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In 1949 Richard Feynman presented to his fellow physicists a strange idea that was meant to explain a feature of Reality that they had observed in their experiments. He published a paper titled "The Theory of Positrons" (in the 1949 Sep 15 issue of Physical Review, Vol. 76, No. 6, pp 749-759) and in that paper he described the creation of an electron-positron pair and the subsequent annihilation of the positron with another electron. He asserted that only a single electron was involved, progressing forward in time until an interaction kicked it backward in time as the positron, which traveled into the past until a second interaction kicked it forward in time again. He then showed how well that assertion worked in the theory of quantum electrodynamics.
Look at that event pair in more detail. An electron occupies a point in space. Then two gamma photons collide somewhere else and an electron-positron pair emerges from the collision. Finally the positron goes to the first electron and annihilates it, releasing a pair of gamma photons. In the Feynman picture a single electron moves forward in time and encounters a photon moving backward in time. The collision kicks the photon forward in time (note that the photon is its own antiparticle) and sends the electron backward in time, making it appear to us as a positron. The positron collides with a photon moving forward in time and gets kicked forward in time, becoming an electron once again, as the photon gets kicked backward in time. What appears to us as four photons and three particles is actually two photons and a single electron.
In the first event two photons come together and remanifest themselves as an electron and a positron. Cause precedes effect, so we regard the photons as the cause and the electron-positron pair as the effect of the collision. But in Feynman’s view the collision occurs because an electron comes from the future (appearing as a positron) and hits a photon coming from the past, resulting in the electron going back to the future and the photon returning to the past. The cause, in this case a forward photon and a backward electron, doesn’t have clear priority over the effect of the collision, a backward photon and a forward electron. The concept of causality seems to have fallen apart, demolished by a simple change of perspective.
As aids to the imagination in his work on quantum electrodynamics, Feynman devised little sketches, in which he drew the world-lines of particles on a space-time diagram. Physicists call those sketches Feynman diagrams and use them to keep track of the factors that go into the state function associated with the interaction among the particles whose world-lines appear on the sketch. In making those sketches, Feynman drew single thin lines to represent the world-lines of the particles and thereby left out a key feature of the quantum view of causality.
In a Feynman diagram a single, stationary particle appears as a vertical straight line. If we draw an horizontal line across the diagram and label it Now, the world-line below that line (that instant in time) doesn’t change, but above that line the world-line must appear as a spreading array of myriads of fuzzy lines. As Now proceeds from past to future it ravels up those indeterminate world-lines into a single, determinate world-thread. That’s quantum causality, the potential becoming the actual.
But the positron proceeds from future to past. Does it unravel a single world-line that extends into the past to the collision that, in our view, created the electron-positron pair? Or does it ravel up a wide-spreading array of potential world-lines that fan out into the past?
Suppose we answer the latter question in the affirmative. In that case the pair creation event becomes indeterminate as seen from the pair annihilation event. Even after the creation event has occurred there’s a chance that it won’t; there exists a possibility that the positron won’t collide with the photon coming from the past and that eventually some other photon will instigate the pair-production event. That’s absurd, of course, so we dismiss it from consideration as part of Reality.
Of course, this talk of drawing lines, sharp or fuzzy, is pure metaphor, an aid to the imagination that’s seeking to understand this phenomenon. There are no actual lines involved. In actuality each particle associates with an aleatric field, which sprawls across space and time to encode the possible states available to the particle and whose mathematical description enable physicists, through Born’s theorem, to calculate probabilities of the particle occupying any given state.
In that picture, from any given Now a particle appears in the future as a field spread widely over space and it appears in the past as a field manifested in a Dirac delta, the field equivalent of certainty. Viewed in light of the full quantum theory, that statement gives us a problem. The fundamental equations of the quantum theory – those of Schrödinger, Klein and Gordon, Dirac, Proca, and Rarita and Schwinger being prime examples – are all wave equations: they describe the space-wide evolution of the aleatric field of a particle with the elapse of time. Our experience with electromagnetism tells us that wave equations work backward in time the same as they do forward in time and produce the same results. That experience leads us to expect that, as seen on a space-time diagram from any given Now, the aleatric field of a particle should expand backward in time as well as forward in time: the past should be as indeterminate as the future.
