# Symmetry, Transactions, and the Mechanism of Wave Function Collapse

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## Abstract

**:**

## 1. Introduction

“I do not believe that this fundamental concept will provide a useful basis for the whole of physics.”

“I am, in fact, firmly convinced that the essentially statistical character of contemporary quantum theory is solely to be ascribed to the fact that this [theory] operates with an incomplete description of physical systems.”

“One arrives at very implausible theoretical conceptions, if one attempts to maintain the thesis that the statistical quantum theory is in principle capable of producing a complete description of an individual physical system …”

“Roughly stated, the conclusion is this: Within the framework of statistical quantum theory, there is no such thing as a complete description of the individual system. More cautiously, it might be put as follows: The attempt to conceive the quantum-theoretical description as the complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems. In that case, the whole ’egg-walking’ performed in order to avoid the ’physically real’ becomes superfluous. There exists, however, a simple psychological reason for the fact that this most nearly obvious interpretation is being shunned—for, if the statistical quantum theory does not pretend to describe the individual system (and its development in time) completely, it appears unavoidable to look elsewhere for a complete description of the individual system. In doing so, it would be clear from the very beginning that the elements of such a description are not contained within the conceptual scheme of the statistical quantum theory. With this. one would admit that, in principle, this scheme could not serve as the basis of theoretical physics. Assuming the success of efforts to accomplish a complete physical description, the statistical quantum theory would, within the framework of future physics, take an approximately analogous position to the statistical mechanics within the framework of classical mechanics. I am rather firmly convinced that the development of theoretical physics will be of this type, but the path will be lengthy and difficult.”

“If it should be possible to move forward to a complete description, it is likely that the laws would represent relations among all the conceptual elements of this description which, per se, have nothing to do with statistics.”

**Transactional Interpretation of quantum mechanics**(TI), developed over several decades by one of us and recently described in some detail in the book

**The Quantum Handshake**[12]. [We note that Ruth Kastner has extended her “probabilist” variant of the TI, which embraces the Heisenberg/probability view and characterizes transactions as events in many-dimensional Hilbert space, into the quantum-relativistic domain [13,14] and has used it to extend and enhance the “decoherence” approach to quantum interpretation [15].

**Collective Electrodynamics**[16], followed by a deeper dive into the electrodynamics of the quantum handshake, and finally includes descriptions of several historic experiments that have excluded entire classes of theories. We conclude that the approach described here has not been excluded by any experiment to date.

#### 1.1. Wheeler–Feynman Electrodynamics

#### 1.2. The Transactional Interpretation of Quantum Mechanics

## 2. Physical Mechanism of the Transfer

“It is generally assumed that a radiating body emits light in every direction, quite regardless of whether there are near or distant objects which may ultimately absorb that light; in other words that it radiates ’into space’…”

“I am going to make the contrary assumption that an atom never emits light except to another atom…”

“I propose to eliminate the idea of mere emission of light and substitute the idea of transmission, or a process of exchange of energy between two definite atoms… Both atoms must play coordinate and symmetrical parts in the process of exchange…”

“In a pure geometry it would surprise us to find that a true theorem becomes false when the page upon which the figure is drawn is turned upside down. A dissymmetry alien to the pure geometry of relativity has been introduced by our notion of causality.”

## 3. Quantum States

“The hypothesis that we have to admit is very simple, namely that the square of the absolute value of Ψ is proportional to an electric density, which causes emission of light according to the laws of ordinary electrodynamics.”

**neoclassical**(NCT) because of its use of Maxwell’s equations. While there was no question about the utility of NCT in the conceptualization and technical realization of amazing quantum-optics devices and their application, there was a deep concern about whether it could possibly be correct at the fundamental level—maybe it was just a clever bunch of hacks. The tension over this concern was a major focus of the 1972 Third Rochester Conference on Coherence and Quantum Optics [31], and several experiments testing NCT predictions were discussed there by Jaynes [30].

