuscle possible positions, giving strength to some kind of encoding process in a quantum wave. It should be emphasized that the fact that the nomological information is retrieved locally by a corpuscle from a pilot-wave does not mean that such information cannot be of a holistic nature. In fact, if one takes seriously the necessitarian imposition, that is, the nomic necessitation, as can be called, of such information upon the corpuscles possible interaction schemas, then non-locality follows in a reasonable straightforward manner.

Another important problem concerns the so-called configuration space representation in standard quantum mechanics. The Pilot wave realistic approach to quantum mechanics, that is, one accepting the existence of a four-dimensional spacetime grounded quantum field, could tentatively be ruled out, given the higher dimensionality of the configuration space. It can indeed be argued that to have a complete description of the n-body physical case in quantum mechanics, one must write down the appropriate expressions in a 3n + 1 dimensional space. This describes a dynamic that cannot possibly be happening in a four-dimensional space, hence meaning that no four-dimensional quantum field exists for the n-body scenario in quantum mechanics. Much the same could be said about state superposition for one particle, but I will be only referring the configuration space objection. I suggest that this conundrum can be overcome if we admit that the 3n+1 dynamics that we are indeed describing concerns, not the actual quantum configuration states of the corpuscles involved, but the informational representation of those states that is encoded in a four-dimensional field. In other words, Bohrian quantum mechanics, the one used in the Copenhagen School interpretation formalism, describes the dynamics of the encoded nomological information about physical states and not the physical states directly. It is thus quite natural that such a dynamic should involve all possible states and configurations of the interacting corpuscles. In fact, the required completeness, characterizing an informational description of all possible states is the reason why such description brings about the so-called superposition of states and configuration space descriptions that dimensionally outnumber usual four-dimensional space descriptions. The former would then entail a kind of apposition of distinct nomological representations corresponding to different states, mutually exclusive or otherwise, which for reasons of informational completeness should be exhaustively enumerated in the description and that standard quantum mechanics takes for state superposition.

From what I have been reasoning, in the wave memory interpretation, one can defend a sort of epistemological completeness, saying that, while Pilot-wave theory describes the dynamic evolution of the nomological information carrier, which is a four-dimensional structure dynamic, standard quantum mechanics formalism describes the dynamics of the nomological information encoded is the wave carrier structures. It is therefore necessary to relate both dynamics in a way that the two theories become one, deriving the four-dimensional quantum field from the nomological information dynamics. Adding to these two dynamics one must also consider the dynamics of the corpuscle itself, interacting with the memory carrier wave, reading and storing information, as it actualizes its possible behaviors in four-dimensional spacetime. This last kinematic description of the corpuscle can be included in the pilot-wave theory description, since this last involves both the corpuscle and the quantum wave, as both physical entities exchanging information. In what follows I introduce an ontology trying to convey the ideas developed so far.

12.3.2 The Ontology of the Wave Memory Interpretation of Quantum Mechanics

In this sub-section I wish to present a first approach to an ontology that can guide further theoretical developments within the interpretation that I am proposing. It is not a short summary of postulates as the one found in Bohrian quantum mechanics. Such a synthetic approach can come later, when one is more assured of all implicit assumptions that, I hope, will serve as building conceptual blocks for a mature theory of this sort. What I am presenting here is more an attempt to cover as much assumptions territory as possible, trying to give solid ground to future theoretical contractions, which will make the description more elegant. I have made some effort to be as complete and thorough as I could, positing in a rather intuitive way the propositions about what I hypothesize things to be. The numerical hierarchical order of the posited axioms reflects, again in a very intuitive way, the relative semantic importance of the propositions, from my own point of view. What is submitted here is intended to serve as a possible roadmap to new ideas, and not as a definitive conceptual chart.

  • The ontology of the wave memory interpretation of quantum mechanics.

  • 1. Elementary particle axiom.

An elementary particle is a system composed by an elementary oscillator, called a corpuscle, and an extended ordered structure in space and time, called a quantum wave, which is finite. The wave is generated and reinforced by the corpuscle, its elementary oscillator.

  • 1.1. Objectivity axiom.

The corpuscle, its measurable properties, and the wave coexist objectively, at all times, independently of any act of measurement made by a conscious observer.

  • 1.2. Quantum field axiom.

A set of particles gives origin to an extended quantum field, which is a composition of quantum waves.

  • 1.3. Identity between spacetime regions and quantum fields.

An extended quantum field represents the most fundamental state of spacetime metrical order, involving corpuscles in space and time. As such it can be identified as a spacetime region populated by its corpuscles.

  • 1.4. Measurement axiom.

A corpuscle behavior, state or property is recognized trough what is informally said to be a measurement. The behavior of a quantum wave can be inferred from the behavior of its corpuscle. The behavior of an extended quantum field can be inferred from the behaviors of the corpuscles populating that field.

  • 2. Nomological encoding axiom.

A quantum wave is a physical structure that encodes the probabilistic distributions of all possible fundamental behaviors in space and time of the corpuscle. The quantum wave thus acts as a nomological memory (nomology, from the Greek: nomos, law + logos, reason, order). The probabilistic distributions are called the nomological information of the corpuscle behaviors and we say informally that these behaviors are encoded in the quantum wave.

  • 2.1. Pilot-wave axiom.

When a corpuscle expresses a given behavior, it does it so in a random and probabilistic way, accordingly with the nomological information encoded in the quantum wave. In this sense, we say informally that the corpuscle extracts nomological information from the quantum wave. Such a procedure is called the guiding effect, or the pilot-wave effect, exerted by the quantum wave on the corpuscle.

  • 2.2. Fundamental behaviors.

There are four fundamental behaviors attributable to a corpuscle by a given observer, which are its position x, its momentum p, its time t and its energy E.

  • 2.2.1. Properties x and p are said to be nomologically independent, that is, a priori, for each value x in a given x-set, any value of p in a given p-set can occur (t and E are also nomologically independent).