But history is not indeterminate. Cast your imagination back over two and a half centuries. If a certain cosmic ray had shattered one atom in the upper atmosphere and not another, it might have instigated a cascade of charged particles that led a lightning bolt to travel down a wet kite string and kill Benjamin Franklin. We know that Franklin survived his stunt (the next experimenter who tried it wasn’t so lucky) and went on to participate in the American Revolution. He doesn’t exist both alive and dead in July 1752, like a bizarre analogue of Schrödinger’s cat.
Of course, Hugh Everett III’s many-worlds interpretation of the quantum theory tells us that Franklin died from a lightning strike in an alternate Universe that split off from ours at that instant as a consequence of the indeterminacy. But, as far as we can see, those other Universes have no contact with ours and are unavailable for us to study in any way, so we can dismiss them as irrelevant to this discussion.
Imagine that we can travel backward in time and that we follow one of the electrons that participated in the spark that flickered between the dangling key and Franklin’s finger. We would see the electron travel unerringly from interaction to interaction, moving from particle collision to particle collision with infinite precision in its motion. There would be no sign of indeterminacy.
At each interaction an aleatric field would spring abruptly into existence (the uncollapse of the wave function) and then condense onto the electron at its previous interaction (the next interaction in our backward-looking view). But a field coming into existence instantly everywhere violates the rules of Relativity as does a field collapsing instantaneously. Such a thing would require a field-changing impulse to propagate at infinite speed, which is impossible. However, if the field-changing impulse were to propagate backwards in time (as in John Cramer’s Transactional Analysis interpretation of the quantum theory), then we have no sudden collapse of the state function or the aleatric field that it describes. We can say that the particle is controlled by an entity that says, "this is what you did (in our perspective)/will do (in your perspective)".
That statement sounds deterministic, as if the particle’s next interaction were predestined. It’s hard to see how that could not be so. It also implies that the next interaction is predestined and the one after that and the one after that and so on. Can we infer that the electron mentioned earlier was, at the instant it attended Franklin’s kite experiment, retroactively predestined to reach my desk so that I could follow it backward? If we answer yes, then the quantum theory becomes effectively meaningless. If we answer no, then how can we have forward indeterminacy and backward determinism?
Deterministic indeterminacy sounds like some hippie-dippie pseudo-philosophical phrase meant to impress the ignorant, but it is, in actuality, a fair description of Richard Feynman’s sum-over-histories interpretation of the quantum theory. In Feynman’s view an interaction produces, for one of the particles, myriads of aleatric fields, each one corresponding to one possible future of the particle and all of them interfering with each other. Sinusoidal functions are major factors in our state functions, so the state functions, according to a theorem of Fourier, can add up to virtually any function, including one that closely resembles the Dirac delta (the quantum analogue of a deterministic location for a particle). The resultant state function looks like the pilot wave in David Bohm’s Implicate Order version of the quantum theory.
Going back to our original thought experiment, we have an electron moving through space as it travels forward in time. It’s heading for an encounter with a photon moving backward in time. Most of the aleatric fields accompanying the electron encode histories in which the electron misses the photon; only a few, or only one, encode histories in which the electron and the photon collide. Yet somehow all of those fields have such a collective shape that their mutual interference leaves only the history/world-line in which the particles collide.
As in classical physics, we take as given the fact that the electron and the photon have been sent in directions that intersect at some point and that the timing of the particles’ motions enables them to reach that point at the same time. In classical physics that’s sufficient for the causality: if the particles reach the same point at the same time, they will collide. In quantum physics it gives us only a probability that they will collide. So dynamic geometry alone is not sufficient for causality.
