1. The Dual Ontology Conjecture and Synopsis

2. The Physical Structure of the Universe

3. Quantum States and the Dual Ontology

3.1. Single Quantum States and 4D Spacetime

The analysis begins with the dynamic evolution of a single quantum state in 4D spacetime. As the quantum state dynamically evolves in 4D spacetime, it occupies Planck Spheres. Each Planck Sphere occupied by a quantum state constitutes a Bell Quantum Sphere, and a one-to-one mapping and identity exists between a Bell Quantum Sphere in 4D spacetime and the same Bell Quantum Sphere in Planck Space (the "Bell Identity"). Consequently, the Bell Identity holds that the x, y, and z coordinates of a Bell Quantum Sphere not only map to the same Bell Quantum Sphere in 4D spacetime and Planck Space but also the physical attributes of the Bell Quantum Sphere are identical in both spaces.³⁷

Cumulatively, all the Bell Quantum Spheres occupied by a quantum state simultaneously form the quantum state's "Bell Quantum Field" in 4D spacetime and its ultra-high dimensional "Bell Quantum Hyper-Point" in Planck Space. For example, in the case of an electron in the ground state of a hydrogen atom, the 1.92 x 10⁷⁴ Bell Quantum Spheres that comprise the electron's Bell Quantum Field in 4D spacetime simultaneously form an ultra-high dimensional (3 x 1.92 x 10⁷⁴) Bell Quantum Hyper-Point in Planck Space.

As a quantum state spreads in 4D spacetime, the number of Bell Quantum Spheres occupied by the quantum state increases. The Bell Identity ensures that the increase in the number of Bell Quantum Spheres that comprise the quantum state's Bell Quantum Field in 4D spacetime is simultaneously reflected in an increase in the number of Bell Quantum Spheres that comprise the quantum state's Bell Quantum Hyper-Point in Planck Space. Accordingly, on a one-to-one basis, as the number of Bell Quantum Spheres occupied by the quantum state in 4D spacetime increases, the number of Bell Quantum Spheres comprising the quantum state's Bell Quantum Hyper-Point also increases.

3.1.1. The Collapse of a Single Quantum State

The Bell Identity not only links the dynamic evolution of a quantum state's Bell Quantum Field in 4D spacetime with its Bell Quantum Hyper-Point in Planck Space, but it also links the collapse of a quantum state's Bell Quantum Hyper-Point in Planck Space with its Bell Quantum Field in 4D spacetime. More specifically, the Bell Identity links the instantaneous decrease in the number of Bell Quantum Spheres that comprise a quantum state's Bell Quantum Hyper-Point in Planck Space with a simultaneous reduction in the number of Bell Quantum Spheres that comprise the quantum state's Bell Quantum Field in 4D spacetime.³⁸

For example, assume Box A is a three-dimensional box with impenetrable walls. Box A is 2 inches wide, 1 inch deep, and 1 inch tall. There is zero potential inside Box A. The instant a single quantum state ("quantum state A") is generated, it simultaneously forms Bell Quantum Field A in 4D spacetime and Bell Quantum Hyper-Point A in Planck Space. Quantum state A is inserted into Box A. The quantum state can spread throughout Box A but never has zero energy.

As quantum state A spreads, the Bell Identity ensures that the increase in the number of Bell Quantum Spheres that comprise Bell Quantum Field A is simultaneously reflected in a one-to-one increase in the number of Bell Quantum Spheres that comprise Bell Quantum Hyper-Point A. The opening of Box A triggers the instantaneous collapse of Bell Quantum Hyper-Point A.³⁹ The collapse instantaneously reduces the number of Bell Quantum Spheres that formerly composed Bell Quantum Hyper-Point A, and the Bell Identity ensures that the collapse is simultaneously linked to the instantaneous reduction in the number of Bell Quantum Spheres that now comprise Bell Quantum Field A. Quantum state A is now generally localized in Box A and begins to spread again.⁴⁰.

3.1.2. The Einstein/de Broglie Box Thought Experiment

The Einstein/de Broglie Box thought experiment is a more complex example of the dynamics of a single quantum state.⁴¹ Assume Box B is initially identical to Box A. A single quantum state ("quantum state B") is generated, simultaneously forming Bell Quantum Field B in 4D spacetime and Bell Quantum Hyper-Point B in Planck Space. Quantum state B is inserted into Box B and begins to spread. As quantum state B spreads, the number of Bell Quantum Spheres that comprise Bell Quantum Field B within Box B increases, and so does the number of Bell Quantum Spheres that form Bell Quantum Hyper-Point B in Planck Space.