  • 2.2.2. Properties x and t are nomologically dependent, that is, for each value x in a given x-set, only one value of t can occur in a given t-subset (p and E are also nomologically dependent).

  • 2.3. Nomological information extremum conditions.

  • 2.3.1. Position nomological information extremum principle.

A corpuscle will tend to be in regions of its pilot wave, where the nomological information encoded about position is higher.

  • 2.3.2. Momentum nomological information extremum principle.

A corpuscle will tend to assume the states of movement within its pilot wave, where the nomological information encoded about momentum is higher.

  • 2.4. Interactions encoding axiom.

An extended quantum field encodes the probabilistic distributions of all possible interactions in space and time between its populating corpuscles. We say informally that the corpuscles interactions are encoded in the extended quantum field.

  • 2.5. Spacetime metric axiom.

An extended quantum field encodes, in the sense taken in 2, 2.1, 2.2, 2.3 and 2.4, all kinematic behaviors of the corpuscles populating the field. Consequently, the extended quantum field may be said to encode the spacetime metric of the region identified with that field.

  • 3. Epistemological completeness axiom.

Quantum Mechanics is a theory that uses, in a mutually non-exclusive way, two distinct descriptions that complement themselves and are consistently related, towards a complete description of quantum reality:

  1. (i)

    Pilot-wave description.

A description that uses the Pilot wave theoretical approach, dealing with the dynamics of the quantum waves, of the corpuscles, and of the interactions between the former and the later in four-dimensional spacetime. As such it is a theory about actual states, properties, and behaviors of quantum waves and corpuscles, and about how nomological information is exchanged between corpuscles and waves.

  1. (ii)

    Bohrian Quantum Mechanics formalism.

A description about the dynamics of the nomological information that is encoded in the quantum waves. As such it is a theory, not about the actual states, properties, or behaviors of corpuscles, but about their encoded representation in four-dimensional wavelike physical structures, the quantum waves, also called nomological structures or nomological wave carriers. The description about nomological information uses the algebra of standard quantum mechanics, with operators and matrix calculus.

  • 3.1. Conformity axiom.

All physical phenomena, including the dynamics of the nomological information, occur in four-dimensional spacetime and the three kinds of dynamics: of the nomological information, and of the corpuscles and quantum waves in four-dimensional spacetime, are consistently related.

  • 4. Necessitarianism axiom or Necessitarian principle.

Necessitarianism of the nomological information encoded in a quantum wave, or encoded in a quantum field, is the metaphysical principle according to which:

  1. (i)

    All possible behaviors of a corpuscle within its wave are the ones that either are already encoded or else can be encoded in a quantum wave,

And to which:

  1. (ii)

    All possible behaviors and interactions of all the corpuscles populating an extended quantum field are the ones that either are already encoded or else can be encoded in a quantum extended field.

  • 4.1. Nomological configuration.

A nomological configuration is a set of measurable values of a property which can be attributed to a given set of corpuscles. For each corpuscle there is a particular value, different from the corresponding values of all remaining corpuscles. A nomological configuration represents a possible interaction between two or more corpuscles.

  • 4.2. Configurational nomological information.

The configurational nomological information about a given property is the probability distribution of a set of possible nomological configurations. The configurational nomological information is encoded in an extended quantum field.

  • 4.3. Pilot-wave field axiom for the configurational nomological information.

Whenever a set of values is measured for a property in a given nomological configuration, we say informally that the corpuscles have extracted configurational nomological information from their populated quantum field. Such a procedure is called the guiding effect, or the pilot-wave effect, exerted by the quantum field on its populating corpuscles.

  • 4.4. Nomological nonlocality axiom.

Nomological nonlocality is the metaphysical principle by which, given a nomological configuration and its related property, all configurational nomological information for that property is always available to all corpuscles in all spatial locations, within the extended quantum field populated by those corpuscles.

  • 4.4.1. Nomological nonlocality axiom for the values in a nomological configuration.

Given that nomological nonlocality holds, nomological nonlocality for the values in a nomological configuration is the metaphysical assertion positing that, given a nomological configuration and its related property, all possible measurable values for that property can be expressed by any corpuscle in any spatial locations within the extended quantum field, populated by the corpuscles.

  • 4.5. Necessitarian enforcing from the configurational nomological information.

Necessitarian enforcing is the metaphysical assertion positing that, given a nomological configuration and its related property, given that necessitarianism holds and that nomological nonlocality holds, then the fact that a corpuscle assumes one possible measurable value for that property, necessarily implies that each of the remaining corpuscles will assume one and only one of the remaining values for the same property in the nomological configuration.

  • 4.6. Nonlocal and local physical phenomena contention.

Aprioristically there is nomological nonlocality, in the sense that all configurational nomological information must be completely available to all corpuscles, anywhere and at all times within an extended quantum field, in order to obtain physical phenomena. However, the way by which necessitarianism enforcing applies can be said to occur in two possible modes:

  1. (i)

    Physical nonlocality mode: the value of a given property assumed by a corpuscle is independent of any travelling signal going into the vicinity where that corpuscle is, and coming from any of the vicinities where the remaining corpuscles are,

  2. (ii)

    Physical locality mode: the value of a given property assumed by a corpuscle depends on at least one travelling signal going into the vicinity where that corpuscle is and coming from at least the vicinity of one of the remaining corpuscles. Each signal can also be considered a corpuscle expressing the signal suitable behavior that must be nomologically encoded in the extended quantum field.

  • 4.6.1. Nonlocality/locality experimental discernibility: in principle, it is possible to verify experimentally if an interaction either complies to Physical nonlocality or to Physical locality.

