NOTE: this is an early-stage work-in-progress, and assertions are made here that are about plans and speculations, without being properly contextualized as such (otherwise it would all be impossible to read).
The Spinfield Model represents an attempt to understand the same phenomenology as the Standard Model from a mechanistic perspective, in terms of the cellular automaton (CA) and pilot-wave frameworks (specifically the de Broglie double solution models; Darrow & Bush, 2024; Darrow, 2025). As in many other cases, this alternative model could provide a more satisfying physical model of the same phenomena described by the calculational tool of the Standard Model. The CA framework introduces a natural ultraviolet cutoff scale, which can potentially explain the success of the renormalization procedure, and the mass hierarchy issues with the Higgs mechanism.
Based on the considerations discussed in mechanisms, the Spinfield Model has discrete massive particles that interact dynamically with distributed quantum wave fields, which in turn interact with distributed force fields like the electromagnetic force. Given that these mechanisms obey all the same conservation laws as the Standard Model, there is some basis for optimism.
The discrete particles are represented using a field of harmonic oscillator values localized to individual CA cells, using complex numbers (i.e., a CVHO) to represent the spin and other associated properties of stochastic motion. These oscillators drive and couple with a physically real quantum wave field (in 3D CA space, not configuration space) and the EM field. Charge and velocity parameters arise in this model in terms of phase differences between CVHOs (as in Standard Model wave functions for charge / probability), and forces can shift these phases to directly affect the velocity.
Much of the rest mass of a particle emerges from these spinning oscillators, providing the potential to explain the mass hierarchy across generations of particles, and between charged and uncharged leptons (i.e., electrons and neutrinos). Furthermore, different configurations of spinning oscillators provide a sufficient basis space for the different known types of particles, providing a framework for understanding how these particles can transmute into each other (e.g., as described by the weak interactions), and can be converted to and from raw energy (i.e., high frequency oscillations in the EM field).
The use of CVHOs to represent discrete particles coupled to radiative wave-based fields implements the essence of the pilot wave framework, while keeping everything within the same overall mathematical framework of oscillatory wave-like dynamics. The localization of the CVHOs to an individual CA cell is essential to keep the “accounting” of particle properties in order: once you let any of these properties leak out across multiple cells, the dissipative nature of wave dynamics irrevocably causes them to dilute away.
However, a system based purely on discrete, cell-localized particles cannot effectively produce force-field interactions among particles: the odds of any other discrete particle “hitting” another are very small, whereas these big sloppy waves provide a continuous, broadly diffusing and saturating medium for force transmission. In effect, the radiative wave field is essential for the particle to be able to properly “sense” the force field effects, functioning like an antenna or “whiskers” in sensing forces over a broader space, beyond its own singular cell.
These “mechanistic” considerations from the CA framework show how the otherwise strange wave-particle duality represents a hybrid “best of both worlds” solution, relative to something based exclusively on discrete particles, or exclusively on continuous wave functions.
Relative to the standard QM interpretations, the pilot-wave nature of this approach avoids the impossible conundrums of instantaneous wave collapse at the point of measurement, because the discrete particles are always localized to a discrete CA cell, even when you’re not looking. The primary challenge for such a framework is dealing with the apparent non locality of quantum entanglement.
The real quantum wave
The solution to the non-locality issue is associated with the physically real quantum wave that is generated by discrete particles, and influences the velocity phase dynamics of the particle CVHOs. The Dirac wave equations for spin 1⁄2 fermions provide the Standard Model representation of such particles. The Spinfield Model effectively factorizes this equation into localized particle CVHOs that capture the internal spin dynamics and charge generation properties, plus a radiative quantum wave that drives interactions with other particles (including itself).
Once we have a solid foundation for the CVHO spin dynamics, it should be possible to essentially subtract that from the Dirac wave function, with the remainder specifying the quantum wave function properties. However, the existing Bohmian pilot-wave model is based entirely on the Schrodinger wave function, operating instantaneously and non-locally over the exponentially-large configuration space (Norsen et al., 2015).
