Solutions to quantum puzzles. The nature of the three forms of quantum matter has explained several quantum puzzles, and Bohm’s interpretation of the Schrödinger wavefunction points the way to a solution of those that remain. We have seen how both photons and particles with rest mass have both a wave-like and particle-like nature, though they are fundamentally different forms of matter on this explanation and have fundamentally different explanations. Photons are waves that have a particle-like nature because each such bit of matter is a complete cycle of quantum events, whereas particles with rest mass have a wave-like nature because their motion is constituted by another form of matter attached to the rest mass that endures through time as a series of cycles of quantum events. This points the way to a certain kind of ontological explanation of quantum mechanics, and in order to test its adequacy, let us consider how it would handle the three quantum puzzles: the structure of the atom, Heisenberg’s uncertainty principle, and the Bell correlations.

Structure of the atom. Bohm’s interpretation of the Schrödinger equation is the key an ontological explanation of the structure of that atom. Schrödinger’s equation determines a wavefunction for the conditions that hold in atoms, with a positively charged nucleus surrounded by electrons (but since it is too complex to solve when many electrons are involved, each electron is usually treated separately, taking the mean position of the other electrons as boundary conditions). The time-independent Schrödinger wavefunction for the atom has an amplitude for the electron that varies with locations in space, and as Max Born suggested, the square of that amplitude (when normalized) in any region of space can be interpreted as the probability of finding an electron located there. The wavefunction describes various orbitals, or regions of space relative to the nucleus where two electrons (with opposite orientations of spin) are most likely to be found. This is the structure that explains the periodic table of elements and is used to explain chemical bonds among atoms.

The orbitals of the atom are identified by quantum numbers, such as the principle quantum number (indicating the energy levels: n = 1, 2, 3  . .), the orbital angular momentum quantum number (l = 0, 1, 2, . . ), and the magnetic quantum number (m, which determines the orientation of the orbital angular momentum as a magnetic moment it has in a magnetic field imposed in some direction). Electrons also have an intrinsic spin quantum number, s = ½, and two electrons, with opposite orientations of spin can occupy each orbital. Here is a rough description of the possible orbitals.

Electrons occupy shells, corresponding to different energy levels, and in the lowest energy shell (n = 1), there is only one orbital (the s orbital), which can contain two electrons (with opposite intrinsic spin). It has no orbital angular momentum (l = 0). The probability of finding the electron in the s orbital is highest at the center of the nucleus, and the probability of finding it farther away falls off exponentially.

In the second shell (n = 2), with the next higher permitted energy, there is not only an s orbital, but also three different p orbitals. The p orbitals correspond to electrons having an orbital angular momentum (l = 1, as if they were in orbit around the nucleus), and each such orbital has a node running through the nucleus, indicating that a p electron will never be found to be located where the nucleus is. Moreover, in the plane in which it has its orbital angular momentum, the real (that is, non-complex) component of the wavefunction’s amplitude has the p electron located in one or another region on opposite sides of the nucleus, that is, 180^o apart. Thus, since there are three p orbitals at the second energy level, atoms in which the second shell of electrons is full have (real valued) orbitals arranged in 3-D space that look like three, mutually perpendicular barbells.

In the third shell, at the next energy level, there is another s orbital, three p orbitals, and five d orbitals with a more complex geometrical structure, and so on through the energy levels of the atom. Since each orbital can contain two electrons (with opposite intrinsic spin orientations), the number of protons in the nucleus determines the structure of the lowest energy atom of each elemental kind.

In order to explain the structure of the atom ontologically, we need to recognize that it is constituted by three forms of matter and an interaction between them that can be seen as involving something in the nature of a photon (that is, virtual photons).

Rest mass matter. The particles with rest mass include the neutrons and protons that make up the nucleus as well as the electrons. But each proton and electron carries an electric charge, which is a form of force-field matter that helps constitute each particle, though as we have seen, the quantity of such matter is already counted in the rest masses of the particles.

Kinetic energy matter. Both the nucleus and the electrons are in motion as a result of their interaction, but the nucleus is so much more massive than the electrons that its quantum kinetic energy cycles are very small compared to those of electrons (and can be ignored in estimating quantities). Bohr assumed that electrons are in motion relative to the nucleus in order to explain the structure of the hydrogen atom, and despite doubts about electrons following determinate trajectories like classical material objects, it is clear that electrons have some kind of motion. (Electrons must move in order to have orbital angular momentum, and unless electrons in the s orbital had some kind of motion, there would be no explanation of how there could be s orbitals at higher energy levels.) Thus, according to this ontological explanation of the forms of matter, the electrons bound to the nucleus in an atom must have kinetic matter in addition to their rest mass matter, that is, the electrons are moved around by quantum kinetic cycles.