“We have not commented on the beautiful experiment reported here by Clauser [32], which opens up an entirely new area of fundamental importance to the issues facing us…”

“What it seems to boil down to is this: a perfectly harmless looking experimental fact (nonoccurence of coincidences at 90°), which amounts to determining a single experimental point—and with a statistical measurement of unimpressive statistical accuracy—can, at a single stroke, throw out a whole infinite class of alternative theories of electrodynamics, namely all local causal theories.”

“…if the experimental result is confirmed by others, then this will surely go down as one of the most incredible intellectual achievements in the history of science, and my own work will lie in ruins.”

**Tests of Bell’s Inequality**[3,4] and/or

**EPR experiments**[6,7,8,9,10,11]. Both the historic EPR experiment and its analysis have been repeated many times with ever-increasing precision, and always with the same outcome: a difinitive violaton of Bell’s inequalities. Local causal theories were dead! [Much of the literature on violations of Bell’s inequalities in EPR experiments has unfortunately emphasized the refutation of local hidden-variable theories. In our view, this is a regrettable historical accident attributable to Bell. Nonlocal hidden-variable theories have been shown to be compatible with EPR results. It is locality that has been refuted. Entangled systems exhibit correlations that can only be accomodated by quantum nonlocality. The TI supplies the mechanism for that nonlocality.]

**Collective Electrodynamics**[16]. As we detail below, the quantum handshake, as mediated by advanced/retarded electromagnetic four-potentials, provides the effective non-locality so evident in modern versions of these EPR experiments. In Section 13, we analyze the Freedman–Clauser experiment in detail and show that their result is a natural outcome of our approach. Jaynes’ work does not lie in ruins—all that it needed for survival was the non-local quantum handshake! What follows is an extension and modification of NCT using a different non-Maxwellian form of E&M [16] and including our non-local Transactional approach. We illustrate the approach with the simplest possible physical arrangements, described with the major goal of conceptual understanding rather than exhaustion. Obviously, much more work needs to be done, which we point out where appropriate.

#### Atoms

**bound states**. The spatial shape of the wave function amplitude is a trade-off between getting close to the proton, which lowers its potential energy, and bunching together too much, which increases its ${\nabla}^{2}$ “kinetic energy.” Equation (1) is simply a mathematical expression of this trade-off, a statement of the physical relation between mass, energy, and momentum in the form of a wave equation.

**eigenstates**are standing-wave solutions of Equation (1) and have the form $\Psi =R{e}^{-i\omega t}$, where R and V are functions of only the spatial coordinates, and the angular frequency $\omega $ is itself independent of time. For the hydrogen atom, the potential $V={\u03f5}_{0}{q}_{p}/r$, where ${q}_{p}$ is the positive charge on the nucleus, equal in magnitude to the electron charge q. Two of the lowest-energy solutions to Equation (1) with this potential are:

**Bohr radius**${a}_{0}$:

**stationary states**. The lowest energy bound eigenstate for a given form of potential minimum is called its

**ground state**, shown on the left in Figure 3. The corresponding charge densities are shown in Figure 4.

“…not at all difficult, but very tedious. In spite of their tediousness, it is rather fascinating to see all the well-known but not understood “rules” come out one after the other as the result of familiar elementary and absolutely cogent analysis, like e.g. the fact that ${\int}_{0}^{2\pi}cosm\varphi \phantom{\rule{4pt}{0ex}}cosn\varphi \phantom{\rule{4pt}{0ex}}d\varphi $ vanishes unless $n=m$. Once the hypothesis about ${\Psi}^{*}\Psi $ has been made, no accessory hypothesis is needed or is possible; none could help us if the “rules” did not come out correctly. However, fortunately they do [22,35].”

## 4. The Two-State System

**normalization condition**:

**uncertainty principle**, which simply restated the Fourier properties of an object described by waves in a statistical context. No statistical attributes are attached to any properties of the wave function in this treatment.