Without becoming entangled with quantum state B, an impenetrable divider is inserted into the middle of Box B, separating it into two identical boxes, C and D. Ignoring the physical dimensions of the divider, Box C and Box D are now 1 inch wide, 1 inch deep, and 1inch tall. Inside Boxes C and D, the potential energy remains zero.

The insertion of the impenetrable divider has separated the quantum state into two equal Bell Quantum Fields in 4D spacetime: Bell Quantum Field C and Bell Quantum Field D. Nevertheless, the quantum state continues to form a single Bell Quantum Hyper-Point in Planck Space, now designated as "Bell Quantum Hyper-Point CD." The mathematical wave function ${\Psi }_{C,D}=\frac{1}{\sqrt{2}}\left({\Psi }_C+{\Psi }_D\right)$ indicates that the quantum state is in a superposition of location with a 50% chance that following the quantum state's collapse, the quantum state is found in Box B and a 50% chance that the quantum state is found in Box C.

Box C remains in Princeton, and Box D is shipped to Copenhagen, 6,252 kilometers away. Although the movement of Box D's physical location has altered the x, y, and z coordinates of the Bell Quantum Spheres that comprise Bell Quantum Field D in 4D spacetime, the Bell Quantum Spheres that comprise Bell Quantum Field C and Bell Quantum Field D continue to comprise Bell Quantum Hyper-Point CD. In other words, so long as Bell Quantum Hyper-Point CD does not collapse, the quantum state remains in a superposition of location with a 50% chance of being found in Box C and a 50% chance of being found in Box D.

Regardless of whether Box B or Box C is opened first, the opening of either box is the physical trigger that causes the collapse of Bell Quantum Hyper-Point CD. In turn, the collapse instantaneously reduces the number of Planck Spheres that formerly comprised Bell Quantum Hyper-Point CD. The reduction in the number of Planck Spheres that formerly comprised Bell Quantum Hyper-Point CD has several significant ramifications.

If Box C is opened first and the entire quantum state ψ_c (x₁,y₁,z₁)_c is found in Box C, 1) the number of Planck Spheres that comprise Bell Quantum Field C and Bell Quantum Hyper-Point C have been instantaneously reduced, 2) Bell Quantum Field C is generally localized in Box C, and 3) there is a one to one mapping and identity between the Bell Quantum Spheres that comprise Bell Quantum Field C in 4D spacetime and Bell Quantum Hyper-Point C in Planck Space. In addition, the mathematical wave function formerly representing Bell Quantum Field D and Bell Quantum Hyper-Point D no longer has meaning. A similar analysis applies if Box C is opened and the quantum state is not found in Box C or if Box D is opened first and the quantum state is or is not found in Box D.

Finally, the collapse of Bell Quantum Hyper-Point CD is not a "jump in 4D spacetime."⁴² Recall that a Bell Quantum Hyper-Point is an ultra-high dimensional point in a Planck Space where time, space, and volume do not exist. When Bell Quantum Hyper-Point CD collapses, there is a reduction in the number of dimensions that comprise the quantum state's new Bell Quantum Hyper-Point, and the Bell Identity ensures that there is a simultaneous reduction in the number of Bell Quantum Spheres that comprise the quantum state's new Bell Quantum Field. What appears to be a "jump" in 4D spacetime is the collapse of Bell Quantum Hyper-Point CD in Planck Space and the reduction in the number of Bell Quantum Spheres that comprise the quantum state's Bell Quantum Field.⁴³

3.2. N-Body Quantum States

The introduction of the Dual Ontology, the dynamic evolution of all quantum states in 4D spacetime, and their collapse in Planck Space sets the stage for re-examining the ontic status of N-body quantum states. Instead of relying on a 3N configuration space to describe the dynamic motion of N-body quantum states, the Dual Ontology conjecture posits that all N-body quantum states are physical entities that dynamically evolve in a 4D spacetime and collapse in Planck Space.⁴⁴

3.2.1. Evolution and Collapse of N-Body Quantum States

The issues raised by the dynamic evolution of an N-body quantum state in 4D spacetime and its collapse in Planck Space are highlighted by a re-examination of the Bohm version of the EPR experiment.⁴⁵ A pair of electrons is prepared in the singlet state. The wave function of the entangled quantum state is in a superposition, and its combined angular momentum is 0. The instant the entangled quantum state is generated, the quantum state simultaneously forms two Bell Quantum Fields in 4D spacetime (Bell Quantum Fields E and F, respectively) and an ultra-high dimensional Bell Quantum Hyper-Point ("Bell Quantum Hyper-Point EF") in Planck Space. Quantum state E is sent toward Princeton, and quantum state F is sent toward Copenhagen. Testing equipment is configured to conduct a Stern-Gerlach experiment in the z direction on either Bell Quantum Field E or Bell Quantum Field F.