12.4 Nomological Encoding Formalism in Quantum Mechanics

In this section I will try to provide motivation for the way nomological information can be encoded in a quantum wave. I will assume that the world initially was or in some conditions can still be a chaotic primeval place where all possible behaviors, either nomologically encoded or encodable in a quantum wave, can occur. Possibly, with the exception of a minimum a priori nomological order ruling quantum wave dynamics and the guidage interaction. Although this minimum nomological order requisite corresponds, once more, to the imperative imposition of a necessitarian order, apparently coming out of nowhere, the metaphysical weight of such imposition is significantly lightened by what I am proposing here. In fact, it may even happen that a sort of nomological feedback process may exist between the corpuscle and the quantum wave, providing the quantum wave with the overall nomological information for its own behavior. A process that somehow would generalize guidage as a nomological informational closed loop circuit. This is a difficult problem that I will postpone for now.

In the first section of this work, I have tried to offer motivation for the memory interpretation of Quantum Mechanics, coming from the corpuscular aspect of quantum behavior, and thus conforming to the pilot-wave approach. In this section, I will try to do the same, starting from the mathematical Bohrian formal side, now conforming to the usual wave packaging in standard quantum mechanics, as it will become clear ahead.

Let us start the argumentation by considering a very simple description involving two measurable properties in the world, let’s say, P1 and P2. Let us also accept that the two properties are not correlated a priori, which means that they can, a priori, assume simultaneously any possible value from each of their respective domains \( {D}_{P_1} \) and \( {D}_{P_2} \). The properties are hence said to be nomologically independent, according to the ontology introduced in the last section (axiom 2.2.1). Let us also posit that such conjoint, although independent expression of pairs of values for (P1, P2), has been such over a period of time that it can be described by a joint probability function f:

$$ f\left({P}_1,{P}_2\right)=\frac{\partial^2F}{\partial {P}_1{\partial P}_2} $$
(12.15)

With the normalization condition for the joint domain \( D={D}_{P_1}\times {D}_{P_2} \), such that:

$$ {\iint}_Df\left({P}_1,{P}_2\right)d{P}_1{dP}_2=1 $$
(12.16)

The marginal probability density functions resulting from (12.16) will then be:

$$ \varPsi \left({P}_1\right)={\int}_{D_{P_2}}f\left({P}_1,{P}_2\right){dP}_2 $$
(12.17a)
$$ \varphi \left({P}_2\right)={\int}_{D_{P_1}}f\left({P}_1,{P}_2\right){dP}_1 $$
(12.17b)

The question now is how can we encode these marginal probability distributions in waves, using for the two properties in question, P1 ≡ x for position and P2 ≡ k for the wave number (related with momentum trough p = ℏk).

One possible way would be to use two distinct types of encoding waves e(x, k) and e′(x, k) as nomological information carriers, modulated in amplitude by each one of the marginal probabilities: h(x, ∆x) for x, and g(k, ∆k) for k. This can be formally represented using the following encoding scheme:

$$ \varPsi (x)=\frac{C\left(\Delta k\right)}{\sqrt{2\pi }}{\int}_{-\infty}^{\infty }g\left(k,\Delta k\right)e\left(x,k\right) dk $$
(12.18a)
$$ \varphi (k)=\frac{C^{\prime}\left(\Delta x\right)}{\sqrt{2\pi }}{\int}_{-\infty}^{\infty }h\left(x,\Delta x\right){e}^{\prime}\left(x,k\right) dx $$
(12.18b)

Where C(∆k) and C′(∆k) are scaling functions that depend on the spreading parameters ∆x and ∆k of the marginal distributions for position x and for the wave number k.

I will call (12.18a) and (12.18b) the encoding transformations for position and wave number (or momentum, using de Broglie’s relation). It must be said that in (12.18a) for each position x, the marginal distribution g(k, ∆k) controls the relative weight of the particular wave, carrying the nomological information about a possible value for k. That is, carrying the probability density value, according to which such value of k (or momentum) will occur for a corpuscle at that same position x.

In an analogous manner, in (12.18b), for each possible value of k (that is momentum, using de Broglie’s relation), the marginal distribution h(x, ∆x) controls the relative weight of the particular wave with that value of k, carrying the nomological information about a particular value for x. That is, carrying the probability density value, according to which such value of x will be the one corresponding to the position that a corpuscle will occupy, while exhibiting the same given value of k (or of momentum).

In all, one expects to have a large number of carrier quantum waves encoding the marginal probabilities for the occurrence of position and momentum. There will be the need for two types of nomological wave carriers, related according to the encoding transformations schema above. I must emphasize that although such a schema for one corpuscle can be imagined to be happening in four-dimensional space, the n-body case will demand a more sophisticated treatment, which I will not attempt here. Again, my main purpose in this work is only to suggest an alternative way of thinking about quantum mechanics.

The first encoding transformation (12.18a), furthermore, offers a correlational view (embedded in the sum provided by the integral) between a given position value and the momentum values nomologically available to that same position value. While the second transformation (12.18b) offers the correlational view between a given momentum value and the position values nomologically available to that same momentum value. By the expression “nomologically available” I mean that the quantum waves do encode what is possible to happen in the world (as stated in axiom 4, Sect. 12.3). It should also be mentioned that the heuristic strength of the encoding schema above is that it uses carrier quantum waves as the encoding building blocks. From (12.18a) one can in principle obtain the spatial geometry of the extended quantum field, since it is a position dependent function resulting from a sum of wave carriers, at each position.

We can now use the prior schema to encode gaussian distributions for the position x and for the wave number k (momentum). The formers are to be understood as the natural distributions for position values and for “state of movement” (or momentum) values that are a priori independent and nomologically available. That is, that can be expressed as actual behaviors by the corpuscles. Of course, one should consider that a position type behavior must equate to the possibility of existence of a corpuscle at some location within the quantum wave. While a state of movement behavior must be equated to the possibility of a corpuscle having a certain momentum, at some location within the quantum wave. In this state of affairs, we are not considering a state of movement as a ratio between space and time. It is to be considered as a fundamental property that has the same “ontological dignity”, so to speak, as position itself. State of movement, that I am equating to momentum, would then be an intrinsic existential property of a physical entity with the same grounding importance as the entity’s location or position. In other words, in the present framework a corpuscle’s existence implies that it has necessarily a position and a state of movement (or momentum), aprioristically and independently of any other condition and, in fact, of each other. That is the true and most profound meaning of axiom 2.2.1 in the ontology offered in Sect. 12.3.