There exists another way to conceive our imaginary experiment and this is a good place to bring it into play. We imagine our particle creation event as beginning with a virtual electron-positron pair, a pattern for particles that don’t actually exist (they are merely potential particles). At some instant they borrow enough energy E to come real for an interval of time t=h/E, in accordance with Heisenberg’s indeterminacy principle. During that interval a gamma photon (or two) gives the particles enough energy to come fully real and enough extra to make the particles fly apart. Later the positron encounters another electron and the two particles release all of their energy into two photons and go into existing as a virtual-particle pair.
This view is not inconsistent with the previous description of the creation and annihilation of the particle pair. On a Feynman diagram the particle pairs brought into brief, temporary existence by the Heisenberg principle appear as tiny loops in spacetime. As with all particles, an aleatric field must associate with the pair and a state function must describe that field.
We gain another perspective on this situation when we look at the quantum representation of a forcefield, an electric field in particular. In that representation an electric charge emits a flux of virtual photons and if those photons travel for forward and backward in time, we don’t even have an implied insult to the conservation laws. In that view a virtual photon coming from the past bounces off the charge and goes back into the past and a virtual photon coming backward in time bounces off the charge and goes back to the future.
When I first heard about the idea that an electric field consists of a flux of virtual photons, I couldn’t understand how particles moving away from a charge could cause another charged particle to be attracted to the first. The idea of particles carrying negative momentum, momentum oriented opposite the particle’s direction of motion, seemed to me absurd. But the concept of particles traveling both backward and forward in time accounts neatly for both attractive and repulsive forces, which form is manifested depending upon which part of the flux a second particle engages. Thus we have another reason to accept the Feynman picture of quantum Reality.
If we see a photon, carrying at least 1.022 Mev of energy, hit an atom and produce an electron-positron pair and if we then see the positron undergo mutual annihilation with another electron, thereby producing two 511-kev photons, we can attribute the scene to pure happenstance. That would be the classical (or, more properly, semi-classical) interpretation. If we conceive the positron as the time-reversed aspect of both electrons, as Feynman hypothesized, pure happenstance doesn’t work so well. A time-forward electron strikes a time-reversed photon and knocks it toward the future while itself becoming a time-reversed electron (a positron). The positron strikes an atom, which emits a 1.022-Mev photon into the past while knocking the positron back forward in time to resume its existence as an electron.
At some instant the positron has already come from the photon hitting the atom but has not yet annihilated the electron. How is its aleatric field evolving in that instant? Seen from past to future, the positron’s field must be determinate at the atom (it definitely came from there) and indeterminate at the annihilation event (even though we know that the event must take place with that particular electron). Seen from future to past, the positron’s field must be determinate at the annihilation event and indeterminate at the atom. That’s what the standard quantum theory requires.
John Cramer’s Transactional Interpretation of the quantum theory, a variation on Feynman’s sum-over-histories version, postulates the existence of aleatric waves traveling both forward and backward in time. If an electron collides with atom-1 and some time later collides with atom-2, Cramer’s interpretation gives us an atemporal state function connecting the two collisions. When the electron bounces off of atom-1, the event creates an aleatric wave moving forward in time, encoding the trajectories available to the electron based on its linear momentum in light of Heisenberg’s principle. Some of those possible trajectories hit atom-2 and many don’t. When the electron hits atom-2, the event creates an aleatric wave moving backward in time and, through interference, erasing the unmanifested possibilities from the time-forward wave. The mathematics is a rabid velociraptor, but Fourier’s theorem assures us that the process will, in fact, transform the original fully-indeterminate aleatric wave into something like the pilot wave of David Bohm’s Implicate Order.
In that perspective the electron hits atom-1 and sets out on an indeterminate course, as required by the quantum theory, but then it hits atom-2 and the course becomes determinate retroactively. In some sense, then, the collision with atom-2 was preordained. In essence we have the collision between the electron and atom-1 sending out a wide beam of probability radiation and a second, later collision transforming that beam into a thin ray of probability radiation, in essence a certainty ray. If we could read the state function directly from the aleatric waves as they occur, would we see it, as the first event happens, describe a wide beam or a thin ray?