**The Electron**, R.A. Millikan [36] anticipates the solution in his comment about the

”…apparent contradiction involved in the non-radiating electronic orbit—a contradiction which would disappear, however, if the negative electron itself, when inside the atom, were a ring of some sort, capable of expanding to various radii, and capable, only when freed from the atom, of assuming essentially the properties of a point charge.”

## 5. Transitions

**dipole strength**for the transition. When one R is an even function of z and the other is an odd function of z, as in the case of the 100 and 210 states of the hydrogen atom, then ${d}_{12}$ is nonzero, and the transition is said to be

**electric dipole allowed**. When both ${R}_{1}$ and ${R}_{2}$ are either even or odd functions of z, ${d}_{12}=0$, and the transition is said to be

**electric dipole forbidden**.

## 6. Atom in an Applied Field

**power**(rate of change of electron energy), which is understood to be an average over many cycles. The time required for one cycle is $2\pi /{\omega}_{0}$, so Equation (25) becomes:

## 7. Electromagnetic Coupling

**Collective Electrodynamics**[16]. The standard treatment is given in Jackson,

**Classical Electrodynamics, 3rd Edition**, Chapter 8 [40].] It is shown in Ref. [16] that electromagnetism is of totally quantum origin. We saw in Equation (22) that it is the vector potential $\overrightarrow{A}$ that appears as part of the momentum of the wave function, signifying the coupling of one wave function to one or more other wave functions. Thus, to stay in a totally quantum context, we must work with electromagnetic relations based on the vector potential and related quantities. The entire content of electromagnetism is contained in the relativistically-correct Riemann–Sommerfeld second-order differential equation:

**retarded**field. Conversely, the four-potential from a current element on the future light cone of the electron ($t+r/c$) will be “felt” by the electron at earlier time t, and is termed an

**advanced**field. Historically, with rare exception, advanced fields have been discarded as non-physical because evidence for them has been explained in other ways. We shall see that modern quantum experiments provide overwhelming evidence for their active role in

**quantum entanglement.**

“There is no such concept as ’the’ field, an independent entity with degrees of freedom of its own.”

**there are no self-energy infinities in this formulation**.

**cosmic microwave background**, to which atoms here and now are coupled by the quantum handshake. For independent discussions from the two of us, see Ref. [16], p. 94 and Ref. [19].

## 8. Two Coupled Atoms

- (1)
- Excited atom $\alpha $ will start in a state where $b\approx 1$ and a is very small, but never zero because of its ever-present random statistical interactions with a vast number of other atoms in the universe, and
- (2)
- Likewise, atom $\beta $ will start in a state where $a\approx 1$ and b is very small, but never zero for the same reason.

**optical system**between the two atoms. Even the simplest such arrangement couples the two atoms orders of magnitude better than the simple $1/r$ dependence in Equation (40) would indicate. We take up the enhancement due to an intervening optical system in Section 10. Any such enhancement merely provides a constant multiplier in ${P}_{\alpha \beta}$. In any case, Equation (39) becomes:

**excite**the atom. When the the electron motion has opposite phase from the field, the electron transfers energy “to the field”, and the process is called

**stimulated emission**.

#### Competition between Recipient Atoms

**Fermi’s “Golden Rule” [42]**, the assertion that a transition probability in a coupled quantum system depends on the strength of the coupling and the density of states present to which the transition can proceed. The emergence of Fermi’s Golden Rule is an unexpected gift delivered to us by the logic of the present formalism.

## 9. Two Atoms at a Distance

“In a pure geometry it would surprise us to find that a true theorem becomes false when the page upon which the figure is drawn is turned upside down. A dissymmetry alien to the pure geometry of relativity has been introduced by our notion of causality.”

“I regard the equations containing retarded functions, in contrast to Mr. Ritz, as merely auxiliary mathematical forms. The reason I see myself compelled to take this view is first of all that those forms do not subsume the energy principle, while I believe that we should adhere to the strict validity of the energy principle until we have found important reasons for renouncing this guiding star.”

