Kantian Quantum Mechanics

The discomfort that I feel is associated with the fact that the observed perfect quantum correlations seem to demand something like the "genetic" hypothesis. For me, it is so reasonable to assume that the photons in those experiments carry with them programs, which have been correlated in advance, telling them how to behave. This is so rational that I think that when Einstein saw that, and the others refused to see it, he was the rational man. The other people, although history has justified them, were burying their heads in the sand. I feel that Einstein's intellectual superiority over Bohr, in this instance, was enormous; a vast gulf between the man who saw clearly what was needed, and the obscurantist. So for me, it is a pity that Einstein's idea doesn't work. The reasonable thing just doesn't work.

John Stewart Bell (1928-1990), author of "Bell's Theorem" (or "Bell's Inequality"), quoted in Quantum Profiles, by Jeremy Bernstein [Princeton University Press, 1991, p. 84]

Nothing exists until it is measured.

Niels Bohr

Thus many quantum-gravity theorists believe there is a deeper level of reality, where space does not exist...

These days, many of us working on quantum gravity believe that causality itself is fundamental -- and is thus meaningful even at a level where the notion of space has disappeared.

Lee Smolin, The Trouble with Physics, The Rise of String Theory, the Fall of a Science, and What Comes Next [Houghton Mifflin Company, 2006, pp.240, 241].

I prefer to call it the dualistic philosophy, since it describes the universe as consisting of two layers. The first layer is the classical world of Einstein, with objects that are directly observable [i.e. phenomenal] but no longer predictable... The second layer is the quantum world, with states that are not directly observable but obey simple [deterministic] laws...

The dualistic philosophy seems to me to represent accurately our present state of knowledge. It says that the classical world and the quantum world are both real, but the way they fit together is not yet completely understood...

I prefer the dualistic philosophy because I give equal weight to the insights of Einstein and Bohr. I do not believe that the celestial harmonies discovered by Einstein are an accidental illusion [the thesis of "decoherence"].

Freeman Dyson (1923-2020), Emeritus Professor of Physics, The Institute for Advanced Study, Princeton, "Einstein as a Jew and a Philosopher," The New York Review of Books, May 7, 2015, Volume XVII, Number 8, p.17

Einstein was willing to concede that quantum mechanics explains the recorded behavior of the subatomic world, but he was convinced it had two flaws. First, it fails to give precise predictions for the outcomes of individual processes. Instead, it gives only statistical predictions. To check them, one must do an experiment many times and compare the resulting distributions of outcomes with the predictions of the theory. Second, quantum theory fails to give an objective picture of the world that is unconnected to our role as observers. The formulas of quantum theory correspond to our actions preparing experiments and measuring their outcomes. Einstein objected to this because he believed strongly that physics should provide a picture of nature “as it is in itself.”

Lee Smolin, "Einstein's Legacy -- Where are the 'Einsteinians'?" Logos 4.3 -- Summer 2005

The question everyone then faced was how to think of the electron wave that de Broglie had invented and Schrödinger had tamed. Schrödinger at first thought that the electron simply is a wave. This didn't hold up because it was easy to show that the wave tended to spread out in space as it traveled, whereas one could always find a localized particle. Max Born then proposed his rule that the wave is related to the probability of finding the particle.

Lee Smolin, Einstein's Unfinished Revolution, The Search for What Lies Beyond the Quantum [Penguin Books, 2019, p.83] [note].

Classic quantum mechanics seems to exhibit some of the characteristics that Immanuel Kant described about the relation between phenomenal reality in space and time and things-in-themselves.

As interpreted by Roger Penrose (and now, we see, by Freeman Dyson), quantum mechanics, first of all, posits a certain metaphysical dualism. In the world as it exists apart from observation, matter and energy consist of waves that are deterministically governed by Schrödinger's Equation. The waves have an undoubted physical reality because of the interference effects that can be observed, and because the three dimensional size of atoms is due to the state of their electrons as three dimensional standing waves -- otherwise there is nothing to "fill the space" of atoms, except particles somehow being everywhere at once, which they can't be because in changing directions to get here and there they would radiate energy. (Fields can be said to fill the space, but this only postpones the problem, since fields in quantum mechanics are exchanges of virtual particles.) On the other hand, the square of the wave function gives a probability distribution for where discrete particles may be found once the wave function is collapsed by an act of observation. The wave function thus contains the sum of all possible states of a system until it is observed. This produces the paradox of Schrödinger's Cat, who is both alive and dead at the same time, in just that proportion as each state is probable.

The act of observation, which collapses the wave function, is conformable to the Kantian act of synthesis, by which phenomenal objects are introduced into consciousness and subjected to the categories of the understanding. Niels Bohr's own Principle of Complementarity was that matter and energy could exhibit wave properties, or particle properties, but never both at the same time. If what Kantian consciousness requires is discrete actual things in space and time, this is exactly what is delivered in quantum mechanics:  Bohr stipulated that observers and their equipment would never be subject to quantum mechanical probability effects. Around us, for Bohr, we maintain a little, discrete, actual, Classical universe.