Force-field matter. Since protons and electrons carry opposite electric charges, they jointly impose a force field on the part of space occupied by the atom. The forces that these particles exert on one another change how they move, and the attraction of positive and negative charges is great enough to bind the electrons to the nucleus (with the negative potential energy representing the loss of some force-field matter that was counted in their rest masses as independent objects). But part of the force-field matter that the particles have given up still exists in the atom as the kinetic energy matter by which the electrons (and the nucleus) move across space as time passes, and the motion of electrons relative to the nucleus entails a change in the force field that is jointly imposed by them.

Virtual photons. The interaction between these particles is a process that is continually converting potential energy into kinetic energy and kinetic energy into potential energy, that is, converting matter between force-field matter and quantum kinetic cycles. Electrons (and the nucleus) are continually either giving up force-field matter and acquiring kinetic energy matter or giving up kinetic energy matter and acquiring force-field matter, and such transfers of matter are represented in the gauge field theory for electrodynamics as bosons, called "virtual photons."

The structure of the atom can be explained by the quantum nature of the kinetic energy matter of the particles with rest mass and the gauge bosons that transfer momentum and energy between them and force-field matter. Both the changes in the locations of the particles and the changes in the motion of the particles occur in a step like way, because they both involve quantum events. That can explain the structure of the atom, because those quantum events must fit together neatly in the spatio-temporal geometry determined by the inherent motion in space in order for them all to coincide with the same part of space. It is as if the quantum events constituting the atom were spatio-temporal bricks, and the existence of an atom were a result of their fitting together both spatially and temporally like a brick wall being built into the future. The masonry is so neat and well organized that the wall can be built indefinitely high, making the atom stable.

The quantum nature of kinetic energy matter means, as we are conceiving it in our possibly too crude way, that electrons (and nucleus) change location in a step-like way, that is, covering some whole distance in a period of time as a single, indivisible event. It is as if the electron must first complete an entire quantum kinetic cycle before it can change its momentum, and when it does change momentum, it must complete another complete quantum cycle before it can change again. Thus, only at certain locations and at certain moments does the electron change how it is moving.

Any changes that occur in an electron’s motion depends on the electric forces being exerted by all the electrons and protons, that is, on the field that they jointly constitute (because they are all made partly of force-field matter). These forces cause electrons to change how they move (that is, change their momentum), and that depends on some kind of (virtual) photon which gives the electron momentum and energy or takes it away. But on this model, such interactions occur only at the end of each quantum kinetic cycle, and it is a step-like change that determines the nature of the next quantum kinetic cycle. The quantum nature of the process makes the quantity of the change clear, because according to Newton’s laws of motion, the amount of energy and momentum that is transferred to the electron each time would depend on how much of energy and momentum the electron picked up from the force field matter in space during its previous quantum kinetic cycle. The change in the electron’s kinetic energy would depend on the distance it covered in the force field during the last kinetic event, and the change in the electron’s momentum (including its change of direction) would depend on the period spent being subject to the force field during the last quantum kinetic cycle. (Or more precisely, since the strength of the force varies over that distance and period, the change in energy would be the integral of the force over that distance, and the change in the electron’s momentum would be the integral of the force over that period of time).

This way of thinking about the quantum nature of the kinetic cycles may be an overly crude way of portraying the electromagnetic interactions, but the step-like changes bring out how the interaction involves not just the electron, but a complete quantum event making up its kinetic matter. The change occurs in a cyclic fashion, in which the last quantum kinetic cycle combines with the force-field matter to determine how much the next quantum kinetic cycle differs in momentum and energy. Such electromagnetic interactions are geometrically complex, because changes in electric forces cause changes in magnetic forces, which affect their motion, and what is more, these particles also have magnetic moments due to their intrinsic spin, which affects them in a different way. The way that these forces work is what is described by the gauge field theory for electrodynamics. The transfer of matter from force-field matter to kinetic matter or back is mediated by the gauge boson for the electromagnetic field, that is, by the exchange of a particle between them. This particle is like a real photon, because it is constituted by electric and magnetic forces interacting in some way. But it is unlike the photons that constitute light, because the momentum and energy it carries is not related by E = pc. They cannot have a constant proportion, because the energy and momentum needed to change the motion of objects with inertial mass as required by Newton’s laws do not have the same proportion in every case. (That is, momentum is a function of velocity, whereas energy is a function of the square of velocity, and so the proportion between them will vary with the velocity involved.) But this is just the nature of virtual photons, as opposed to real photons, which can exist independently and make up ordinary light. The matter constituting virtual photons can come from the force-field matter included in the rest masses of the particles, but they must have whatever unit-like nature is required to transfer all of the momentum and energy picked up from the force-field matter during the last quantum kinetic cycle at the moment that cycle ends, whatever the real nature of this possibly crude representation may be.