“Setting $f(x,y,z,t)={f}_{1}$ amounts to calculating the electromagnetic effect at the point $x,y,z$ from those motions and configurations of the electric quantities that took place prior to the time point $t.$ "Setting $f(x,y,z,t)={f}_{2},$ one determines the electromagnetic effects from the motions and configurations that take place after the time point $t.$”

“In the first case the electric field is calculated from the totality of the processes producing it,

and in the second case from the totality of the processes absorbing it…

Both kinds of representation can always be used, regardless of how distant the absorbing bodies are imagined to be.”

**solid angle**containing paths from one atom to the other. The effect of the optical system is to replace the $1/r$ dependence with Solid Angle/$\lambda $, as described in Section 10. Thus, the 0.04 s transition given by Equation (54) for two isolated atoms 1 m apart becomes $2\times {10}^{-9}$ s when a 1 steradian optical system is used.

## 10. Paths of Interaction

“Grand Principle: The probability of an event is equal to the square of the length of an arrow called the ’probability amplitude.’…”

“General Rule for drawing arrows if an event can happen in alternative ways: Draw an arrow for each way, and then combine the arrows (’add’ them) by hooking the head of one to the tail of the next.”

“A ’final arrow’ is then drawn from the tail of the first arrow to the head of the last one.”

“The final arrow is the one whose square gives the probability of the entire event."

**phasor**, introduced in 1894 by Steinmetz [45,46] as an easy way to visualize and quantify phase relations in alternating-current systems. In physics, the technique is known as the

**sum over histories**and led to

**Feynman path integrals**. His “probability amplitude” is the amplitude of our vector potential, whose square is the

**probability**of a photon.

**Thus, this is the fundamental origin of the $1/r$ law for amplitudes.**

**We deal with a realm of which statistical QM denies the existence.**

## 11. Global Field Configuration

**All waves propagating from atom $\alpha $ to atom $\beta $ along high-amplitude paths arrive in phase!**

## 12. Relevance to the Transactional Interpretation

## 13. Historic Tests

#### 13.1. The Hanbury–Brown–Twiss Effect and Waves vs. Particles

#### 13.2. Splitting Photons

A source emits a single photon isotropically, so that there is no preferred emission direction. According to the quantum formalism, this should produce a spherical wave function Ψ that expands like an inflating bubble centered on the source. At some later time, the photon is detected. Since the photon does not propagate further, its wave function bubble should “pop”, disappearing instantaneously from all locations except the position of the detector. In this situation, how do the parts of the wave function away from the detector “know” that they should disappear, and how is it enforced that only a single photon is always detected when only one photon is emitted?

- The handshake goes to completion and the partner atom is in one of the detectors, in which case only the chosen detector registers an output.
- The handshake goes to completion, but the partner atom is not in one of the detectors. In this case, no output is registered from either detector.
- The initial stages of a handshake begins in two partner atoms, one in each detector. When the source atom has de-excited, both of the detector atoms are left in mixed states with roughly equal components of ground-state and excited state wave functions, as was illustrated in Figure 7. This is not a stable configuration because both of the detector atoms have oscillating dipole moments sending strong “unrequited” advanced confirmation waves. These waves are in phase at and focused on the source, and they are likely to find another well-phased excited-state atom there or nearby that will complete the four-way transaction. Thus, a four-atom HBT event should be created, in which there are two emissions and two detections. Such an unlikely event would register as an “accidental” case of two simultaneous emissions in the same time window. There will never be an event with a single emission and two detections.