Kant did not view things-in-themselves as containing the sum of all possibilities, and phenomena all actualities; but this duality is conformable to Kant's metaphysics as to none other. As a contribution to the metaphysics of possibility, the quantum mechanical wave function can easily be seen as complementary to Kant's idea of things-in-themselves, where various kinds of things can happen (like free will) that are not comprehensible in terms of phenomenal reality. Kant would just have to allow that characteristics of physical reality can intrude some depth into things-in-themselves, which he would not have considered -- though we can also handle this by positing an intermediate level of reality, between true unconditioned things-in-themselves and true discrete phenomenal objects -- as Kant otherwise actually does himself for space and time as "pure intuitions." The wave function straddles the classic Kantian boundary, sharing some properties with phenomena, others, as underlying phenomena, with things-in-themselves.

Thus, where Kant would have considered all of phenomena governed by determinism, we now see the wave function as deterministic, while the collapse of waves into particles is random. Although chance in quantum mechanics has often been argued as allowing for free will, a free cause is still a very different thing from a random cause, which doesn't need mind or self or intention. Moral freedom is thus still left among things-in-themselves. Kant himself would have had difficulty placing randomness in his ontology, if he, like Hume, believed that chance violates determinism. Since chance is now part of the physics, it cannot be denied; but it also still remains a different matter from purposive freedom.

Kant's idea that space and time do not exist among things-in-themselves has been curiously affirmed by Relativity and quantum mechanics. In Relativity, time simply ceases to pass at the velocity of light:  for photons that have travelled to us as part of the Cosmic Background Radiation, time has stood still for most of the history of the universe. On the other hand, quantum mechanics now posits "non-locality," i.e. physical distances, and so the limitation of the velocity of light in Relativity, don't seem to exist. This means that although time may apply to the wave function, space may not. The full empirical reality of space is only found among discrete particles and objects.

This curious result is the consequence of the Einstein-Podolsky-Rosen (EPR) Paradox, which was intended by Einstein as a reductio ad absurdum of quantum mechanics. If, for instance, a positron and an election are both created from an energetic photon, the conservation of angular momentum requires that one be spinning one way, and the other the other. But the complementary spins are equally probable for each particle. Thus, in quantum mechanical terms, the wave functions of each particle divide without a discrete state being determined. The particles might then separate to even cosmological distances, but as soon as the spin of one particle is observed, the other particle must have the opposite spin, which means that the wave function has collapsed across those cosmological distances and caused the other particle to assume a predictable spin. If this occurs instantaneously, it would violate the limitation of the velocity of light in Special Relativity.

This has now been shown to actually occur on the basis of Bell's Theorem (from John Bell, 1928-1990), meaning that Quantum Mechanics does violate Special Relativity by allowing instantaneous interactions across even cosmological distances. However, once observed, processes must still obey Special Relativity and the limitations of spatial distance, creating the kind of duality described by Kant. Bell himself found this result disturbing, but to Kant it would fit in with his own theory that space is only imposed by the representation of phenomenal objects.

Einstein always objected to quantum mechanics because his metaphysical realism recoiled from the idea that observation would create a different kind of reality than what existed independently. At first Heisenberg's Uncertainty Principle could be interpreted as meaning that the act of observation would physically disturb a system in an ordinary and realistic way, but then it soon became evident that strange things were allowed to happen in the wave function that not only could not be observed but could not even be conceived in ordinary and realistic ways. Reality existed in a different way while under observation than it did in itself.

Now, the original philosophical theory which advocated something of the sort, that observation (the synthesis of objects in consciousness) imposes certain forms and rules before things can appear as phenomenal objects, was indeed that of Kant. Einstein and all his contemporaries must have been aware that there was something familiar about the emerging quantum world. The outright anti-realism of Bohr's Copenhagen Interpretation, although the focus of conflict, was only one historical possibility. Kant's empirical realism and transcendental idealism was another. But I have not noticed Kant receiving any kind of notice or credit for a theory that would address some of the paradoxes produced by quantum mechanics, denying the independence of physical reality from the presence of human consciousness. While recognizing the ontological dualism, Freeman Dyson now says (above) that "the way they fit together is not yet completely understood...," without acknowledging that just such a dualism, conformable to quantum mechanics, already exists in Kant's phenomenalism. Since nothing is so characteristic of Kantian philosophy than that dualistic principle, perhaps it is only a matter of time before philosophers pull their heads out of the "post-modernist" hole in the ground and pay attention. Physicists, of course, don't have to care, unless they hear the call of metaphysics as well as physics.

Since this page was originally posted, one of the most notable responses was from a correspondent who was indignant that the views of David Bohm (1917-1992) and other alternative theories about quantum mechanics were not presented. The purpose of this site, however, is to develop and apply Kantian and Friesian philosophy, and not necessarily to examine every other theory that other people may find important or definitive. Since Kant's was the original philosophical theory in which the observer imposes conditions on the nature of objects, it is arguably an interpretation with historical and conceptual priority. Thus, since it usually is not given much credit for this, it deserves some extra attention, as provided here.

Now, however, some additional comment may be in order, after I was struck by the treatment of recent developments in quantum mechanics in a centennial article in the February 2001 Scientific American, "100 Years of Quantum Mysteries," by Max Tegmark and the historic physicist (e.g. a teacher of Richard Feynman) John Archibald Wheeler (pp.68-75 -- Wheeler died, aged 96, in 2008). According to Tegmark and Wheeler, the recent trend is to try and preserve the determinism of the wave function, substituting for discrete particles more localized waves whose interference or interaction has been aborted by "decoherence," in which superpositions of wave functions are "dissipated" by "tiny interactions with the surrounding environment."