We would see a thin ray, but not necessarily one coming from atom-2. The aleatric wave emanating from the electron’s collision with atom-1 encodes all of the possible trajectories that the electron could follow, most of them missing a collision with atom-2. Only one possibility comes manifest when the electron hits an atom and triggers the emission of the time-backward wave that erases all of the other possibilities. Of course, we don’t actually see aleatric fields or waves, wide beam or thin ray, but we only know what atom the electron has hit when we detect its recoil or emission of a photon.
How does our positron fit into this kind of picture? It necessitates a picture that looks perfectly deterministic: the aleatric waves involved appear as the thinnest rays of probability, the picture of certainty. We have four particles (an electron, an atom, a time-forward 1.022-Mev photon, and a time-reversed 0.511-Mev photon) participating in two events separated from each other in time and space. If we look at our scene at some instant prior to both events, we would see the 1.022-Mev photon headed toward a collision with the atom and the electron headed for a later collision with the 0.511-Mev anti-photon. If one of the collisions doesn’t occur, then the other can’t: if one occurs, then the other must.
We can conceive the emission of the positron and its annihilation as a quantum transaction whose counter-wave condenses the time-forward aleatric wave into a thread-like probability distribution, a narrowly defined world-line for the positron. That transaction depends on a 1.022-Mev photon hitting an atom. Certainly that event can happen at random, but once it does occur the second event becomes necessary. The electron must collide with a 0.511-Mev photon traveling backward in time in order that the electron goes backward in time as the positron.
Assume the existence of an aleatric aether permeating the Universe. Events create ripples in that aether, both forward and backward in time, and the existence of a particle is a continuous series of events. If we move forward in time, we see a ripple contracting toward the point where the 1.022-Mev photon will hit the atom. The ripple becomes a point at the instant of electron-positron creation and expands outward from there. The electron and the positron leave their own trails of ripples, interference between the time-forward and time-reversed ripples generating plane waves that track the particles’ motions. The electron-positron annihilation event produces its own ripple. Because the positron attends both events, we know that there exists an inertial frame in which both events occur at the same point in space. In that frame the aleatric waves emitted by the events expand and contract in concentric spherical shells. The time-forward wave from the pair production event and the time-reversed wave from the pair annihilation event cross each other at an instant midway between the events; otherwise, there’s no contact among the ripples and, thus, no interference that would account for the collapse of the wave function. Cramer’s transaction doesn’t seem to take place.
Now let’s ask a question that no one seems to have asked before: why does the state function have to collapse? We often metaphorize the wave/particle duality of quantum mechanics by comparing a particle to a surfer riding a wave. But we know that if a surfer gets knocked off a wave (such as by colliding with a seal), the wave continues onward. Wouldn’t the same thing happen with an aleatric wave if the particle riding it were removed by a collision?
We note that the aleatric field of a particle consists of a spherical standing wave centered on the particle (its frequency determined by the particle’s rest mass-energy) and a set of plane waves representing the particle’s linear momentum (if any). The plane waves are not extensive: their amplitudes have such a form that they differ substantially from zero only short distances from the point at which we expect to find the particle at any given instant. Our description of the particle’s momentum is manifested in a small pulse of aleatricity that spreads outward as it propagates.
When the particle is removed from it, by a collision, the pulse travels onward without carrying any material properties: linear momentum, energy, spin, and so on stay with the particle. The pulse carries in itself only the probabilities that those properties will come manifest at some place at some time. As that wave moves onward it may wash over some second particle. All the wave will do to that particle is to impose a brief perturbation in the probabilities of when and where the particle’s properties come manifest. Then the wave moves on and the perturbation goes away with it. After its particle is stripped from it, the wave has no discernible effect on any other particles: it has effectively ceased to exist. Thus, we don’t need a collapse of the wave function.