Thus, we need to adopt a pragmatic approach to the properties of the residual quantum wave for the Spinfield Model, such that it can account for the nature of the interactions otherwise modeled in configuration space (see Norsen, 2022 for ongoing work in this area). For example, if it needs to propagate at some multiple of the speed of light in order to capture the seemingly non-local interactions, then that is far more acceptable and easily implemented within the CA framework relative to actual non-local interactions, which are completely intractable. See non-locality for more discussion.
Weak interactions
The initial source of constraints on the particle aspect of the Spinfield Model derives from the complex and indeed somewhat bizarre phenomenology of neutrinos and the weak interactions more generally. For example, the weak force is understood to drive the decay of a heavy muon lepton into a lighter electron, emitting two neutrinos in the process. It also drives beta decay of neutrons (made of quarks).
The massive bosons of the weak force quickly decay into various fermion particles, and they only interact infrequently, and at very short distances, with anything else. Thus, it is unclear exactly what these things really are. There are also lots of strange angles and other parameters associated with the weak force, along with parity violations that may reflect the fundamental ordering of harmonic oscillator updating.
If it is possible to capture the behavior of these weak interactions in a more principled manner, that would certainly represent a major advance, even though we already have very accurate descriptions of this behavior. In general, the Standard Model framework of spontaneous symmetry breaking and electroweak integration seems like an attempt to dress up something that is fundamentally complex and “ugly”. And further attempts to push supersymmetry as a general principle have failed. Thus, perhaps the Spinfield Model can provide a way of seeing why these weak interactions are actually functionally essential, within the constraints of the CA framework.
Stochastic particle motion
A major challenge for the CA framework is representing the motion of discrete particles. First, there is a basic coordination problem for the parallel update process, where the destination cell needs to receive the particle properties from the source, while the source needs to clear out these properties. Second, the cubic lattice makes it impossible to achieve isotropic motion along any vector direction using deterministic, local (Markovian) patterns of discrete jumps.
Stochastic motion is required to accomplish isotropic motion, using random jump probabilities along each axis to accomplish isotropic motion at a macroscopic level. As discussed in that page, certain forms of stochastic motion naturally produce the Schrödinger wave function as an expectation distribution for the location of a discrete particle (Nelson, 1966).
This intrinsic stochasticity accounts for an important component of the nondeterministic behavior of quantum physics, as reflected in the probabilistic nature of the Copenhagen interpretation and the Born probability rule. However, an additional source of apparent stochasticity is due to the inability to accurately measure the initial state of any quantum system, as captured by the Heisenberg uncertainty principle.
Both the intrinsic stochasticity and the uncertainty in initial states represent purely epistemic components of the Standard Model wave function. Thus, these components must also be taken into account in formulating the nature of the residual physically real quantum wave.
Particles making waves
Within the Standard Model, the electron and other charged particles are described as point particles, and yet these infinitesimal points must somehow generate the corresponding EM wave field. This is known as the back-reaction problem from the particle onto the field, and it remains a thorny and unsolved problem (Rohrlich, 2000; Rohrlich, 1999; Rohrlich, 1997; Ford & O’Connell, 1991; Ford & O’Connell, 1993; Baez, 2020). It has also been discussed within the pilot-wave framework, for the generation of the real quantum wave field (Holland, 2006). The phenomenon of zitterbewegung, where the particle spin interacts with the generated waves, is also relevant (Hestenes, 2008; Hestenes, 1990; Sidharth, 2009; Roman et al., 2003; Barut & Bracken, 1981; Wang & Zhang, 2001).
The discretization of space imposed by the CA framework helps resolve the infinity problem, but creates another problem at the same time, by virtue of the discrete, stochastic motion dynamics within the CA grid. These discrete jumps create high-frequency perturbations from the driving of the wave by the particle, and due to the Planck relation, such high-frequency perturbations have very high energy levels.