This is the complete opposite of an approach like that of Bohm, who, like Einstein, believed that discrete particles with definite locations are always present. That would now be called a "hidden variable" theory, i.e. that the quantities for the location of the particles are there, but are hidden from observation. It is frequently said that the success of Bell's Theorem rules out all hidden variable theories, but Bohm's is an exception to this. The clarification seems to be that local hidden variable theories were ruled out; but Bohm's is a non-local theory. Bohm postulated a new force, the "quantum potential," to account for the wave-like and interference effects between particles. The "quantum potential" itself takes the form of a wave, a "guide wave" or "pilot wave," which directs the motion of the particles. Bohm thus combines wave and particle theory, preserving the realism of particles but accounting for wave effects with the action of the pilot wave. The quantum potential field and pilot waves, however, have unique properties. They act non-locally, so that the wave does transmit the EPR information instantaneously without losing any strength, and the wave is able to guide the motion of particles without any energy being involved. As a good quantum theory, the "quantum potential" field theory also might be expected to postulate the existence of particles that mediate the field, but this does not seem to be part of it -- and it would be awkward, because even virtual particles, which mediate forces in quantum mechanics, observe the limitations of Relativity. Later, Bohm assimilated the quantum potential into a larger theory of the "implicate order," in which a hidden order, unity, and wholeness underlies all reality and accounts for all quantum effects, including the non-locality evident in the result of Bell's Theorem. Science writer Martin Gardner [cf. "David Bohm," in Did Adam and Eve Have Navels? Debunking Pseudoscience, W.W. Norton & Company, 2000] describes this as asserting that the separating particles in the EPR case are actually the same particle in a higher dimension [p.77].

Now, it is a respectable and venerable practice in physics to postulate new forces. For such theories to gain popularity, however, there is a great deal to overcome, not the least of which is just Ockham's Razor. If the main reason to have the "quantum potential" is just to preserve a realism and determinism about particles, then most physicists are not going to get too excited. The "implicate order," on the other hand, is a large dose of metaphysics. Just as that may make the theory more attractive to theosophists, it is going to turn off mainstream physics, which is probably why Bohm's name is not even mentioned in Tegmark and Wheeler's article. The implication there is that the hidden variable theories are finished and that the hope for a deterministic quantum mechanics will be found in dealing with the wave function, eliminating discrete particles altogether.

However, it is evident in the article's own terms that even "decoherence" doesn't help much with the basic quantum mechanical dilemmas about possibility. Thus, although it does not occur in the main text, the insert on page 73 contains the telling admission, "Decoherence does not completely eliminate the need for an interpretation such as many-worlds or Copenhagen." Indeed. This is because even the "dissipation" of superpositions still leaves alternative "classical" probabilities. The alternative possibilities are either going to have to separate into different worlds, or they are going to have to collapse into just one particle. The insert on page 74 extends the decoherence of different worlds to the mental states of the observer, who can be both happy and sad about the fall of a playing card without the happy or the sad person being aware of the other. This does not seem to help much in eliminating the strangeness of quantum mechanics or the vast metaphysical overkill of the "many worlds" interpretation. If the wave function collapses into one particle, or one mental state, however, then this maintains the metaphysical dualism between wave function and particle that both Bohm and the decoherentists want to eliminate.

Recently, as noted, Freeman Dyson expresses a preference for dualism, mainly for philosophical reasons. He describes the "quantum only" world posited by decoherence but does not identify any conceptual or physical shortcomings with the theory. I think he is too kind.

If dualism survives, and a dose of metaphysics is in order, then Kant still provides a good alternative. Indeed, Kantian things-in-themselves can provide a modest "undivided wholeness" not unlike Bohm's theory, though with no more than is necessary to explain non-locality, as considered above. While Bohm's attraction to metaphysical synthesis led to a long association with Indian theosophical teacher Jiddu Krishnamurti, the appeal of Bohm's theory to many seems largely based on its elimination of consciousness as a factor in the physics. Thus, Martin Gardner [op.cit.], finds Bohm's realism attractive and says, "Human consciousness is not essential, as von Neumann, Eugene Wigner, and others supposed, to collapse the wave functions" [p.79]. This curiously would represent a triumph, the elimination of consciousness from the fundaments of reality, that Indian philosophy, whether formulated by Krishnamurti or any other popular teacher, would not have found congenial. So perhaps Bohm was not attracted enough to Indian ideas. At the same time, I have not been aware of recent physicists paying attention to Kant [note].

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Kantian Quantum Mechanics, Note 1

The third epigraph above by the physicist Lee Smolin, a quick dismissal of the physical reality of the wave function, ψ(x), in quantum mechanics, suffers from a difficulty. If it is supposed to be an argument, the conclusion does not follow.

The question everyone then faced was how to think of the electron wave that de Broglie had invented and Schrödinger had tamed. Schrödinger at first thought that the electron simply is a wave. This didn't hold up because it was easy to show that the wave tended to spread out in space as it traveled, whereas one could always find a localized particle. Max Born then proposed his rule that the wave is related to the probability of finding the particle.