[When electrons do finally exchange a photon with the nucleus and their next quantum kinetic cycle is changed, they have a different location from where they were at the end of the last quantum kinetic cycle and their motion is changed for the next quantum kinetic cycle. This step-like change in their motion is the effect of virtual photons on the electron, but since the electron is a charged particles, it is also helping to impose the force field from which the virtual photons arises. And that is something that we must assume the electron does constantly, not just at the end of each quantum kinetic cycle, for as we shall see when we take up the gauge field theory, the electric charge is explained ontologically as a pulsation of electric forces emanating from the center of rest mass that is synchronized with electric charges throughout the universe. That is, all negative electric charges exert their maximum electric force at the same time in a cyclic way, and what makes positive electric charges opposite is that they exert their maximum electric force 180⁰ out of phase. (The synchronization of their pulsations is what is represented by their "orientation in complex vector fields," and the virtual photons of the gauge filed theory are the forces that must be exerted on charged particles as a result of their motion in order to conserve electric charge, that is, to keep their pulsations in synch with the universal pulsation of electric charges everywhere despite their motion.) In any case, in order to be able to explain quantum electrodynamics in this way, we will assume that electron is exerting its electric force in synch with the universal pulsation, and thus, it must occurs constantly during each quantum kinetic cycle. And that means that we are assuming that the electron has a determinate location at each moment during each quantum kinetic cycle. (For furthere discussion, see Change: Basic Objects: Gauge Field.)]

If the interactions among these forms of matter must have the unit-like nature that we are assuming explains Planck’s constant ontologically, the structure of the atom can be explained as a result of how all the kinds of quantum events involved fit together in the spatio-temporal geometry determined by the inherent motion in the part of space where they exist. This means that the interactions between the electrons and the nucleus would have a cyclic character, and all the interactions between electrons and the nucleus (as well as between the electrons themselves) would give them quantum kinetic cycles that are synchronized and related spatially, so that they fit together neatly in space and time like spatio-temporal brick in the atom as a brick wall being built into the future. But since there are slightly different combinations of momentums (quantum kinetic cycles) and positions (the locations where one quantum kinetic cycle ends and another begins), there is no way to say precisely where any particular electron is at any time.

Without trying to explain the orbitals in detail, it is clearly possible that the electrons are following determinate pathways as a result of interactions of this kind, changing their quantum kinetic cycles in a step-like way while all the time helping constitute the electromagnetic force field by way of their (pulsating) electric charges.

Though electrons in the s orbital are most likely to be found in the nucleus, that does not mean that they do not have a regular motion at all. Assume that each such electron is in a cyclic interaction with the nucleus in which it is accelerated first in one direction across the nucleus and then back in the opposite direction. The changes in how it moves come at the end of each lap when it is maximally far away from the nucleus, and it does not change its velocity during the trip, because it is a single quantum kinetic cycle (at least in the lowest energy state). That is, where one quantum kinetic cycle ends and another one begins, the electron changes its momentum all at once, without slowing down or speeding up. The reason it is most likely to be found at the center of the nucleus is that it can have any direction of back and forth motion through the nucleus, and the nucleus is the one part of space traversed by every possible pathway. At higher energy levels, the electron would be moving faster, and thus, it would have quantum kinetic cycles that are shorter and quicker, and at the n=3 energy level, it means that the electron has a good chance of being located either with the nucleus or at a distance from it, but not in between.

Electrons in the p orbital at the n=2 higher energy level have an orbital angular momentum. But it may seem that they cannot have a circular orbit around the nucleus in the relevant plane, because its orbital is usually represented as being a sphere located mostly on opposite sides of the nucleus. But the regions on opposite sides of the nucleus are just the real component of the amplitude its Schrödinger wavefunction, and the complex component puts it on opposite sides of the nucleus in the same plane, except for being rotated by 90⁰. The p orbital could, therefore, be a result of two quantum kinetic cycles, each trying to pull it back and forth across the nucleus in perpendicular directions (as in the s orbital), but perpendicular to one another. The quantum interactions with the nucleus that keeps changing their quantum kinetic cycles would have to be synchronized to occur 90⁰ out of phase to have this result, but that could be just the condition of such quantum events being able to coincide with the same part of space at all.

Even though different p orbitals are rotating electrons in independent planes of three dimensional space, they may also be synchronized in a certain way so as to keep the electrons from exerting too great a repulsive force on one another. (The general synchronization of these quantum kinetic cycles and changes in them is evident in the s electron at the third energy level, for its probable location is either outside the lower level shell or at the nucleus, suggesting that it is continually moving through those shells in some way.)

The reason that two electrons can fit into each orbital is that, with opposite orientations of spin, they can be synchronized in exactly the same way, but 180⁰ out of phase or in the opposite directions. Their opposite orientations of their intrinsic spins would exert a force (a "magnetic moment") that lines them up in opposite ways in the magnetic field, and that suggests that the magnetic fields plays the central role in making it possible for the exchange of virtual photons to generate such a neat pattern in the spatio-temporal geometry of the inherent motion.

Much more needs to be worked out in order to show how all the electrons in the orbitals could be following determinate trajectories determined by quantum kinetic cycles, but there seems to be no reason to deny that they have such step-like trajectories, even if they cannot be measured precisely. And it could be extended to include the other orbitals of atoms and the molecular orbitals that explain chemical bonds among atoms and groups of atoms.