Thus, it is necessary to impose some kind of local smoothing and distribution of particle properties across a local neighborhood, to eliminate these high-frequency perturbations.
Interestingly, the properties of the weak force appear particularly well-suited for solving this problem, because they are very short range (on the order of \(10^{-17}m\)) and drop off exponentially. These are precisely the properties that would enable a local smoothing of the area surrounding a discrete particle, so that the entire neighborhood of cubes on the lattice is already prepared in advance of wherever the particle might move. This also provides an important constraint on the length scale for the CA cube sizes.
The uncertainty principle
The discrete particle localization of the pilot wave framework already violates the strict interpretation of the uncertainty principle, in that the particle position is always precisely determined at the underlying CA level. However, any observer within this CA system cannot directly access this position information. Instead, the purely epistemic factors due to stochastic motion and uncertain initial state, combined with the influence of the quantum wave on particle motion, result in a level of uncertainty that should be generally consistent with the uncertainty principle.
Nevertheless, for the specific case of the conjugate variables of momentum and position, these are both represented at the level of the discrete particle CVHOs, which raises the theoretical possibility of some kind of violation of the most strict form of the uncertainty principle. Although the quantum wave directly influences the particle velocity (momentum) in a manner consistent with standard wave functions, perhaps there might be some way to access the underlying particle-level values. Perhaps weak measurements might be related?
Discrete photons
Although there are compelling arguments in favor of the strong distinction between discrete localized fermion particles and purely wave-based radiative waves for the EM field, there remains a non-zero possibility that some kind of discrete particle-like aspect of the EM field, i.e., the photon, might actually be necessary to account for all of the known physics.
In this case, the photon could be an excitation in a harmonic oscillator field, like it is in the QED field theory model. However, so many aspects of EM radiation, e.g., very long wavelengths distributed over huge macroscopic length scales, appear to be strongly incompatible with this framework. By contrast, particles have Compton wavelengths in the \(10^{-14}m\) length scale, and the stochastic motion of low-momentum particles creates a corresponding wide distribution of positions. As the momentum increases, the frequency also increases, which narrows the corresponding waves.
Thus, there would have to be a very compelling reason to explore a discretized basis for photons, whereas it is very natural for fermions.
Summary of model constraints
To summarize the above, the fundamental constraints shaping this model are as follows:
• A discrete particle such as an electron is the source of radiating wave fields, as described by the Maxwell EM equations for charged particles, and a physically real quantum wave field, which accounts for the wave nature of particles, as revealed in the double-slit experiment, for example.
• A particle must be discretely localized to a single CA cell at all times, in order to make the accounting work.
• Discrete particle localization, combined with the cubic lattice grid of the CA, requires stochastic motion to achieve isotropic motion, where the discrete CA grid effectively disappears and supports motion in any continuous direction.
• Stochastic motion in a discrete grid results in discrete jumps of the particle at a discrete time step, which generally results in a high-frequency perturbation in any wave fields that the particle is the source of.
• Therefore, the discrete particle must continuously distribute its source-generating properties across surrounding cells, thereby creating a smooth distributed source density, where the discrete jumps thus have minimal impact, only shifting the source. This spreading of particle properties should not have wave-like oscillatory dynamics, but instead should be diffusive in nature, and should have an exponential falloff to retain particle localization. These properties apply to the weak force. This distribution process must take place at a supraluminal update rate, to accommodate particle motion that is close to the speed of light.
• The discrete particle properties (charge, 3 velocity vector) are represented using the complex phase relationships among multiple CVHOs (see charge from phase), so the distribution process involves synchronizing these phase relationships to those of the discrete source particle.
• In effect, a particle thus appears as a distributed blob of phase offsets among the spinfield of CVHOs.
• There is one reference source CVHO, that entrains the others: R0. The real and imaginary components of R0 can have one of two different directions of rotation around the imaginary plane: clockwise or counterclockwise. This differs for particles vs. antiparticles, especially of relevance to the neutrino, which has no other charge values, and thus this rotation direction represents the only distinction between a neutrino and an antineutrino.