Indeed, that proposition gives us a good description of an indeterminate future and a determinate past. As we progress toward the future, the patterns of matter and radiation that comprise the world evolve in a probabilistic manner, as in the conventional quantum theory. If we were to move backward in time, we would see aleatric waves returning to their particles, catching them up, and guiding them back to the events that created the aleatric waves. The aleatric waves then act much like the pilot waves in David Bohm’s Implicate Order version of the quantum theory.
That view doesn’t seem to offer any help in analyzing our positron experiment. It doesn’t tell us which of the events is the cause and which is the effect. The cause is the sine qua non of the whole scenario: if it doesn’t happen, the effect doesn’t happen. The cause can be entirely indeterminate. It may or may not occur. But the effect occurs if and only if the cause occurs. The pair annihilation happens after the pair creation, so we think that it must be the effect. But if the pair annihilation doesn’t occur, sending the electron backward in time as a positron, then the pair creation won’t occur, so we must now conceive the pair annihilation as the cause.
We have no problem with the concept of a hard gamma photon striking an atom and generating an electron-positron pair. We also have no problem with the idea of the positron meeting a random electron and undergoing a mutual annihilation that sends out two semi-hard photons. We get into conceptual trouble when we assert that the positron must meet a specific, preordained electron at a locus where and when it also meets a 511-kev photon coming from the future. It looks too much like a deus ex machina in a Universe where gods do not pop out of machines.
Did Feynman follow a wrong path with his idea of particles traveling backward in time? Though he pursued the idea for mathematical convenience, it seems to create a conflict with our basic notion of causality. But there’s a way to fix that problem.
We can still conceive the positron as an electron traveling backward in time, but it doesn’t have to be the same electron that comes to the annihilation event. Conceive the vacuum as being filled with virtual particles, ghost particles that lack only the conserved properties (energy, momentum, electric charge, etc.) that would make them real. When the two leptons (electron and positron) come together, they become virtual particles that go on their separate ways. The energy, momentum, and spin that they brought to the event go into realizing two virtual photons that move away from each other at an angle that gives them the combined linear momentum that they got from the leptons while also taking the energy (the photon relation E=pc necessitates some cancellation of the photons’ momenta). The electric charges carried by the leptons cancel each other out. The positron continues to exist, albeit as a ghost, so the identity of the electron that annihilates it is irrelevant.
Thus we now see a 1.022-Mev photon strike an atom and realize an electron and a positron out of the virtual realm. The photon becomes a virtual particle after giving its leftover properties to the atom in accordance with the conservation laws. The electromagnetic force, mediated by an exchange of unrealized virtual photons, attracts the positron and a random electron to each other. The particles annihilate each other, becoming virtual particles while their energies and momenta realize a pair of virtual photons.
We have evidence for the veracity of that picture. It appears in Feynman diagrams as tiny loops. Those loops represent the Heisenberg effect, in which electron-positron pairs achieve reality for brief instants of time in accordance with Heisenberg’s indeterminacy principle. Any action less than Planck’s constant effectively does not exist, so the particle pairs can flicker in and out of existence for intervals equal to or less than Planck’s constant divided by 1.022 Mev, the energy that must be "borrowed" to realize the particles.
Causality, then, has two aspects. An observer moving forward in time will see a hard photon hit an atom at random and produce an electron-positron pair in being absorbed. The positron asserts its electromagnetic field as it flies away from the event and engages a random electron in an act of mutual annihilation, producing a pair of medium-hard photons. Of course the particles are accompanied by the spreading aleatric waves that denumerate the probabilities of the particles occupying certain locations at certain times. An observer moving backward in time will see those aleatric waves narrow down, becoming pilot waves that carry their particles back to the events that produced the waves.
Thus, the time-forward observer sees events occur probabilistically. Particles may or may not interact with each other, depending upon the probabilities encoded in their aleatric waves. And a time-reversed observer sees the same events occurring in reverse order at precisely determined loci in time and space, part of a perfectly deterministic Universe, where the aleatric waves become the pilot waves of David Bohm’s Implicate Order. Causality is simply a matter of temporal perspective.
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