Quantum jumps. Finally, the puzzle about the electron jumps entailed by the step-like changes in the energy level of atoms would be solved. All the changes in momentums of electrons, even those within its energy level, are step-like jumps. They occur at the end of one quantum kinetic cycle (in our possibly crude way of thinking about it) and before the next quantum kinetic cycle begins. It is clear that the change in energy state is a change in the orbital occupied by an electron is a step-like change, because it occurs with the absorption and emission of a single photon of the appropriate energy (and momentum). But that is just what would be expected, if the atom has a structure that is determined by the way that the quantum events of the various forms of matter constituting the atom must fit together in order to coincide with the same part of space given the spatio-temporal geometry determined by the inherent motion in space. The electron absorbs or emits a real photon, which changes its next quantum kinetic cycle so that it is part of a different orbital. The only possible changes are step-like changes, because they are changes in the structure of the atom.

The structure I have tried to describe here is the same structure that is determined by the “quantum potential” that David Bohm found in the Schrödinger wavefunction by mathematically separating out the classical forces. That left a force with a localized effect that did not decline with distance in the way electric forces do, but spread throughout space. Though Bohm thinks of it as “active information” which tells the electron how to play out its classical role, it can be explained, as I have suggested here, by recognizing that kinetic energy exists as a form of quantum matter by which objects with rest mass coincide with space, because that determines the same structure in the inherent motion in space. Quantum kinetic cycles and the inherent motion in which they are fit together are, in other words, another ontological explanation of Bohm’s quantum potential.

Lorentz distortions. By the way, this explanation of the structure of the atom affords an of the inevitability of the Lorentz distortions. In explaining the truth of Einstein’s special theory of relativity, I showed that the Lorentz time dilation and length contraction would be inevitable in the atom, if the electrons were bound to its nucleus by a unit-like two-way electromagnetic interaction. (See Change: Special theory of relativity.) That is apparently the implication of the Schrödinger wavefunction that describes the motion of such an electron subject to the positive charge of the nucleus, as can be seen in the s orbital.

The s orbital corresponds to a standing wave (as in a plucked string) without a node, and that means that the path of the electron is only half the total Schrödinger wavelength. (A standing wave of the complete cycle would have a node, because one half would be positive amplitude and the other half would have negative amplitude.) Since the momentum of an electron cannot change during a quantum kinetic cycle, it seems that either a single cycle of the wavefunction must be responsible for both legs of its trip across the nucleus, or else a complete cycle of the wavefunction is responsible for each leg. In either case, the electromagnetic interaction between the electron and the nucleus involves a two-way motion across the s orbital.

Such a two-way, unit-like interaction would cause Lorentz distortions in the atom, as explained in the discussion of special theory of relativity, because the inherent motion is what mediates changes in the force field (and quantum potential) caused by the electron motion. Thus, when the atom has a high velocity relative to the inherent motion, the periods of the cyclic interactions between the electrons and the nucleus increases (causing a time dilation), and the difference between the one-way velocity of light in opposite directions in space changes in the longitudinal distance across which they act (causing a length contraction).

As we have seen, the relativistic increase in inertial mass is simply the addition of quantum kinetic cycles to the rest mass cycles, which determines the scaling factor for quantum kinetic cycle and determines the force required to change its momentum. Thus, the quantum nature of matter affords an ontological explanation of the Lorentz distortions, which should eliminate the suspicion that they are simply ad hoc assumptions contrived to defend classical physics from the Einsteinian revolution.

Heisenberg’s uncertainty principle. The Heisenberg uncertainty principle holds that it is not possible to measure both  the position and momentum of a particle, or indeed both members of any pair of complementary variables, with arbitrarily high precision. According to the Copenhagen interpretation, this is because these classical properties do not describe the real nature of what exists at the most elementary level. Position and momentum are just properties we read into the world by using instruments designed to measure material objects according to principles of classical physics. Since both position and momentum are needed to predict what a classical particle will do, the Heisenberg uncertainty principle entails, at least, a limitation in what can be known, and it can be taken to mean that what exists at one moment does not determine what happens the next moment, or the denial of determinism.

The Heisenberg uncertainty principle is equivalent, as mentioned above, to the non-commutability of operators on the Schrödinger wavefunction:

When the Schrödinger equation is set up for a given situation, such as an atom or the two-slit experiment, the time-dependent Schrödinger wavefunction is a complete description of how interactions unfold over time. They unfold in a completely deterministic way, just like a classical wave function, except that the Schrödinger wavefunction uses complex numbers to describe the wave and it describes a wave in a configuration space with as many dimensions as three times the numbers of particles involved.

In order to make predictions from the Schrödinger wavefunction, mathematical operators must be applied. They generate real numbers as expectation values for the relevant property. But what is predicted is either just a mean value for many such measurements, that is, a probabilistic prediction, or if it does predict a precise value for the property involved, that property is one of a pair of complementary properties, and the other member of the pair cannot be predicted precisely.