What Smolin says is that the electron cannot be a wave because a wave will "spread out," while a particle is always "localized." However, where do we find the "localized" particle? Wherever, as it happens, the wave has "spread out." For, as Smolin says himself, the probability of finding the particle is a function of the square of the wave, wherever it is that the wave has "spread out" to.

So, this spreading out is no argument against its being the electron, since the electron, as a particle, might be found anywhere the wave has gone, when the wave function collapses. For the argument to work, the wave would need to "spread out" to some point where we could not ever find the particle. But this is not what he says, or implies.

Thus, Smolin, who is so skeptical of standard quantum mechanics, seems to join with those anti-realists who dismiss the "physical significance" of the wave function. As we see at the link, fasionable physicists like to say things like "in quantum physics unobserved things have no properties whatsoever." Smolin keeps telling us he is a Realist, but this seems to tell us that he is not a Realist when it comes to the wave function.

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Kantian Quantum Mechanics, Note 2

A correspondent helpfully provided some references to recent treatments of Kant in relation to Quantum Mechanics, apparently as closely related to Bohr's antirealism -- references which mostly turned out to be listed at "Copenhagen Interpretation of Quantum Mechanics." In should be remembered in this, however, that Kantianism means empirical realism, while, if the wave function is to be placed at a new level among things in themselves, that introduces another element of realism, i.e. the realistic wave function of de Broglie and Schrödinger. Bohr, who says "Nothing exists until it is measured," does not have a realistic interpretation of the wave function -- or, for that matter, of reality in general:  that is precisely what the "Copenhagen Interpretation" is. This is starkly different from what Kant's view would be of things in themselves. Bohr was not a Kantian. That Bohr, or even Einstein and Gödel, knew his Kant does not mean that he (or Einstein and Gödel) agreed with him. Indeed, even apart from Kant, Bohr's dictum is profoundly paradoxical, since, if nothing exists before we measure it, we would not be able to find anything to measure. This is like the Logical Positivists asserting that nothing has meaning unless it can be verified, even though we would need to understand that meaning before we could know whether it could be verified or not. (And since the Positivists supposedly accepted Hume's critique of Induction, they should have known that science cannot actually verify much of anything, leaving all of it meaningless -- no less than the conclusion of the Deconstructionists.)

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Richard Feynman's
Quantum Mechanics

Now I know that other old men have been very foolish in saying things like this, and, therefore, I would be very foolish to say this is nonsense. I am going to be very foolish, because I do feel strongly that this is nonsense! I can't help it, even though I know the danger in such a point of view. So perhaps I could entertain future historians by saying I think all this superstring stuff is crazy and is in the wrong direction....

I don't like that they're not calculating anything. I don't like that they don't check their ideas. I don't like that for anything that disagrees with an experiment, they cook up an explanation -- a fix-up to say, "Well, it still might be true." For example, the theory requires ten dimensions. Well, maybe there's a way of wrapping up six of the dimensions. Yes, that's possible mathematically, but why not seven? When they write their equation, the equation should decide how many of these things get wrapped up, not the desire to agree with experiment. In other words, there's no reason whatsoever in superstring theory that it isn't eight of the ten dimensions that get wrapped up and that the result is only two dimensions, which would be completely in disagreement with experience. So the fact that it might disagree with experience is very tenuous, it doesn't produce anything; it has to be excused most of the time. It doesn't look right.

Richard Feynman, quoted in Not Even Wrong, The Failure of String Theory and the Search for Unity in Physical Law, by Peter Woit [Basic Books, 2006, p. 174-175]

In Richard Feynman's lectures on Quantum Electrodynamics (QED, The Strange Theory of Light and Matter, Princeton University Press, 1985) we get a certain picture of the behavior of sub-atomic particles and its analysis in quantum mechanics. While there may be other sources expressing Feynman's views, these lectures were delivered in 1983, just five years before his death, and so I take them as representative of his mature thought. While Feynman's theory is sometimes described as an alternative to other attempts to work out the metaphysics of quantum mechanics, I do not think that it is complete or consistent enough to qualify for that description -- though it is definitely a distinct mathematical approach. Nevertheless, it illustrates the approach of a great physicist, one who admittedly was very impatient with philosophical issues and who got a charge out of how odd it all was:

I'm rather delighted that we must resort to such peculiar rules and strange reasoning in order to understand Nature, and I enjoy telling people about it. [p.78]

Of course, Feynman admits that we don't "understand Nature." On the page before the quote given he frankly says, "...the way we have to describe Nature is generally incomprehensible to us" [p.77]. And at the very beginning he warns the reader/auditor of his presentation, "You're not going to be able to understand it" [p.9]. We can easily say about this that we understand some things and don't understand others, which is the normal situation in science. With Feynman, however, we get the picture that the stuff we don't understand he has stopped trying to understand:

I have pointed out these things because the more you see how strangely Nature behaves, the harder it is to make a model that explains how even the simplest phenomena actually work. So theoretical physics has given up on that. [p.82, boldface added]

Is it normal in science for someone to say that a major branch of science, even the queen of the sciences, has "given up on" a major scientific question? When theory and research reach a dead end, certainly, nothing more can be done for the time being, which can certainly be the case with quantum mechanics; but Feynman seems unusually complacent about this. If you reach a dead end, there may be nothing more to do, but you are ready to leap back at it with the slightest hint of a new approach. As a "curious character," Feynman is someone we would expect to be ever ready and eager for that hint.