#### 13.3. Freedman–Clauser Experiment

#### 13.3.1. Dynamics of the Transaction

**A quantum handshake is an antisymmetric bidirectional electromagnetic connection between two atoms on a light cone, whose direction of time is the direction of positive energy transfer.**

#### 13.3.2. The NCT-Killer Result

## 14. Conclusions

**The Quantum Handshake**[12], Einstein’s effort was doomed from the start and was also beside the point. The uncertainty principle is simply a Fourier-algebra property of any system described by waves. Both parties to the Solvay argument lacked any real clarity as to how to handle the intrinsic wave nature of matter. In the introduction, we quoted Einstein’s deepest concern with statistical QM:

"If I had to choose now between your wave mechanics and the matrix mechanics, I would give the preference to the former because of its greater intuitive clarity, so long as one only has to deal with the three coordinates x,y,z. If, however, there are more degrees of freedom, then I cannot interpret the waves and vibrations physically, and I must therefore decide in favor of matrix mechanics. However, your way of thinking has the advantage for this case too that it brings us closer to the real solution of the equations; the eigenvalue problem is the same in principle for a higher dimensional q-space as it is for a three-dimensional space.

"There is another point in addition where your methods seem to me to be superior. Experiment acquaints us with situations in which an atom persists in one of its stationary states for a certain time, and we often have to deal with quite definite transitions from one such state to another. Therefore, we need to be able to represent these stationary states, every individual one of them, and to investigate them theoretically. Now a matrix is the summary of all possible transitions and it cannot at all be analyzed into pieces. In your theory, on the other hand, in each of the states corresponding to the various eigenvalues, E plays its own role.”

- Advanced waves were not explicitly used as part of the process.
- The “focusing” property of the advanced-retarded radiation pattern had not been identified.

**What if Einstein was right?**

**Density Functional Theory**(DFT), and are responsible for amazingly successful analyses of an enormous range of complex chemical problems. The original Thomas–Fermi–Hohenberg–Kohn idea was to make the Schrödinger equation just about the 3d density. The practical implementations do not come close to the original motivation because half-integer spin, Pauli exclusion, and 3N dimensions are still hiding there. DFT, as it stands today, is a practical tool for generating numbers rather than a fundamental way of thinking. Although it seems unlikely at present that a more intuitive view of the multi-electron wave function will emerge from DFT, the right discovery of how to adapt 3D thinking to the properties of electron pairs could be a major first step in that direction.

**Pauli Exclusion Principle**, most commonly stated as: Two electrons can only occupy the same orbital state if their spins are anti-parallel.

"We live on an island surrounded by a sea of ignorance.

As our island of knowledge grows, so does the shore of our ignorance."

- We do not yet understand the mechanism that gives the 3D wave function its “identity”, which causes it to be normalized.
- We do not yet have a physical picture of how the electron’s wave function can be endowed with half-integer “spin”, which why it requires a full 720° (twice around) rotation to bring the electron’s wave function back to the same state, why both matter and electron antimatter states exist, and why the two have opposite parity.
- We do not yet have a physical understanding of how two electron wave functions interact to enforce Pauli’s Exclusion Principle.

“However, the Real Glory of Science

is that we can Find a Way of Thinking

such that the Law is Evident!”

**Conceptual Physics**.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DFT | density functional theory |

EPR experiment | Einstein, Podolsky, and Rosen experiment demonstrating nonlocality |

NCT | neoclassical theory, i.e., Schrödinger’s wave mechanics plus Maxwell’s equations |

QM | quantum mechanics |

TI | the Transactional Interpretation of quantum mechanics [12] |

WFE | Wheeler–Feynman electrodynamics |

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**Figure 1.**Model of transaction formation: An emitter atom 2 in a space-antisymmetric excited state of energy ${E}_{2}$ and an absorber atom 1 in a space-symmetric ground state of energy ${E}_{1}$ both have slight admixtures of the other state, giving both atoms dipole moments that oscillate with the same difference-frequency ${\omega}_{0}={\omega}_{2}-{\omega}_{1}$. If the relative phase of the initial small offer wave $\psi $ and confirmation wave $\psi *$ is optimal, this condition initiates energy transfer, which avalanches to complete transfer of one photon-worth of energy $\hslash {\omega}_{0}$.