In other words, classical properties come in complementary pairs that do not commute. The values predicted for such properties depend on which complementary operator is applied first. The application of an operator changes the wavefunction, so that the next operator is actually applied to a different wavefunction.

When a measurement is actually made, the quantum system turns out to have a property with a determinate value. The standard interpretation of what happens in such an measurement is called the “collapse of the wavefunction.”

What actually exists in the system represented by a Schrödinger wavefunction is assumed to be a superposition of all the states that might be revealed by a measurement. That is, states corresponding to all possible outcomes of measurements actually exist at the same time. Thus, what happens when a measurement is actually made is that the wavefunction collapses into one of those superposed states. The system is changed, and then another wavefunction describes the system, representing a different superposition of states.

Since there is nothing to determine which way the wavefunction collapses, this view denies determinism. In effect, it explains the truth of the Heisenberg uncertainty principle by the actual indeterminacy about what happens. 

There is, however, no collapse of the wavefunction, according to ontological explanations of quantum mechanics along the lines presented here. In any quantum system, every particle with rest mass has a determinate position and momentum and follows a classical trajectory, and measurements reveal properties that the system actually has. Instead of giving the system the measured property, as the “collapse of the wavefunction” interpretation implies, measurement discovers which property the system already had.

This way of interpreting measurements of quantum systems is entailed by an ontological explanation, because it explains the properties and regularities described by physics as aspects of the substances constituting the world (and if it is to be genuinely explanatory, it cannot depend on any randomizing factor assumed as part of the basic nature of the substances constituting the world). But the price of holding such a view is explaining why the Heisenberg uncertainty principle is true. And that can be accomplished by explaining why the operators corresponding to complementary variables are non-commutable.

The ontological explanation of complementarity is just the quantum nature of matter. What “quantum” refers to ontologically are the elementary events of which everything but space is composed. Each quantum event is a unit, which either occurs as a whole or not at all, and every such quantum event has the size of a single quantum of action, denoted by h, Planck’s constant. This explains, as we have seen, both the particle-like nature of photons as well as the wave-like nature of particles with rest mass. In the case of such particles, the complementarity comes from the quantum nature of their kinetic energy, that is, from the nature of the form of matter that changes the locations of particles with rest mass. Kinetic energy is constituted by quantum kinetic cycles, implying that the motion of a rest mass involves a series of cyclic quantum events, each of which is a unit of action that moves the rest mass across space a certain distance during a certain period of time.

What ultimately causes the Heisenberg uncertainty is the quantum kinetic cycle. The velocity of a particle with rest mass moving through space depends on the wavelength of its quantum kinetic cycle, but the particle can have a range of different positions in space at the beginning and end of each quantum kinetic cycle. That is, each quantum kinetic cycle involves a certain phase as well as a certain wavelength. But since the particle is located in a potential field, in order for its energy level to be fixed, a different location at the end of each cycle may require a slightly different wavelength the next cycle. Thus, the quantum state of the particle is some combination of wavelength and phase at its energy level, but there are many combinations that might satisfy those conditions.

Both complementary properties cannot be measured with arbitrary precision at the same time, because they are different aspects of the same bit of matter, which is a series of cycle of quantum events, each of which can interact only as a whole. Either it interacts in a way that reveals the wavelength of quantum kinetic cycle, which leaves its phase undetermined, or else it interacts in a way that determines its phase (that is, the position of the rest mass at the beginning or end of a quantum kinetic cycle), and its wavelength is undetermined. But both cannot be measured at the same time, because a quantum event interacts only as a whole. And complementary aspects cannot be measured is succession, because such interactions change the cycles of quantum events.

The Schrödinger equation describes the motion of particles with rest mass in a potential field where there is a continual exchange between kinetic energy and potential energy, and on this ontological explanation, the wavefunction that holds for any given system describes the quantum kinetic cycles that result for such an interaction. I have suggested what such an explanation implies about the atom and the two-slit experiment, but it can be generalized.

One way that the Schrödinger wavefunction is different from a classical wavefunction is that it is complex. There are complex numbers, involving the square root of minus one, that cannot be eliminated, and that makes its relationship to the actual world problematic. On this ontological interpretation, however, they represent the different possible phases of the quantum kinetic cycles constituting the momentum of the rest mass cycles. That is, on our crude interpretation, the starting points and ending points of the quantum kinetic cycles can have different locations in space and time and still be quantum kinetic cycles of the kind that can exist under those circumstances. The complex numbers are a mathematical device for representing all those different possible phases and keeping track of how they affect one another.

The other way in which the Schrödinger wavefunction is different from a classical wavefunction is that it describes a wave in a configuration space with three times as many dimensions as there are particles in the system, and that also makes its relationship to the actual world of three dimensions problematic. On this interpretation, however, each of the 3-dimensional spaces is used to keep track of how the phases of the quantum kinetic cycle a particle involved in the system unfolds in time. Though the quantum kinetic cycles of all the particles depend on classical forces and laws, each particle needs a 3-D space of its own in order to represent all its possible phases separately.