The truth, I think, is that Feynman seems relatively complacent about this because it involves more of a philosophical issue, more of a metaphysical problem, than he is comfortable with or interested in. He likes what he does, and what he does is physics. You don't need the metaphysics to do the physics, and so Feynman does not feel much loss. And, being a bit of an iconoclast, he delights in the strangeness of it all. On the other hand, Feynman is not some kind of positivist or non-cognitivist. If we don't understand how Nature works, this is not because science is some kind of autistic, masturbatory exercise -- something that we just make up, something that is a mere human convention. If our theories about Nature are strange, it is because Nature is strange. Feynman was also pleased, to be sure, with quantum electrodynamics because of its predictive success (cf. the discussion of the observed and calculated magnetic moment of the electron, pp.6-7), which is what really distinguishes him from the "theorists" who don't believe that there is any real evidence for any scientific theory, except the forms of social control that impose theories on the mystified and oppressed (betraying the Marxist origin of this kind of stuff).

What is the most striking in Feynman's version of quantum mechanics is his impatience with the wave-particle duality:

For many years after Newton, partial reflection by two surfaces was happily explained by a theory of waves, but when experiments were made with very weak light hitting photomultipliers, the wave theory collapsed:  as the light got dimmer and dimmer, the photomultipliers kept making full-sized clicks -- there were just fewer of them. Light behaved as particles. [pp.23-24, boldface added]

This is the key to Feynman's views:  he likes particles and is not interested in waves. This puts him more in the metaphysical camp of Einstein and the older realists and out of step with the developments detailed above which try and preserve the determinism of the wave function. He definitely doesn't like duality:

You had to know which experiment you were analyzing in order to tell if light was waves or particles. This state of confusion was called the "wave-particle duality" of light, and it was jokingly said by someone that light was waves on Mondays, Wednesdays, and Fridays; it was particles on Tuesdays, Thursdays, and Saturdays, and on Sundays, we think about it! It is the purpose of these lectures to tell you how this puzzle was finally "resolved." [p.23, note]

The puzzle, however, was not "resolved," which may be why Feynman here carefully puts that word in "scare" quotes. We get a fuller statement here:

...the wave theory cannot explain how the detector makes equally loud clicks as the light gets dimmer. Quantum electrodynamics "resolves" this wave-particle duality by saying that light is made of particles (as Newton originally thought), but the price of this great advancement of science is a retreat by physics to the position of being able to calculate only the probability that a photon will hit a detector, without offering a good model of how it actually happens. [p.37]

It is worse than that, since Feynman himself must say that the light goes everywhere at once, follows all possible paths, which is something a single finite particle can't do, regardless of the probability of where it may be found by a detector (cf. p.46, about diffraction gratings). So the wave-particle duality is not so easily "resolved." Indeed, Feynman himself later describes rather well how the wave-particle duality works:

Nature has got it cooked up so we'll never be able to figure out how She does it:  if we put instruments in to find out which way the light goes, we can find out, all right, but the wonderful interference effects disappear. But if we don't have instruments that can tell which way the light goes, the interference effects come back! Very strange, indeed! [p.81]

With this, we don't need the "Monday, Wednesday, and Friday" rule. If we know where the particle is, then clearly it can't be everywhere, and the effects that depend on it being everywhere (interference, diffraction), disappear. If we don't know where the particle is, then all the effects explicable by wave mechanics appear. When the cat is away, the mice will play.

But Feynman also retreats occasionally from his flat "light is made of particles" assertion:

In fact, both objects [i.e. electrons and photons] behave somewhat like waves, and somewhat like particles. In order to save ourselves from inventing new words such as "wavicles," we have chosen to call these objects "particles." [p.85]

So now they aren't really particles, we have just "chosen" to call them that, just to avoid irritating neologisms. This is rather different from the "the wave theory collapsed" stage of the account. But are we really dealing with something like "wavicles"? No, because these things actually don't behave "somewhat like waves, and somewhat like particles" -- they behave entirely like waves in some situations, and entirely like particles in others. And what is the difference? As Feynman understands quite well himself, we get particles with localizing detectors, waves without.

Given his preference for particles, what Feynman does is create a mathematical means of duplicating the effects of wave mechanics. His system is called "summing over histories." The "histories" are all the possible tracks that a particle can take, like a photon reflecting off of a surface. The Classical rule is that light follows the shortest path, and that the angle of incidence is equal to the angle of reflection. Most possible paths violate both these rules. What Feynman does is that each possible path is represented by a vector. The length of the vector can be the square root of the probability of the particle going that way, but for reflections (in Chapter 2), Feynman makes the arrows of "arbitrary standard length" (p.41). What is important is the direction of the arrow, and that is determined by a little "stopwatch," which runs, with the arrow rotating as the hand of the watch, as the particle travels. The direction of the arrow when the watch stops gives us what we need to work with. The vectors of a number of possible paths are then put end to end ("summed"). It turns out that vectors for lengthy and improbable paths point in many different directions and result in little net length when put end to end. When we look at the area representing the least distance and the least time, confirming to the classical rules, the vectors point in more or less the same direction; and when they are put end to end add up to a substantial vector, whose square is the overall probability of the particle taking that path. What we get is therefore more or less the Classical result.