Regardless of the space-like separation of Bell Quantum Fields E and F in 4D spacetime, the Bell Identity ensures that the Bell Quantum Spheres that comprise Bell Quantum Fields E and F also comprise Bell Quantum Hyper-Point EF. As Bell Quantum Fields E and F dynamically spread in 4D spacetime, the number of Bell Quantum Spheres that comprise their respective Bell Quantum Fields increases, as does the number of Bell Quantum Spheres that comprise Bell Quantum Hyper-Point EF.

Bell Quantum Hyper-Point EF collapses following a Stern-Gerlach experiment in the z direction in either Princeton or Copenhagen.⁴⁶ The collapse instantaneously reduces the number of Bell Quantum Spheres that formerly composed Bell Quantum Hyper-Point EF. The Bell Identity ensures that the collapse is simultaneously linked to the reduced number of Bell Quantum Spheres that comprise Bell Quantum Field E and Bell Quantum Field F, respectively. Unlike the case with a single Bell Quantum Hyper-Point, Bell Quantum Hyper-Point EF now forms two independent Bell Quantum Hyper-Points designated as Bell Quantum Hyper-Point E and Bell Quantum Hyper-Point F. Bell Quantum Hyper-Point E shares a one-to-one mapping and identity with the Bell Quantum Spheres that now form Bell Quantum Field E, and Bell Quantum Hyper-Point F shares a one-to-one mapping and identity with the Bell Quantum Spheres that now form Bell Quantum Field F. Following the collapse of Bell Quantum Hyper-Point EF, Bell Quantum Hyper-Point E and Bell Quantum Hyper-Point F form a product state rather than an entangled state ψ_1,2((x₁,y₁,z₁)_1,(x₂,y₂,z₂)₂) → ψ₁(x₁,y₁,z₁)₁ ⊗ ψ₂(x₂,y₂,z₂)_2,t). Finally, the reduction in the number of Bell Quantum Spheres that comprise Bell Quantum Field E and Bell Quantum Field F instantaneously causes the generalized localization of both Bell Quantum Field E and Bell Quantum Field F regardless of how spread out each of the individual Bell Quantum Fields may have been before the collapse of Bell Quantum Hyper-Point EF.

If the Stern-Gerlach experiment in the z direction is conducted in Princeton on quantum state E, and the quantum state is found in the z spin-up direction, quantum state F is instantly in the z spin-down direction in Copenhagen. Conversely, if the Stern-Gerlach experiment in the z direction is conducted in Princeton on quantum state E and the quantum state is found in the z spin-down direction, quantum state F is instantly in the z spin-up direction in Copenhagen. The same analysis applies if the Stern-Gerlach experiment in the z direction is conducted on quantum state F in Copenhagen instead of quantum state E in Princeton.

3.2.2. The Double-slit Experiment

The double-slit experiment and its multiple incarnations have been reviewed extensively by physicists and philosophers. Notwithstanding the numerous controversies surrounding the experiment, the experimental results of the various double-slit experiments are subject to the same ontological structure and physical laws that govern the collapse of all quantum states under the Dual Ontology conjecture.

Individual electrons ("quantum states") are sent one at a time towards "Wall (W)" with two narrow Gaussian slits (A) and (B). In comparison with a quantum state's wavelength, slits (A) and (B) are sufficiently narrow so that a quantum state diffracts if it passes through slits (A) and (B). Some of the individual quantum states that move towards Wall (W) hit the wall, while others pass through slits (A) and (B). Once a quantum state passes through slit (A), it spreads uniformly as a spherical wave toward a detector "Detector (D)." Similarly, as the quantum state passes through slit (B), it too spreads uniformly as a spherical wave toward Detector (D). The two spherical waves interfere as they spread toward Detector (D). A single flash occurs whenever a quantum state hits Detector (D). Only one flash is detected per quantum state. After many flashes, Detector (D) indicates an interference pattern.

When each quantum state is generated, it forms a Bell Quantum Field in 4D spacetime and an ultra-high dimensional Bell Quantum Hyper-Point in Planck Space. Accordingly, when a quantum state is fired toward Wall W, the quantum state is simultaneously comprised of a Bell Quantum Field in 4D spacetime and a Bell Quantum Hyper-Point in Planck Space. As the quantum state travels toward Wall W, it spreads spherically in 4D spacetime but continuously forms a single Bell Quantum Hyper-Point in Planck Space. A one-to-one mapping and identity exists between the Bell Quantum Spheres that comprise the quantum state’s Bell Quantum Field in 4D spacetime and the quantum state's Bell Quantum Hyper-Point in Planck Space.