When a mathematical operator is applied to the wavefunction and a prediction is made about the value of some property, the different possible phases for all the particles are all reconciled with one another, working out the interference effects they have on one another. And the prediction is still usually just a mean value for many experiments, because there is a range of different states in which the system might be at that point, depending on which precise phases and wavelengths the quantum kinetic cycles actually had.

The reason that operators on the Schrödinger wavefunction do not commute is that they predict two aspects of the same quantum event, such as the wavelength and phase of the quantum kinetic cycle (as in the explanation of the Heisenberg uncertainty above). It is possible to predict a property precisely when it has already been measured once. But the wavefunction that represents the quantum system as having that precise property cannot be used to predict the complementary property of the particle precisely. For example, when a measurement of the momentum has been made, there is an operator that can be applied to the wavefunction that will predict the momentum precisely. But then the phase cannot be predicted precisely, because quantum kinetic cycles with that wavelength can have different phases. The same holds in reverse if the phase of the cycle is measured.

The “hidden variable,” on this explanation, is space and how bits of matter coincide with it, because the quantum nature of the kinetic energy of the particles is the factor that determines what happens to the particles. They need a complete quantum kinetic cycle to get from one place to another, and thus, at the end of each quantum kinetic cycle, the forces picked up during that cycle are what determines the next complete quantum kinetic cycle. The interaction is step-like, and though I may be portraying it too crudely by thinking of the quantum events as having definite beginning points and ending points, the requirement that particles travel across space by such cyclic quantum kinetic events is what needs to be added in order to see how what happens to the particle is determined.

On this ontological explanation, therefore, the quantum system is deterministic, and we can understand in principle how it is determined. But it is not possible to overcome the Heisenberg uncertainty because of the nature of the quantum kinetic cycles that constitute the motion of particles with rest mass. They exist only as a whole or not at all, and thus, they are the smallest unit that can interact with other bits of matter as a unit, which means in only one way at a time. That is, the uncertainty comes from an incompleteness about the representation of the Schrödinger wavefunction: it represents quantum kinetic cycles, but it does not reveal which of all possible combinations of wavelengths and phases is actual.

This incompleteness interpretation of the Heisenberg uncertainty solves the problem of Schrödinger’s cat. Such cases arise when the phases of the quantum cycles interfere in such a way that the system can unfold in radically different ways. For example, in one case Schrödinger’s cat is alive and well, and in the other case  it is dead. On the collapse of the wavefunction view, the Schrödinger wavefunction is a complete description of the situation, implying that what exists is a superposition of all the possible outcomes, and thus, since it turns out one way or another when someone looks, which one actually happens must depend on the measurement. But if which of the radically different alternatives is actual depends on the phases of their quantum cycles at the beginning, it is determined, and the uncertainty about what happens comes from that information not being included in the wavefunction representing the system. The incompleteness is inevitable, but that does not mean that it is indeterministic.

The phenomenon of tunneling can also illustrate the uncertainty. In tunneling, a charged particle moves past a force field that is classically strong enough to contain it. It occurs, for example, when there is a potential barrier separating electrons from protons attracting them that is just large enough to overcome the attractive force between them. But different electrons have different quantum kinetic cycles, setting up different patterns of spacetime cells in the inherent motion, and depending on whether they reinforce or cancel out the waves set up by the kinetic cycles of the protons, the force of attraction will sometimes be great enough for the electron to tunnel across the barrier.

The situation can be described by a Schrödinger wavefunction, which represents it as a packet of waves, each standing for a different possible combination of positions of the particles. As the situation evolves, however, the packet splits into two different parts, one in which electrons escape and one in which they do not. Thus, the equation represents two distinct channels, which subsequently do not interact. Which member of the packet is actual depends on precise locations and kinetic cycles of the particles (both wavelengths and phases). But they behave in a way described by the Schrödinger wavefunction because they follow the wave pattern set up by their kinetic cycles (See Bohm Ch. 5).

Bell correlations. The final quantum puzzle is the violation of the “Bell inequality” by certain quantum systems. John Bell pointed out that quantum theory predicts that there are correlations between distant events that cannot be explained without supposing that there is a causal influence of some kind that travels between them faster than the velocity of light.

Bell correlations occur when symmetrical particles, with opposite spin orientations, travel apart from one another in opposite directions and the spin of each is measured far away from the other. They always have opposite spin orientations when measured by imposing a magnetic field in the same direction in space. When one is up, the other is down. But the spin orientation they have in one direction of three dimensional space should not affect the spin orientation in either of the other two independent directions of space. And thus, the measurement of the spin of one of the separated particles in one direction should not affect the spin measured in the other particle in a different direction. Nevertheless, it is possible to use the measurement of the spin orientation of one of the particles in one direction to predict better than expected what spin the other particle will have when it is measured in an independent direction. That would be impossible, if spin orientation is a property that each particle has from the moment they separate and carry with them.