Let us then write the marginal distributions for x and k with a gaussian form, according to:

$$ h\left(x,\Delta x\right)=A{e}^{-\frac{x^2}{2\Delta {x}^2}} \vspace*{-\baselineskip}$$
(12.19a)
$$ g\left(k,\Delta k\right)=A{e}^{-\frac{k^2}{2\Delta {k}^2}} $$
(12.19b)

The encoding carrier quantum waves will be:

$$ e\left(x,k\right)={e}^{ikx} \vspace*{-\baselineskip}$$
(12.20a)
$$ {e}^{\prime}\left(x,k\right)={e}^{- ikx} $$
(12.20b)

These are plane waves, infinitely spread throughout all space, in the same spirit held in Bohrian quantum mechanics formalism.

The scaling functions being:

$$ C\left(\Delta k\right)=1/\Delta k \vspace*{-\baselineskip}$$
(12.21a)
$$ C\left(\Delta x\right)=1/\Delta x $$
(12.21b)

To this we must now add supplementary encoding conditions, thus formally completing the encoding schema, which are given by:

$$ \varPsi (x)=h\left(x,\Delta x\right) \vspace*{-\baselineskip}$$
(12.22a)
$$ \varphi (k)=g\left(k,\Delta k\right) $$
(12.22b)

More generally, to obtain (12.22a) and (12.22b) one would have to take the modulus of Ψ(x) and φ(k), since their sums are waves and not real valued distributions. For the present purpose, however, what has been written will suffice.

After the proper substitutions we finally get:

$$ \varPsi (x)=\frac{1}{\Delta k\sqrt{2\pi }}{\int}_{-\infty}^{\infty}\varphi (k){e}^{ikx} dk $$
(12.23a)
$$ \varphi (k)=\frac{1}{\Delta x\sqrt{2\pi }}{\int}_{-\infty}^{\infty}\varPsi (x){e}^{- ikx} dx $$
(12.23b)

These are recognizably the Fourier transformations usually applied in quantum mechanics that inevitably lead to the Heisenberg uncertainty relations.

At this point, I should emphasize that the use of plane waves as nomological carriers in the proposed scheme is only a direct consequence of the fact that these are the natural solutions of the Schrödinger linear differential equation. If we were to consider a nonlinear differential equation, such as the one proposed by Croca in 2003 (Croca 2003, p. 72) (Eq. (12.3), Sect. 12.2), then a different type of nomological wave carriers would be formally feasible for the same encoding schema. We could, in fact, and alternatively use Morlet wavelet solitons as nomological information carriers, since these are the natural solution of Croca’s nolinear master equation.

Illustrating this possibility, let us now use wavelets of the form depicted in Eq. (12.4) with null average velocity and written at instant zero, contending that such simplifications will not stress the main ideas suggested. Still considering the same gaussian distributions (12.19a) and (12.19b), for x and k, we will now use as carrier quantum waves:

$$ e\left(x,k\right)={e}^{-\frac{x^2}{2{\sigma_x}^2}+ ikx} \vspace*{-\baselineskip}$$
(12.24a)
$$ {e}^{\prime}\left(x,k\right)={e}^{-\frac{k^2}{2{\sigma_k}^2}- ikx} $$
(12.24b)

Applying the same scaling functions (12.21a) and (12.21b) and the same encoding conditions as provided by (12.22a) and (12.22b), we will get, after the proper substitutions:

$$ \varPsi (x)=\frac{1}{\Delta k\sqrt{2\pi }}{\int}_{-\infty}^{\infty}\varphi (k){e}^{-\frac{x^2}{2{\sigma_x}^2}+ ikx} dk $$
(12.25a)
$$ \varphi (k)=\frac{1}{\Delta x\sqrt{2\pi }}{\int}_{-\infty}^{\infty}\varPsi (x){e}^{-\frac{k^2}{2{\sigma_k}^2}- ikx} dx $$
(12.25b)

It must be realized that the carrier quantum wavelets will have a gaussian shape controlled by the parameters σx and σk. That is, they are not idealized and (might I add, strictly metaphysical), plane waves. On the contrary, these quantum wavelet solitons represent finite physical four-dimensional entities, and their finitude is a realistic and natural attribute very much common to any kind of wave phenomena. Of course, we also must expect to find quantum wavelet solitons with several values of σx and σk, meaning that the encoding scheme is more complex than is being proposed here. However, for the purpose of this analysis, I will confine myself to the simplest case, where average constant values for σx and σk will be taken.

In the same spirit of the nomological interpretation of quantum mechanics that I have been advocating thus far, I will now make a formal connection between the encoding transformations (12.25a) and (12.25b), using mother wavelet solitons, and the position-momentum generalized uncertainty relation (12.5a) suggested by Croca (Croca 2003, p. 104). I will only consider the position-momentum uncertainty relation, presupposing that one can draw similar conclusions for time and energy, something that however I will not attempt here but that can also be found in (Croca 2003, p. 104). What follows is a somehow different version of the original derivation formulated by Croca (Croca 2003, p. 94).