As the quantum state passes through slits (A) and (B), both slits independently diffract the quantum state, and the quantum state begins to spread from both slits towards Detector D in a spherical pattern. The Bell Quantum Spheres of the quantum state that passes through slit A forms Bell Quantum Field A in 4D spacetime, and the Bell Quantum Spheres of the quantum state that passes through slit B form Bell Quantum Field B in 4D spacetime. The quantum state forms a single Bell Quantum Hyper-Point AB in Planck Space. Mathematically, the quantum state's wavefunction is typically described as ${\Psi }_{ A+B}=\frac{1}{\sqrt{2}}\left({\Psi }_{ A }+{\Psi }_{ B}\right)$.⁴⁷

Before reaching Detector D, the two wave functions combine linearly, and the linear combination causes interference. The modulus squared of the wavefunction is greatest where the linear combination is constructive and approaches zero where the interference is destructive. Whether or not the mathematical wavefunction of the quantum state forms a peak or a trough, the quantum state always continues to form a single Bell Quantum Hyper-Point AB in Planck Space.

In 4D spacetime, a flash anywhere on Detector (D) indicates that a quantum state's Bell Quantum Hyper-Point has physically collapsed. The collapse of a quantum state is an instantaneous physical process in Planck Space. Following its collapse, the quantum state forms a single Bell Quantum Hyper-Point. However, the number of dimensions that comprise the quantum state's new Bell Quantum Hyper-Point is reduced to a subset of the Bell Quantum Spheres that formerly comprised its Bell Quantum Hyper-Point. Simultaneously, the Bell Identity ensures that the collapse of the quantum state's Bell Quantum Hyper-Point is, on a one-to-one basis, linked to an instantaneous reduction in the number of Bell Quantum Spheres that now comprise the quantum state's single Bell Quantum Field in 4D spacetime.

Regardless of how spread out the quantum state's Bell Quantum Field may have been in 4D spacetime before its collapse, and regardless of the interference caused by the interaction of the diffracted waves spreading from slits (A) and (B), the collapse of a quantum state's Bell Quantum Hyper-Point ensures that the entire quantum state is generally localized in 4D spacetime. The collapse of the quantum state to a generally localized position anywhere on Detector (D) is instantaneous, and the mathematical wave function of Bell Quantum Spheres that are no longer occupied by the quantum state following its collapse has no meaning.

Following many flashes caused by the collapse of individual electrons, an interference pattern emerges on the Detector (D). The interference pattern is the combined result of the collapse of individual Bell Quantum Hyper-Points in Planck Space and the simultaneous reduction in the number of Bell Quantum Spheres that comprise each quantum state's Bell Quantum Field.

3.2.3. A Simple Which-Way Experiment

Historically, "which way" experiments have compounded the theoretical issues of double-slit experiments. Once again, however, from the perspective of the Dual Ontology conjecture, the theoretical analysis of a which-way experiment is the same as it is for all other collapse-based quantum events.

The baseline experiment repeats the double-slit example in section 3.2.2 but is modified by inserting a proton into an empty rectangular box at the center of Wall (W).⁴⁸ The proton is positively charged, and each electron ("quantum state") fired towards Wall (W) is negatively charged. Assuming the experiment is 100% reliable, slit (A) flashes if the proton is attracted towards slit (A), and slit (B) flashes if the proton is attracted towards slit (B). Experimentally, 50% of the flashes are recorded at slit (A) and 50% at slit (B) 50%.

As an electron spreads in 4D spacetime, it forms a Bell Quantum Field in 4D spacetime and simultaneously forms a single Bell Quantum Hyper-Point in Planck Space. If slit (A) flashes, the electron's Bell Quantum Hyper-Point collapses, and the number of Bell Quantum Spheres that comprise the electron's Bell Quantum Hyper-Point is reduced. Simultaneously, on a one-to-one basis, the Bell Identity ensures that the number of Bell Quantum Spheres comprising the electron's Bell Quantum Field is reduced. The electron is now generally localized at slit A. The analysis is the same if slit B flashes rather than slit A.

Once the electron is generally localized at either slit (A) or slit (B), it will once again continue to spread towards Detector (D). Critically, however, since the electron passed through either slit (A) or slit (B), but not both, an interference pattern is not formed at detector D. Conversely, if neither slit (A) nor slit (B) flashes, as the electron passes through slits (A) and (B) the analysis of the double-slit experiment described in Section 3.2.2 above is once again applicable.

4. Physical Considerations

5. Special Relativity and Quantum Mechanics

6. Quantum Path Irreversibility and The Arrow of Time

7. Quantum Cosmology, Cosmogony, and the Dual Ontology

8. Conclusion

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