The greater than expected correlations are predicted by quantum theory. The prediction is made by applying the appropriate operators to the Schrödinger wavefunction for the system, and so the measurements are usually interpreted as involving a collapse of the wavefunction. That makes it seem as though the measurement of the spin of one of the particles helps determines which orientation of spin the other particle will have.

The Bell correlation is not only a prediction of quantum mechanics, but it has been confirmed by experiments.

Bohm (1993, Ch. 7) treats Bell correlations like any other puzzling phenomenon predicted by quantum mechanics, that is, as an indication of the quantum potential. Bohm is also giving an ontological explanation, but on his theory, the quantum potential is just a “non-local” aspect of the processes themselves, as if the common pool of information were broadcast faster than the velocity of light. Indeed, Bohm takes the world as a whole to have such a non-local aspect to it.

Non-locality seems to deny substantivalism about space, and that would make it incompatible with spatiomaterialism. If space is a substance, then what separates one part of space (and what happens there) from any distant part of space (and what happens there) are parts of space between them that have an existence that is distinct from both of them. Thus, the only way that this real separation between the parts of space can be overcome is by something traveling across space as time passes. To put it negatively, immediate action at a distance would seem to deny that there really is any substance between the distant points of interaction that is enduring through time distinct from them.

The inherent motion in space is a dramatic way of representing this fact about space as a substance. It is, perhaps, conceivable that Bohm’s non-locality is compatible with spatiomaterialism, because I have been speaking of the inherent motion in a more realistic way than may be necessary. That is, instead of thinking of space as containing an inherent motion, we can think of space as having a spatio-temporal geometry. Thus, what I have described as waves laid out in space by the inherent motion could likewise be just an aspect of the essential nature of space everywhere that always exists at the moment. That is, when the quantum kinetic cycle of a rest mass coincides with space, it has a certain wavelength and phase, and that wavelength and phase give it a different relationship to other parts of space with the same wavelength that are in phase with it than it does with those that are not in phase. Thus, what I have described as a particle “broadcasting” its wavelength and phase throughout space would be just a relationship that always already exists in the spatio-temporal geometry of space. If that were how the quantum potential is mediated, as Bohm assumes, it would explain the Bell correlation in the same way as other quantum phenomena.

I doubt that any such ontological explanation is adequate, however, because in order to explain interference phenomena in the two-slit experiment, for example, the quantum potential at any point in space would have to depend not only on the wavelength and phase of the particle, but also on the geometrical structure of the wall with two-slits. The waves laid out by the inherent motion that guide the particle to one of the fringes of the interference pattern must be singled out from all the other spacetime cells by the structure of the apparatus and how it fits together with the wavelength of the particle, and that would also have to be something about each location in space that always already exists for each possible arrangement of particles and wall with two-slits. This would be to attribute an enormously complex structure to the essential nature of space at every moment of its existence, and the complexity of such an explanation makes it look rather ad hoc. It would be a much simpler ontological explanation if the quantum potential were determined by an actual wave from the moving particle in the inherent motion that interacts with the two-slit wall, but that is not action at a distance.

There is, however, another explanation of the Bell correlations which is compatible with the principle of local action. Contrary to what many philosophers and physicists assume, what is actually known about this phenomenon does not force us to believe that the principle of local action is violated. There is a way of interpreting these phenomena that is compatible with explanation of the quantum potential by waves laid out in space by the inherent motion.

The predictions from quantum mechanics have to do with measurements of spin orientation, and they cover only those cases in which both events are actually measured. As a matter of fact, however, every experiment that can test Bell’s theorem involves many, many runs in which a measurement is simply not successfully made of one or the other particle (or of both particles). It is possible, therefore, that the cases in which both measurement are made are a biased sample. That is, if we could know the spin orientations in all the cases in which two particles split, it could turn out that their spin orientations in different directions were indeed independent and there is no Bell correlation.^[1]

Such a bias in the experiment cannot be just an accident. The many cases that must be ignored because no measurement was made must, for physical reasons, be mostly of a kind that, if included, would wipe out the improbable correlation between the distant events.

It may not seem like there can be any such factor, because the Bell correlations are predicted by quantum theory. That makes it seem that the Bell correlations are just another puzzling quantum phenomena, which manifest the same underlying mechanisms (whatever they are) as in any case of measurement. This is the assumption that is made in taking the correlation to involve the collapse of the wavefunction, except that unlike the other puzzling phenomena, it cannot be explained by the kinds of ontological causes described above, because Bell inequalities show that the collapse of the wavefunction involves action at a distance. That is, the hidden variable cannot be a local property, but must be a property that somehow holds of the whole system, including both particles, regardless how far they are apart at the time.