I do not mean that to be a comprehensive explanation, just enough to give us a picture here. Anyone wanting more detail should consult QED itself. The key element is the "stopwatch," which gives us the direction of the vector, which makes it possible that the vectors are going to add up to something or cancel each other out. But this is a very unusual stopwatch. It does not measure time. Feynman says, "the stopwatch hand turns around faster when it times a blue photon compared to a red photon" [p.47]. What is it that is "faster" about blue light than red light? Not the velocity, not the rate of time itself (no Relativistic effect here), just the frequency. The rate of the "stopwatch" is determined by the frequency of the photon. But particles do not have frequencies. Waves do. Feynman's stopwatch corresponds, not to time, like ordinary watches, but to the phase of a wave function. The summing of the vectors reproduces the interference effects of waves.

Thus Feynman is able to smuggle characteristics of waves into a theory that is supposed to be about particles. There seems little pretence here that the "stopwatch" represents anything the particle is actually doing, as a particle. It is simply a mathematical device that gets us good results, and its very abstraction and dissociation obscures its correspondence to the natural characteristics and behavior of waves. Feynman, again, seems to rather enjoy the peculiarity of it. As he says elsewhere:

...adding arrows for all the ways an event can happen -- there is no need for an uncertainty principle! [p.56]

But the uncertainty principle is not eliminated by the little arrows. Not only does it remain uncertain where particles are when they are behaving like waves, but it remains impossible that they should be any one place in particular to do what they do. Feynman's enthusiasm mistakes a mathematical abstraction for a substantive conclusion -- a precise and excellent example of the Sin of Galileo. But Feynman knows better than this. He knows that successful mathematics in a successful theory does not mean that we understand what is going on. But he gets carried away, and it is always a temptation to explain what is not understood as something that cannot and need not be understood. We see that in the following passage:

I am not going to explain how the photons actually "decide" whether to bounce back to go through; that is not known. (Probably the question has no meaning.) [p.24]

Not only does it have meaning but there is even an answer:  the photons both bounce back and go through, just as Schrödinger's Cat is both dead and alive. They can do that as waves. They can't do that as particles (unless we use an indefinite number of particles, even in single particle experiments, as Feynman does). Which is the problem. What the photon must "decide" is where it is going to be when the wave function collapses. This is the crux of quantum indeterminacy, and Feynman simply doesn't want to deal with it.

But it is a problem larger than quantum mechanics. It is a problem of the metaphysics of possibility and probability, which no physicist or metaphysician (or physician) has done a very good job of dealing with. When we have just rolled three or four "boxcars" in a row (i.e. 12 on the dice), how do the dice then "know" that it is time to even out the statistical average by avoiding boxcars for a while in the future? Of course, the dice can't "know" because the earlier rolls of the dice have no physical effect on the later ones. But we have similar situations with sub-atomic particles. How do atoms of Uranium "know" that enough other atoms have decayed to account for the statistical half-life of the isotope, and that they need to wait, perhaps for millennia, to decay? Again, it looks like they can't "know," but the individual events just happen to conform to the statistical average. The standard response of the mathematician to give up at that point (or say that it is a question that "has no meaning") is now troubled by the results of the Einstein-Podolsky-Rosen (EPR) Paradox, discussed above. Distant particles, whose properties have some indeterminate quantum correlation, "know" instantaneously what happens to the other particles, if this implies determinate states. This happens without "hidden variables," i.e. without determinate but unknown properties of the particles. What it means is that the wave function is a physical connection and that its collapse is instantaneous, violating Special Relativity. If we apply this to dice throwing, it could mean that the dice do "know," without any Classical physical connection, what has happened in the past.

Richard Feynman, of course, has no intention, and really no interest, in getting into such territory. His statement that the question of how particles "decide" where to go probably "has no meaning" is an afterthought in which fragments of philosophical theory bob like flotsam in the flood. What it would get Feynman, as a theory, is that Nature is incomprehensible because it cannot be understood, i.e. there is no meaning for understanding to get. If true, this would certainly justify his phlegmatic disinterest -- "theoretical physics has given up on that." But Feynman really expresses it more as a wishful thought, or as a decision, than as a real conclusion. It is a limit that he is willing to accept, because what interests him is the mathematical technique that is productive of the predictive results. That is the real stuff, and the philosophical questions, the metaphysics, are less important -- as, in physics, they actually are less important.

Feynman's quantum mechanics in the end benefits from the abstraction that is possible in scientific theories. Not all questions need to be answered, understood, or even addressed to have a successful theory with dramatic results. It is therefore no disqualification to Feynman's greatness that he didn't resolve the basic philosophical problems of quantum mechanics. Certainly nobody else has. What is of interest is his theory as an example of one direction in which we can go with particles alone, avoiding wave mechanics, even though, in the end, characteristics of wave mechanics (the "stopwatch") must be attached, rather extraneously, to the particles. This is revealing -- namely that the physical waves in fact cannot be dispensed with, as Feynman himself occasionally seems aware, as when he says that his "particles," hitherto the vindication of Newton, are actually "somewhat" like waves.