• According to the (second-order) Dirac equation, the Pauli spinors only affect the interaction with the EM Maxwell field vectors (\(\vec{E}\) and \(\vec{B}\)). Thus, the particle state variables themselves do not directly spin on their own.
• The frequency of rotation of R0 could be driven by the relativistic total energy of the particle (rest mass + kinetic energy), in accordance with Planck relation and the Compton wavelength. Thus, R0 drives the fundamental frequency of the corresponding quantum wave. Alternatively, it may be the case that we need to fix this frequency for some reason(s), e.g., to rotate entirely between a to b in one step, which is what the Pauli spinors seem to do.
• The charge CVHO T0 oscillates in either -90, +90 or 0 phase relation to R0 (for leptons), producing -1, +1, or 0 charge from the resulting charge computation between R0 and T0. For neutrinos, the phase is always 0. For electrons / positrons, there is a redundancy between the direction and phase values, with both being opposite in the antiparticle – it is theoretically possible that there could be clockwise vs. anticlockwise versions of electrons with the same phase relations, for example. But the neutrino case seems to anchor the need for a fundamental spin direction in R0 itself. Maybe it would be possible for R0 and T0 to spin in opposite directions too? That would seem to imply graded charge values, except perhaps in the full-step spin (spinor) case? This can be easily tested and resolved. It is a bit awkward that R0 needs to entrain the rate of T0 in the total energy case.
• The momentum CVHOs X, Y, Z oscillate with the corresponding phase relationship to R0 to represent the normalized velocity 0-1 relative to c. With R0 spinning at the relativistic energy frequency, the ability to drive changes in this phase could potentially naturally produce the relevant relativistic momentum effects? And a key principle is that each value is phase-boosted as a function of the local quantum wave function, as the main force conveyance effect.
• This distributed spin-up of phases also has qualitative similarities to the Higgs field in the Standard Model, which is likewise associated with the weak dynamics. Therefore, all of these known weak / Higgs properties, which are also where the CP asymmetries arise (consistent with the basic asymmetry in integrating CVHO variables), provide the constraining data that the Spinfield Model must explain.
• What is the ground, zero-point state of the spinfield? It consists of CVHO oscillators that are presumably continuously oscillating, while decaying away whatever particle states they last encountered. As a discrete particle moves through, it spins-up these oscillators, entraining them to represent its momentum and charge properties, which then serve as the source of the radiative wave fields.
• This zero-point field can potentially account for the virtual particles that are only transiently extant, within the QED Standard Model framework, i.e., the “quantum foam” associated with Feynman diagrams and the path integral framework.
Source particle diffusion
Standard diffusion equations in 3D result in a significant dissipation and dilution of the source signal across space, as they effectively take the average across all of the neighborhood states. As a new source signal spreads, it quickly gets averaged out with all the existing residual state values.
One way to combat this is to also propagate a distance from the source signal, and then to preferentially use those states that are closest to the source. This leads to a rapid, preferential spread of the source signal.
The basic wave equation represents a fundamental interaction between space (the spatial gradient, i.e., the momentum operator in the Hamiltonian) and time (acceleration within a single grid element, i.e., the Hamiltonian energy operator). In this context, consistent with special relativity, a massive particle lives strictly in the time component. This can be seen in the Klein-Gordon equation:
\[ \frac{\partial^2 {\phi}}{\partial t^2} = c^2 \nabla^2 \phi - \frac{c^2 m_0^2}{\hbar^2} \phi \]
where the mass-dependent factor enters without any spatial gradient operator, and directly subtracts away from the energy factor (acceleration) on the left-hand side. This lack of spatial extent is consistent with a discrete particle entity, that somehow consumes energy from the wave field in proportion to its rest mass. This is the qualitative picture for how our discrete particles emerge in the Spinfield Model, with the energy consumption propelling an internal oscillation within the discrete cell in the cubic lattice of the cellular automaton (CA) framework.