The prediction of the Bell correlation by quantum mechanics shows, however, only that some quantum phenomenon is involved. It may not, however, be the kind of phenomenon it is seems to be. The nature of intrinsic spin is not well understood, and it is treated as though it were completely described by the outcome of a measurement.

In the case of fermions, of particles with ½ spin, such as electrons, spin is measured by imposing a magnetic field and measuring the magnetic moment, that is, the force. The orientation of spin is simply the sign of that force, positive or negative: if the force is in one direction, it is spin up, and if it is in the other direction, it is spin down. Though that is how spin is measured, it is possible that particles have a more determinate spin orientation that is not measured in that way. An electron, for example, could have a precise orientation in three dimensional space, and though that is what determines the result of the measurement in the one direction that is singled out by the magnetic field applied, it also has other, more subtle effects on how the particle interacts.

In the case of photons, which are what has been used in the experiments that confirm the Bell correlations, spin is even more puzzling. Since the photon is a boson with a spin of 1, it should have three different possible orientations in a magnetic field, but since it moves through space with the velocity of light, one theoretically possible way of interacting is eliminated, leaving two possible orientations of spin. Opposite orientations of spin in the case of photons can be understood as opposite ways in which their electric force rotates as they move across space, one clockwise in the direction of motion and the other counterclockwise. However, it is usually measured by the polarization of the photon as it passes through a polarizer which is at rest and in which perpendicular directions, usually called vertical and horizontal, correspond to the two orientations of spin. But it is not clear why a rotation through a right angle would change whether a photon with, say, a clockwise rotation of its electric force, would pass through the polarizer.

In the case of both electrons and photons, there is enough uncertainty about the nature of spin and what is being measured that it is possible that the Bell correlations depends in some way on how spin orientation is measured. In either case, the three independent directions in which spin orientation (up or down or vertical or horizontal) can be measured are measured by an apparatus that is rotated in a two-dimensional plane perpendicular to the pathway of the particle. Thus, what may be a three-way symmetry among spins in three dimensional space is, in effect, reduced to a three-way symmetry in a two-dimensional plane. It is possible that in projecting that the three dimensional structure of spin orientations onto the two-dimensional plane of the measuring apparatus, some orientations of spin are more likely to pass by undetected than others, and they could be ones that would destroy the Bell correlation.

The selectivity may depend, furthermore, on an interaction between the actual orientation of spin in three dimensional space and the phase of its quantum kinetic cycle. Though the quantum potential that is responsible for interference and other real quantum phenomena requires a real effect propagating through space with the inherent motion, there could be an aspect of the waves set up in space by the inherent motion that makes all wavelengths with the same size and phase, wherever they exist in space, relate in a special way to the three dimensions of space. For example, the two particles have quantum kinetic cycles that are not only of the same wavelength, but also in phase with one another, and thus, if certain phases make it easier for them to interact from certain directions in three-dimensional space than others, the direction used by the detectors to test for spin orientation could result in a biased sample, making it appear that distant events are correlated. Such a factor would bias the sample in a way that makes it seem there are effect traveling faster than the velocity of light. And it would be local, because it depends only on the two particles having kinetic cycles that are in phase.

There is reason to think that some such explanation is correct, because Bell correlations occur only with measurements of spin orientation and the non-locality exhibited by the Bell correlations in measurements of spin is not an essential part of any other quantum phenomena. If it really were a result of action at a distance, it should be possible to make what happens at one location determine what happens elsewhere. But Bell correlations are not of a kind that can be used even to send signals from one place to another. In short, the Bell correlations are such a limited, subtle and questionable violation of the principle of local action that it would be foolish to use it as a reason for denying that spatiomaterialism can be used as an ontological foundation for a new way of doing philosophy, especially when that foundation works out so well in every other way.

Though much more would have to be said to show that this kind of ontological explanation of the nature of matter and space accounts for all the phenomena described by quantum mechanics, including quantum field theory and what it says about the nature of spin, this is enough to show that there is no good reason to believe that it is impossible to reduce quantum mechanics to spatiomaterialism. What is known by physics does not force us to give up the principle of local action entailed by this ontology, because neither experiment nor quantum mechanics is sufficient to demonstrate that the principle of local action does not hold. But this particular ontological theory is just a possibility introduced in order to speculate about a deeper explanation of the nature of matter and space, and what is relevant here is that, even this first approximation shows that there is no reason to believe that anything established empirically by quantum physics forces us to give up spatiomaterialism. There is at least one way that a two-substance ontology like ours can account for the quantum mysteries.

Let me emphasize, however, that it is not necessary to believe that what has been described here is completely accurate. It is only one of a family of ontological interpretations of quantum theory. What is common to the family is that the essential nature of matter involves the ability of bits of matter (of the same form) to exist independently of one another so that they can acquire spatial relations by being contained by different parts of space. There may be reasons for preferring another member of that family to this one. But this explanation of the quantum mysteries is enough to show that we do not have to give up the belief that space and matter are substances that exist continuously over time.