Using the distributions (12.19a) and (12.19b), the encoding conditions (12.22a) and (12.22b), and the encoding transformation (12.25a), we can thus write:

$$ A{e}^{-\frac{x^2}{2\Delta {x}^2}}=\frac{1}{\Delta k\sqrt{2\pi }}{\int}_{-\infty}^{\infty }A{e}^{-\frac{k^2}{2\Delta {k}^2}}{e}^{-\frac{x^2}{2{\sigma_x}^2}+ ikx} dk $$
(12.26)

After integrating this gives:

$$ {e}^{-\frac{x^2}{2\Delta {x}^2}}={e}^{-\frac{x^2}{2/\left(\Delta {k}^2+\frac{1}{{\sigma_x}^2}\right)}} $$
(12.26a)

Which leads to:

$$ \Delta {x}^2=\frac{1}{\Delta {k}^2+\frac{1}{{\sigma_x}^2}} $$
(12.26b)

Solving for ∆k, we then obtain:

$$ \Delta k=\frac{1}{\Delta x}\sqrt{1-{\Delta x}^2/{\sigma}_x^2} $$
(12.26c)

From which, using ∆p = ℏ∆k, we finally get Croca’s position-momentum generalized uncertainty relation:

$$ \Delta p=\frac{\hslash}{\Delta x}\sqrt{1-{\Delta x}^2/{\sigma}_x^2} $$
(12.27)

If we use the encoding transformation (12.25b), following an analogous procedure we get:

$$ \Delta x=\frac{\hslash}{\Delta p}\sqrt{1-{\Delta x}^2/{\sigma}_k^2} $$
(12.28)

This last expression is not the same as (12.27), since the mother wavelet parameters σx and σk, in general, should differ. To deal with that difference, and seeking consistency, we must introduce yet another restriction that I will call the encoding density condition, reading:

$$ \frac{\Delta x}{\sigma_x}=\frac{\Delta k}{\sigma_k}=\rho $$
(12.29)

This condition will make (12.27) and (12.28) equivalent.

In expression (12.29), ∆x and ∆k are the intervals of all possible values for x and k, in a given physical situation for which position and momentum values can occur simultaneously. While σx and σkrepresent the diversity ranges of the carrier wavelets for all possible values of x and k, that can be encoded in the wavelets. Of course, formally, one always has ∆x < σx and ∆k < σk, making ρ < 1. We can therefore see ρ as a kind of encoding density or memory storage capacity for encoding nomological information in the wavelets, as it will become clear ahead.

As ρ → 1, that is, as ∆x → σx and ∆k → σk we can assume that the nomological memory gets saturated, and thus that a higher ordered “law-like” regime almost deterministic, will hold. On the other extreme, as ρ → 0, that is, as ∆x ≪ σx and ∆k ≪ σk, one can assume that the nomological memory is poorly populated, corresponding to a random like regime, with high indeterminacy for the possible pairs position-momentum assumed by the corpuscles in their behaviors. We will shortly return to these interpretations. The present understanding of the position-momentum generalized uncertainty relation presupposes that there is always a dynamic encoding-decoding process, happening between the corpuscles and the quantum carrier wavelets. One that alters the amount of nomological information encoded in the quantum wave carriers.

In what now follows I will use the position-momentum uncertainty relation to show that the encoding transformations (12.25a) and (12.25b), along with the encoding density condition (12.29), imply what can be called a nomological regime spectrum. This having pure chaos and utter determinism as two idealized extreme behavior situations for the corpuscles.

To make the exposition clearer, let us start by a discrete situation idealization, applying the intended interpretation to this situation as an example.

Consider two measurable properties P1 and P2 and their corresponding sets S1 and S2of possible values v1 and v2. The first set will have, by hypothesis, a number #S1 of possible values, while the second set will have #S2 possible values. If #S1 # S2 = N1N2 = M is the total number of possible values for P1 and P2 that can occur simultaneously, then for any two subsets S′1 ⊆ S1 and S′2 ⊆ S2, for the pairs of possible values (v1, v2) ∈ S′1 × S′2 one will always have:

$$ 1\le \#S{\prime}_1\#S{\prime}_2\le M $$
(12.30)

Now, in a more “orderly” situation, given any set S12 = S′1 × S′2, the number of possible pairs of values (v1, v2) should be less than M, since given a value v1 in S′1, not all values v2 in S2 will be available. In other words, given any subset S′1, only the values of a subset S′2 of S2 will be available, where #S′2 <  # S2. There will be “law-like” restrictions imposing that some pairs of values are possible, while others are not. Furthermore, the nomological strength of such restrictions will have an impact on the number #S′1 # S′2 of possible pairs of values, lowering that number in general, for any given set S12 = S′1 × S′2 of possible pairs (v1, v2).

If there is a one-to-one, biunivocal relation between the values in S1 and S2, then for any subset S′1(min) with just one value v1 (the minimum number of possible values for v1), there will be only a subset S′2 with only one value v2 for the only possible pair of values (v1, v2). And since each subset \( {S}_{1\left(\mathit{\min}\right)}^{\prime } \) and S′2 will have only one element, we will have in this situation:

$$ \#{S}_{1\left(\mathit{\min}\right)}^{\prime}\#S{\prime}_2=1 $$
(12.31a)

One can also write the complementary situation, for which, given a one element subset S′2(min), we will have:

$$ \#{S}_1^{\prime}\#S{\prime}_{2\left(\mathit{\min}\right)}=1 $$
(12.31b)

In fact, (12.31a) and (12.31b) express a situation of maximum nomological order or correlation between the values in available pairs of values (v1, v2). A situation that one could call deterministic, given that there is a one-to-one biunivocal relation between v1 and v2.

On the other extreme end of possible nomological scenarios, we can consider that given any subset S′1(min) with just one value v1 (the minimum number of possible values for v1), there will be several possible subsets S′2 of values for v2, such that we can have for the largest subset (coinciding with S2): #S′2(max) = N2,. In this case, one writes:

$$ \#{S}_{1\left(\mathit{\min}\right)}^{\prime}\#S{\prime}_{2\left(\mathit{\max}\right)}={N}_2 $$
(12.32a)

And of course, for the complementary situation, given a one element subset S′2(min), we can have:

$$ \#{S}_{1\left(\mathit{\max}\right)}^{\prime}\#S{\prime}_{2\left(\mathit{\min}\right)}={N}_1 $$
(12.32b)

Both expressions (12.32a) and (12.32b) mean that for any value of v1 in S1, all values for v2 in S2 are nomologically available, and vice-versa. In other words, all pairs of values (v1, v2) are possible to occur, meaning that there are no “law-like” restrictions imposing that some pairs of values are possible, while others are not. In this case, one should expect to observe high indeterminacy in the behaviors of physical entities, described by the measurable properties P1 and P2. This would therefore correspond to the extreme indeterministic situation.