**Figure 2.**Angular dependence of the spatial wave function amplitudes for the lowest (100,

**left**) and next higher (210,

**right**) states of the hydrogen atom, plotted as unit radius in spherical coordinates from Equation (2). The 100 wave function has spherical symmetry: positive in all directions. The 210 wave function is antisymmetric along the z-axis, as indicated by the color change. In practice, the direction of the z-axis will be established by an external electromagnetic field, as we shall analyze shortly.

**Figure 3.**Wave function amplitudes $\Psi $ for the 100 and 210 states, along the z-axis of the hydrogen atom. The horizontal axis in all plots is the position along the z-axis in units of the Bohr radius.

**Figure 4.**Contribution of $x-y$ “slices” at position z of wave function density ${\Psi}^{*}\Psi $ to the total charge or mass of the 100 and 210 states of the hydrogen atom. Both curves integrate to 1.

**Figure 5.**

**Left**: Plot of the three terms in the wave-function density in Equation (11) for an equal $\left(a=b=1/\sqrt{2}\right)$ superposition of the ground state (${R}_{1}^{2}$, blue) and first excited state (${R}_{2}^{2}$, red) of the hydrogen atom. The green curve is a snapshot of the time-dependent ${R}_{1}{R}_{2}$ product term, which oscillates at difference frequency ${\omega}_{0}$.

**Right**: Snapshot of the total charge density, which is the sum of the three curves in the left plot. The magnitudes plotted are the contribution to the total charge in an $x-y$ “slice” of ${\Psi}^{*}\Psi $ at the indicated z coordinate. All plots are shown for the time such that $cos\phantom{\rule{-0.166667em}{0ex}}({\omega}_{0}t+\varphi )=1$. The horizontal axis in each plot is the spatial coordinate along the z-axis of the atom, given in units of the Bohr radius ${a}_{0}$. Animation here [37] (see Supplementary Materials).

**Figure 6.**Squared state amplitudes for atom $\alpha $: ${b}_{\alpha}^{2}$ (blue) and ${a}_{\alpha}^{2}={b}_{\beta}^{2}$ (red) for the Photon transfer of energy ${E}_{0}=\hslash {\omega}_{0}$ from atom $\alpha $ to atom $\beta $, from Equation (44). Using the lower state energy as the zero reference, ${E}_{0}{b}^{2}$ is the energy of the state. The green curve shows the normalized power radiated by the atom $\alpha $ and absorbed by atom $\beta $, from Equation (43). The optical oscillations at ${\omega}_{0}$ are not shown, as they are normally many orders of magnitude faster than the transition time scale $\tau $. The time t is in units of $\tau $. In the next section, we will find that atoms spaced by an arbitrary distance exhibit transactions of exactly the same form.

**Figure 7.**Squared excited state amplitudes for recipient atoms $\beta 1$ and $\beta 2$: ${b}_{\beta 1}^{2}$ (blue) and ${b}_{\beta 2}^{2}$ (red), for the photon transfer of energy ${E}_{0}=\hslash {\omega}_{0}$ from atom $\alpha $. The time t is in units of $\tau $. The left plot, for $\Delta \omega =0.3/\tau $, shows both recipient atoms being equally excited at the beginning, but the slip in phase of the red $\beta 2$ atom causes it to rapidly lose out, so the blue $\beta 1$ atom hogs all the energy and proceeds to become fully excited, much as if it were the “only atom in town”. Its curve is nearly identical with that for the isolated recipient atom in Figure 6. The right plot, for $\Delta \omega =0.15/\tau $, shows a totally different story. Its red $\beta 2$ atom transition frequency is just close enough to that of source atom 0 that it “hangs in there” during the transition period and ends up partially excited, leaving blue $\beta 1$ atom partially excited as well. The smaller the “slip” frequency $\Delta \omega $, the closer are the post-transition excitations of the two recipient atoms. “Split-photons” of this kind are observed in the Hanbury–Brown–Twiss Effect described in Section 13. As noted above, the frequency window for such events is extraordinarily narrow, typically of order ${10}^{-7}{\omega}_{0}$. Doppler shift of this magnitude requires velocity ≈ 30 m/s. Room temperature thermal velocities of gasses are typically tens to hundreds of times this value, which would eliminate such competition. Thus, complete transactions are the most common, with “split” transactions relatively rare and are likely to end as HBT-type four-atom events.