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Note on the Metaphysic of Space;
the Paradoxes of the Ether

When Thomas Young (1773-1829) showed that light created interference patterns and so could only consist of waves, and when James Clerk Maxwell (1831-1879) generalized this into the theory of electromagnetic radiation, both assumed the ontological principle that, since a wave was the deformation of a medium, there would have to be a medium for light or electromagnetic radiation. That was called the "luminiferous" (light bearing) "ether" (Latin aether, Greek αἰθήρ, aithér, Aristotle's fifth element, of which the heavens consisted). Since the speed of a traveling wave is measured in the inertial frame of reference of its medium, this implies that the velocity of light occurs relative to the velocity of the ether. Ideas about the "ether wind" and how the velocity of the earth through the ether could be measured by measuring the velocity of light in different directions (which was the Michelson-Morley experiment) could then be inferred from this theory.

There were, however, logical problems with the idea of the ether. Electromagnetic waves are transverse or shear waves, not like the longitudinal or pressure waves of sound. Transverse waves only exist in a solid medium. Since the displacement of the medium in a transverse wave is perpendicular to the direction of motion, the medium must be elastic so that the deformation will correct itself. Otherwise, the displacement dissipates motion in the medium and the energy is lost. There are no transverse waves in air or water (except on the surface of water, where gravity corrects the deformation); but they do exist in the earth itself. Indeed, the basic bit of evidence that the (outer) core of the earth is molten comes from the fact that transverse (S, shear or secondary) waves from earthquakes do not pass through it, while pressure (P, or primary) waves do. It also happens that the velocity of a transverse wave through its solid medium is proportional to the rigidity (and elasticity) of the medium -- the more rigid (without being brittle), the faster. Since light is extremely fast, the ether must be extremely rigid.

So notice:  How can there be an "ether wind," and how can the earth "move through the ether," when the ether actually must be both solid and extremely rigid -- indeed, the most rigid thing, since light is to Maxwell just about the fastest thing and to Einstein the fastest thing of all? This circumstance is not noticed when people continue to refer to the ether as some kind of gas, or, with at least a recognition that it would have to be a solid, as something like "jello" (as in Richard Wolfson's video physics lectures from The Teaching Company [1996]). Jello is a solid but certainly not very rigid (to say the least).

Now, it turned out (from the Michelson-Morley experiment) that the velocity of light in a vacuum was the same however it was measured, which made it look like any inertial reference frame was at rest with respect to the ether. The velocity of the earth, or anything else, could not be measured relative to the ether. (And the earth could not simply be carrying a patch of ether along with it, since the phenomenon of the aberration of starlight shows that the earth intercepts starlight at an angle, which would not happen if light were passing through the earth's fixed path of ether.) Albert Einstein then made it the basic postulate of Special Relativity that the velocity of light, which was implied by Maxwell's equations, would be the same in any inertial frame of reference, just as all the consequences of Newton's equations were the same in any inertial frame of reference. "Galilean Relativity," which abolished the absolute velocity of rest, is thus following by Einstein's Relativity, which posits the absolute velocity of light. We could forget about the ether.

Yes, we could forget about ether when it came to providing a frame of reference for motion, but there was still the original consideration that "since a wave was the deformation of a medium, there would have to be a medium for light or electromagnetic radiation." Meanwhile, Einstein had proposed, to solve the problem of the Photoelectric Effect, that light came in packets of energy and so were particles, as Newton had believed, rather than waves -- now to be called "quanta" and then "photons." He figured that photons as particles existed independently and so, naturally, did not need a medium as waves did. So Newton was both wrong and right -- wrong about space, right about light. Yet it was, paradoxically, Newton's view of light that really made the metaphysical principle of waves and their medium superfluous. And Einstein himself realized this created a problem with his theory:  The interference effects observed in Young's experiment could only be explained on the basis of a theory of waves. Einstein's difficulty here is rarely noted in discussions of these issues, but he struggled with it for years and never came up with a solution. His desire to somehow combine waves and particles sounds a bit like the theory of David Bohm, but then the Wave-Particle Duality of Bohr was generally accepted to have rendered such efforts unnecessary -- except that it involved a denial of ontological realism [note].

Now, when we consider that it would be impossible for the earth to move through a solid, rigid medium, we could say that this of itself made the ether metaphysically untenable. However, Louis de Broglie later proposed that matter behaves like waves also:  particles create the same interference effects as light, and in quantum mechanics particles can "mix" in ways that can only be explained by summing or subtracting their wave functions. Then, in Paul Dirac's theory of the particle, it turns out that particles, as particles, only have location, not extension. They are "point particles," like the atoms found in some schools of Kalâm. So particles, as particles, pace Einstein, do not fill space. They do not preclude the existence of a rigid medium of ether. What fills space are fields; and fields are, well, a good question.

But, if particles are themselves waves, then they also require a medium. The earth would then not be moving through the ether, it would be, like light, a deformation of the ether. If, that is, a wave is a deformation of a medium. Now, according to Einstein's approach, fields are curvatures in the space-time continuum. That makes space-time sound, not just substantial, but malleable; and all this makes it sound as though the only thing that fills space is space itself, which is the substantial substratum of all matter and energy. This would not be a disagreeable thought to Classical metaphysicians such as Descartes, Spinoza, or Parmenides. What space would then be was thus originally answered by Parmenides:  Being itself.