If N1 = N2 = N, one can condense the situations in (12.31a), (12.31b), (12.32a) and (12.32b) and write:

$$ 1\le \#S{\prime}_1\#S{\prime}_2\le N,\kern0.75em \textrm{if}\kern0.5em {S}_1^{\prime }=\#{S}_{1\mathit{\min}}^{\prime }=1\ \textrm{or}\ {S}_2^{\prime }=\#{S}_{2\mathit{\min}}^{\prime }=1 $$
(12.33)

This expression would only be valid for any subsets S′1 and S′2 if some permanent nomological restriction would be at play, fine tuning the values of #S′1and #S′2 so that (12.30) would always get. That is:

$$ \left(\forall {S}_1^{\prime}\subseteq {S}_1,\kern0.5em \exists {S}_2^{\prime}\subseteq {S}_2\right) or\ \left(\forall {S}_2^{\prime}\subseteq {S}_2,\exists {S}_1^{\prime}\subseteq {S}_1\right):\kern0.5em 1\le \#S{\prime}_1\#S{\prime}_2\le N $$
(12.34)

This not being the case, the general expression describing all situations for S′1 and S′2 would be (12.30).

We will now consider the continuous case, that is, intervals of possible of values for the two measurable properties P1 and P2. Not surprisingly, we will also consider P1 and P2 to be position and momentum. At this point, it is quite straightforward to interpret the position-momentum generalized uncertainty relation (12.27) as an expression similar to the nomological case as expressed by (12.34). We can formulate the similar continuous situation, writing first:

  • For ∆x, ∆p intervals of possible values for x and p that are nomologically independent:

$$ \begin{aligned} &{Either}\kern0.75em \left(\forall \Delta x\subseteq {\mathbb{R}}^{+},\exists \Delta p\subseteq {\mathbb{R}}^{+}\right)\kern0.5em \\&{or}\kern0.75em \left(\forall \Delta p\subseteq {\mathbb{R}}^{+},\exists \Delta x\subseteq {\mathbb{R}}^{+}\right)\kern0.5em {such\ that{:}}\kern0.5em 0\le \Delta x\Delta p\le {\hslash}\end{aligned} $$
(12.35)

Using (12.27) to make the proper substitutions, we write secondly:

  • For ∆x, ∆p intervals of possible values for x and p that are nomologically independent:

$$ \begin{aligned} &{Either}\kern0.5em \left(\forall \Delta x\subseteq {\mathbb{R}}^{+},\kern0.5em \forall \rho \in {\mathbb{R}}_0^{+},\kern0.5em \exists \Delta p\subseteq {\mathbb{R}}^{+}\right)\\ &{or}\kern0.5em \left(\forall \Delta p\subseteq {\mathbb{R}}^{+},\kern0.5em \forall \rho \in {\mathbb{R}}_0^{+},\kern0.5em \exists \Delta x\subseteq {\mathbb{R}}^{+}\right)\kern0.5em {such\ that{:}}\kern0.5em \vspace*{-\baselineskip}\end{aligned}$$
(12.36)
$$ \Delta x\Delta p={\hslash}\sqrt{1-{\rho}^2}, \rho =\frac{\Delta x}{\sigma_x}=\frac{\Delta k}{\sigma_k},0\le \rho \le 1\kern0.5em and\kern0.5em 0\le {\hslash}\sqrt{1-{\rho}^2}\le {\hslash} $$

From (12.36) it is now possible to derive two extreme nomological situations. The first one, as before, in the discrete case, is expressed by (12.31a) and (12.31b), identified as the deterministic situation. This extreme nomological scenario corresponds to a very high encoding density ρ, and therefore to a scenario of high behavioral order. One that can ideally be equated with a physical situation of determinism. Still using (12.36), a possible formulation would be the following.

The deterministic nomological situation or Newton’s nomological situation.

  • For ∆x, ∆p intervals of possible values for x and p that are nomologically independent:

$$ \begin{aligned} &{Either}\kern0.5em \left(\forall \Delta x\subseteq {\mathbb{R}}^{+},\kern0.5em \forall \rho \in {\mathbb{R}}_0^{+},\kern0.5em \exists \Delta p\subseteq {\mathbb{R}}^{+}\right)\kern0.5em \\&{or}\kern0.5em \left(\forall \Delta p\subseteq {\mathbb{R}}^{+},\kern0.5em \forall \rho \in {\mathbb{R}}_0^{+},\kern0.5em \exists \Delta x\subseteq {\mathbb{R}}^{+}\right) \kern0.5em {such\ that{:}}\kern0.5em \vspace*{-\baselineskip}\end{aligned}$$
(12.36a)
$$ \Delta x\Delta p={\hslash}\sqrt{1-{\rho}^2},\rho =\frac{\Delta x}{\sigma_x}=\frac{\Delta k}{\sigma_k},\rho \approx 1\kern0.5em and\kern0.5em \Delta x\Delta p\approx 0 $$

Considering position, this proposition can be interpretated saying that whatever may be the interval of possible values for position (that can occur simultaneously with values of p) and given any available nomological wave carrier (characterized by the parameters σx or σk), the encoding density ρ will always be almost at its maximum (that is, near 1).