**Figure 8.**All the paths from coherent light source S to detector P are involved in the transfer of energy. The solid curve on the “TIME” plot shows the propagation time, and hence the accumulated phase, of the corresponding path. Each small arrow on the “TIME” plot is a

**phasor**that shows the magnitude (length) and phase (angle) of the contribution of that path to the resultant total vector potential at P. The “sea horse” on the far right shows how these contributions are added to form the total amplitude and phase of the resultant potential. (From Fig. 35 in Feynman’s

**QED**).

**Figure 9.**Situation identical to Figure 8 but with a piece of glass added. The shape of the glass is such that all paths from the source S reach P in phase. The result is an enormous increase in the amplitude reaching P. (From Fig. 36 in Feynman’s

**QED**).

**Figure 10.**Normalized vector potential along the x-axis (in wavelength/2$\pi $) between two atoms in the “quantum handshake” of Equation (63). The wave propagates smoothly from atom $\alpha $ (

**left**) to atom $\beta $ (

**right**). Animation here [37]. (see Supplementary Materials).

**Figure 11.**Two atoms in the “quantum handshake” of Equation (63). Animation here [37]. (see Supplementary Materials).

**Figure 12.**Poynting vector stream lines of the “quantum handshake” of Equation (63).

**Figure 13.**The zero crossings of the handshake vector potential at $t=0$. Paths near the axis (between the two red lines) are responsible for the conventional $1/r$ dependence of the potential. Paths shown through high-amplitude regions have an even number of zero crossings, and thus the potentials traversing these paths all arrive in phase, thus adding to the central potential.

**Figure 14.**Schematic diagram of the Hanbury–Brown–Twiss effect, with excited atoms 1 and 2 in distant separated sources simultaneously exciting ground state atoms A and B in two separated detectors.

**Figure 15.**

**Left**: Schematic of the photon-splitting experiment.

**Right**: Plot of the number of coincidences vs. time delay between the arriving pulses. The source on average generates a photon per ≈12 nsec. The finite number counted at zero delay is consistent with the accidental presence of two excited atoms.

**Figure 16.**Left: Schematic of the polarization-correlation experiment. Right: Energy levels of the Ca atoms used in this experiment. From Freedman and Clauser 1972 [6].

**Figure 17.**

**Left**: Superposition contributions of the upper state ${c}^{2}$ (red), shared middle level ${b}^{2}$ (green) and ground state ${a}^{2}$ (blue).

**Right**: Amplitude of dipole oscillations due to upper transition at ${\omega}_{23}$ (red) and lower transition at ${\omega}_{12}$ (blue), from. The horizontal axis in both plots is time in units of ${\tau}_{\alpha}=1.5\phantom{\rule{0.166667em}{0ex}}{\tau}_{\beta}$.

**Figure 18.**Three-vertex transaction formed by a detection event in the Freedman–Clauser experiment. Linked transactions between the source and the two polarimeters cannot form unless the source boundary condition of matching polarization states is met.

**Figure 19.**Coincidence rate vs. angle $\varphi $ between the polarizers, divided by the rate with both polarizers removed. The solid line is the prediction by quantum mechanics, calculated using the measured efficiencies of the polarizers and solid angles of the experiment.

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Cramer, J.G.; Mead, C.A.
Symmetry, Transactions, and the Mechanism of Wave Function Collapse. *Symmetry* **2020**, *12*, 1373.
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