We can see σx and σk as nomological memory capacity parameters for, respectively, position and momentum. Those parameters, however, must be finite and not too large, when compared to the possible intervals ∆x and ∆p of a priori nomological independent values. If σx and σk happen to be too large, that would mean that in the given physical situation the majority of the nomological information will remain unencoded. Geometrically speaking, we can see from (12.4), Sect. 12.2, that when σx (or similarly σk) is very large, we obtain plane waves for the memory wave carriers. That is, the diversity of values encodable in the carrier shape becomes almost null, since we will no longer have a gaussian outline for the wave’s amplitude, this being, in fact, constant. Extended plane waves therefore correspond to low efficient nomological information wave carriers. In such a situation, the behavioral order will indeed be very low, giving way to indeterminacy. Alternatively, a situation of high behavioral order would mean that the available nomological memory has already reached its maximum, and hence that only a few values of position and momentum remain to be encoded. In other words, in that particular physical scenario, the occurrence of pairs of values position-momentum is under strong nomological restriction, carrying a lot of nomological weight upon the corpuscles behaviors, so to speak. The fact that ρ ≈ 1 then implies that for any pair of intervals, that were nomologically independent, one must have ∆x∆p ≈ 0, which in turn implies ∆x ≈ 0 and ∆p ≈ 0. These formulas express the fact that for any pair of nomologically independent intervals, both intervals must be exceedingly small. We can understand this by noting that in a situation of high nomological order, a few values for x will only pair with a few values of p, and vice versa. And, furthermore, that in the extreme possible case one would have a one-to-one biunivocal relation between position and momentum, corresponding to the idealized deterministic situation advocated in Newtonian mechanics.

The second possible extreme nomological situation corresponds to the indeterministic or Heisenberg case, where there are only a few nomological restrictions ordering the corpuscles behaviors, the nomological strength of such restrictions being exceptionally low. This will be the situation similar to the one, in the discrete case above, expressed by (12.32a) and (12.32b). From (12.36) we see again that the meaningful parameter influencing the nomological scenario is the encoding density ρ. So that if such density is exceptionally low, the nomological memory will be poorly populated, implying a low amount of encoded nomological information and, therefore, allowing for the occurrence of indeterministic behavior. We can formulate this situation, using a particular version of (12.36) as follows.

The indeterministic nomological situation or Heisenberg’s nomological situation.

  • For ∆x, ∆p intervals of possible values for x and p that are nomologically independent:

$$ \begin{aligned} &{Either}\kern0.5em \left(\forall \Delta x\subseteq {\mathbb{R}}^{+},\kern0.5em \forall \rho \in {\mathbb{R}}_0^{+},\kern0.5em \exists \Delta p\subseteq {\mathbb{R}}^{+}\right)\\ &{or}\kern0.5em \left(\forall \Delta p\subseteq {\mathbb{R}}^{+},\kern0.5em \forall \rho \in {\mathbb{R}}_0^{+},\kern0.5em \exists \Delta x\subseteq {\mathbb{R}}^{+}\right) \kern0.5em {such\ that{:}}\kern0.5em \vspace*{-\baselineskip}\end{aligned}$$
(12.36b)
$$ \Delta x\Delta p={\hslash}\sqrt{1-{\rho}^2},\rho =\frac{\Delta x}{\sigma_x}=\frac{\Delta k}{\sigma_k},\rho \approx 0\kern0.75em \textrm{and}\kern0.5em \Delta x\Delta p\approx {\hslash} $$

This proposition can be interpreted in the following way. Taking once more position as the property to reason with, whatever may be the interval of possible values that we consider for position x and given the available nomological memory (characterized by the parameters σx and σk), the encoding density ρ will always be low. This will be so, because the average length ∆x for the intervals of possible values, occurring for position, is still very small, once compared with the memory capacity of the nomological wave carrier (again characterized by σx or σk). That is, the nomological situation is such that a low amount of nomological information has been or is being encoded, in that particular physical situation. In such a case we have for the encoding density ρ ≈ 0, implying that ∆x∆p ≈ ℏ. From this we see that if ∆x is extremely small, the interval ∆p will be extremely large. This may be understood saying that whatever the value for position we may consider, this value can occur simultaneously with any value for momentum within an extremely large set of values for momentum. In other words, for a given value of position, any value of momentum goes. This nomological scenario corresponds to a situation of high indeterminacy. On the other hand, if ∆x is extremely large, the interval ∆p will be exceedingly small. This again can be similarly interpreted saying that, for any given value of momentum, any value of position can occur. This, again, translates into a situation of exceptionally low nomological order and high indeterminacy, as the one preconized in Bohrian quantum mechanics.

It should finally be reemphasized that when ρ ≈ 0, we get from (12.5a) (or else from (12.27)) and from (12.5b)) the standard Heisenberg uncertainty relations. In the present nomological interpretation, these last correspond to a situation where the nomological memory carrier waves are plane waves, according to what was indicated in (12.4), Sect. 12.2. Once more, in the given framework, this means that when only nearly plane waves are available as memory quantum wave carriers, the diversity of the encodable nomological information is very low. That is, to have high order correlation between position and momentum, there must be high nomological memory capacity, which corresponds to a nonhomogeneous amplitude outline shape of the carriers, as depicted in (12.24a) and (12.24b), and which depends on the parameters σx and σk. It should finally be mentioned that the usual writing of the Heisenberg position-momentum uncertainty relation is given by:

$$ \Delta x\Delta p\ge {\hslash} $$
(12.37)

In the context of the forwarded interpretation, this can be understood as the statement that in the Heisenberg case, the minimum measurement error of either position or momentum complies respectively either to:

$$ \Delta x\approx \frac{\hslash}{\Delta p} $$
(12.37a)

Or to:

$$ \Delta p\approx \frac{\hslash}{\Delta x} $$
(12.37b)

Only being larger due to experimental noise error. More to the point, for null memory storage density, one gets position-momentum uncertainty to be strictly due to lack of nomological information. Position-momentum uncertainty due to both lack of nomological information and other sources of measurement error will then provide the usual uncertainty relation (12.37).

In other words, we can also use the nomological interpretation applied to the introduced generalized uncertainty relations to distinguish between an idealized pure informational situation, involving quantum wave carriers, and the situation where one should also consider a noise error condition adding to the measurement.

12.5 Conclusion

In this article I have tried to provide motivation for a nomological interpretation of quantum mechanics, what I have called the “Wave memory interpretation of quantum mechanics”. A proposal holding that corpuscles and quantum waves do exist, simultaneously, at all times, formulated in the context of a possible improved Pilot-wave theory, where quantum waves are thought to be real physical four-dimensional structures, serving as encoding carriers for a fundamental type of information in Nature, that I have called nomological information (from the Greek: nomos, law + logos, reason, order).

To give a brief account of the main ideas involved, four-dimensional quantum wave carriers encode the probabilistic distributions of all possible fundamental behaviors of the corpuscles populating those waves. The particles then behave accordingly to the probabilistic distributions encoded in the waves, exchanging nomological information with them, trough what I have equated as a pilot-wave (guidage) effect. While the waves pilot the corpuscles, these generate and reenforce the former, hence acting as moving oscillators. Another foundational idea of this interpretation is that the complete description of quantum phenomena implies the use of two dynamics that are complementary in a on exclusive way and consistently related. One is the dynamic describing both the behavior of four-dimensional quantum carrier waves in spacetime, the movement of their populating corpuscles, as well as the interaction between waves and corpuscles. This is done by means of a Pilot-wave description model, where there is a guidage effect and where one uses a non-linear Schrodinger equation to predict the evolution of the quantum wave. The other is a dynamic describing the evolution of the nomological information encoded in the four-dimensional quantum waves, where one uses the algebra of standard quantum mechanics, applying operators functional calculus, matrix calculus, and the Schrodinger linear equation.

Getting more in detail about what has been devised, in the first section I have offered motivation for the wave memory interpretation from a realistic point of view, adopting the pilot-wave framework. As already stated, this theoretical approach implies accepting the existence of quantum waves along with their corpuscles, at all times, without the need for a consciousness collapse mechanism, bringing the corpuscles into existence. It also implies understanding the guidage or piloting effect exerted by the quantum wave upon the corpuscle, as a kind of nomological information transfer process, where the corpuscle acquires nomological information from the wave, about possible positions and possible momentum values. This information is stored at the location where the corpuscle happens to be.

In Sect. 12.3, I introduced what I have called a strong nomological constructivism, associated with probabilistic necessitarianism in natural laws. This implies upholding a picture of the world where behaviors initially occur at random, progressively informing the available nomological memory and being, in turn, enforced by it. As the nomological memory builds up, it then exerts a probabilistic necessitarian imposition upon the corpuscles behaviors.

In the same section, I have also defended what I have called an epistemological completeness owing to quantum mechanics. The idea that a complete theory about quantum phenomena should envisage, in a mutually non-exclusive way, two complementary descriptions. A description about the dynamics of the memory quantum wave carrier and a description about the dynamics of the nomological information stored in the wave carrier. The first description would also include the dynamics of the corpuscle, since this depends on the nomological information exchange between the wave and the corpuscle occurring in four-dimensional spacetime. The second description would concern the dynamics of the nomological information encoded in quantum waves. Such description, of a purely mathematical formal nature, would involve all possible physical states informationally encoded in the wave, demanding higher order dimensional spaces to accomplish informational representation completeness. Both descriptions should be thought as mutually non-exclusive and consistently related. In principle, it should be possible to derive the four-dimensional pilot-wave field from the Bohrian dynamics in a way that would be consistent either with state superposition description, or with the 3n + 1 description for the n body case in standard quantum algebra.

Also in Sect. 12.3, I have introduced a tentative ontology for the wave memory interpretation, trying to capture all ideas involved and, in particular, a feature that I have called “nomological non-locality”. The idea that a kind of necessitarian enforcement coming from the memory wave carriers, should be universally acknowledged by all corpuscles, independently of their spacetime location.

In Sect. 12.4, I have tried to give an account of a possible encoding schema for nomological information in quantum wave carriers. To that purpose I have suggested that the usual Fourier treatment applied to plane waves, in standard quantum mechanics, can be used to define such a schema. From there, I have applied the same encoding schema to Morlet solitons as quantum wave carriers, instead of using plane waves. This allowed me to derive Croca’s position-momentum generalized uncertainty relation (12.29), having Heisenberg’s uncertainty relation as a particular case. At the end of Sect. 12.4, I have reasoned in favor of the nomological interpretation of the generalized position-momentum uncertainty relation. The relation can be viewed as a measure of encoded nomological order, characterizing the nomological correlation strength between position and momentum, depending on the amount of encoded probabilistic information in a particular physical situation.

12.6 Future Work

To conclude this article, I wish to suggest future research topics that, in my view, can contribute to the development of the nomological interpretation of quantum mechanics. First, there is the problem of actually detecting a quantum wave, the so-called empty wave. That is, a quantum four-dimensional wave without any corpuscle. This, I believe, would be a crucial experimental result paving the way to new scientific findings and philosophical thinking about quantum phenomena. A set of experiments covering several predictable conundrums, performing such task, have been proposed in a recent paper, thus providing several experimental setups for a four-dimensional quantum wave detection (Croca et al. 2023).

A second major problem solution in favor of the present interpretation would be to determine what is the proper formal representation of how probabilistic nomological information is encoded in four-dimensional quantum waves. Critically, one has to formulate the proper representation of how nomological information concerning mutually exclusive physical states is encoded. That is, and putting it in a simpler form, it is necessary to determine how the so-called superposition of states, which consequently can now be understood as a sort of logical conjunction of information about such states, can be encoded in the pilot-wave carrier. In fact, and more generally, one has to determine how can any representation in configuration space be encoded in four-dimensional quantum wave carriers.

Finally, another very important task, bringing strength to the interpretation, implies determining the proper formal relation between the dynamic describing the pilot-wave and the corpuscle behaviors, and the dynamic describing the evolution of the probabilistic encoded information in the pilot-wave.

With the present work I hope to have provided some contribution, however minor, to the understanding of quantum phenomena and the possible formulation of an overall theory that can unify and improve the available formalisms describing quantum behavior.