Parmenides did not necessarily identify Being with space himself, but he did think of Being as extended, and his denial of the existence of nothingness, which could mean empty space or the vacuum, became a dominant consideration in the history of philosophy. When either Empedocles or Descartes denied the vacuum, they were following Parmenides. However, where Empedocles had filled space with four elements, Descartes gave matter just one attribute, extension. This unified matter more like Democritus than like Empedocles. Ironically, the modern conception of matter, which in a sense begins with the atomic theory of John Dalton (1766-1844), modified Democritus in the direction of Empedocles, positing discrete and independent atoms in space, but with the provision that these atoms are of different elements -- not the four (or five) classical elements, but the substances experimentally separated by both alchemists and modern chemists.

The empty space of Democritus and the absolute space of Newton seemed vindicated by 19th century physics. When Einstein Relativized motion and allowed the theory of ether to be dropped, however, this was interpreted by many who were aware of the philosophical debates between Newtonians and Leibniz as a refutation of independent Newtonian (or Democritean) space altogether. Leibniz's theory of space, however, hardly seems suitable to modern physics. Leibniz denied the existence of space because he denied the existence of matter and even of real extension. All that existed for Leibniz were the "monads," which were essentially little point particles of consciousness. This does not sound like anything that Einstein, or his interpreters, would have had in mind. Yet it has drawn decades of enthusiasm from confused physicists, commentators, and even philosophers and historians of science.

What the philosophical interpreters may have had in mind, really, was not Leibniz, but Hume -- not a different metaphysic of space but simply skepticism, that space has no independent reality because it is subjectively "constructed" (which is not what Hume said himself, but is how Hume has tended to be read in the 20th century). Not even Kant's theory, that space is a subjective, or phenomenal, but fixed condition of perception, was seriously entertained. On the other hand, as has been noted, Einstein's view of space-time could just as easily be taken to imply that space is substantial in a fashion closer to Parmenides or Descartes. Again, this direction seems to have been largely shunned in philosophical intepretation, which was driven by skeptical presuppositions.

An interpretation of Relativity, however, let alone an entire philosophy of science, founded upon skepticism, always ran the risk of the slide down the logical slope. If space is subjectively constructed, and perhaps arbitrarily so, then so can be all of science. The skeptical tendency of 20th century philosophy then logically led to deconstruction and all the ways in which all of science can be dismissed as an artifact of quasi-Marxist "power" relationships. This has not helped, to say the least, in understanding the nature of space. But if neither Leibniz nor Hume are the answer to Einstein, then the whole matter must be reconsidered in a way that was really never done in the 20th century.

Three Points in Kant's Theory of Space and Time

The Clarke-Leibniz Debate (1715-1716)

Immanuel Kant (1724-1804)

Einstein's Equation for Gravity

Rotation as Gravity

A Metaphysic of the Forces of Nature in Multiple Dimensions


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Note on the Metaphysic of Space; the Paradoxes of the Ether; Note;
Bose-Einstein Statistics

While Einstein realized that to abandon the wave nature of light meant to leave interference effects unexplained, he also ended up forging ahead with an insight for quantum mechanics that would also depend on the wave nature of light. This was the principle of the mixing or "superposition" of particles. The path into it involved the development of the Bose-Einstein Statistics for particles, which Einstein pursued on the basis of a paper sent to him by Satyenedra Nath Bose (1894-1974).

The diagram illustrates the Bose-Einstein Statistics in the way they were explained by Albert Einstein himself in a 1925 letter to Irwin Schrödinger [cf. A. Douglas Stone, Einstein and the Quantum, The Quest of the Valiant Swabian, Princeton, 2013, p.239]. Ordinarily, if we toss two coins in the air, half the time (2/4) we will get one coin heads and the other tails. We get this result in two out of four tosses because one coin is one way and the other the other, but which is which can be switched. Now, Einstein decided that the "switched" cases are not going to be different in quantum mechanics because the absolutely identical nature of the coins (or of particles) means, not only is there no way to distinguish them, but that in principle they cannot be reckoned as distinct. The cases where one is one way and the other is the other are consequently the same case as far as quantum mechanics is concerned. The means that the probability of tossing the coins where one is heads and one is tails only happens one out of three times rather than two out of four times. This form of probability applies today to particles of even spin, which carry the energy of a particular kind of field, which are consequently called "bosons."

How is it possible that Coin A and Coin B, when one is heads and the other is tails, are in the identical state regardless of which one is heads and which one is tails? Well, if they are distinct particles, we ought to be able to keep track of which is which by their location in space. On the other hand, if they are waves, then they can partake of the nature of waves, which is that they can be at the same place in space, which means that they add together and form a single, combined wave. This is the mixing or superposition of waves/particles, by which the individuality of the particles is lost, and the wave embodies both of them.

Thus, one of Einstein's most profound inquires resulted in something that one of his other most profound inquires, the particle nature of light, could not explain. When Richard Feynman casually asserted that "the wave theory collapsed" through the consideration of the Photoelectric Effect, he failed to note that the old nature of the interference effects and Einstein's new formulation of probability both still relied on waves. If the old metaphysical requirement of waves that they occur in a medium was then not to persist, something would have to done. As it happened, what was generally accepted as having been done was Bohr's rejection of realism:  The wave function represented probability, and this did not translate into physical reality until the wave function collapsed, and particles appeared, because of observation and measurement. However, both interference effects and the Bose-Einstein Statistics involve physical realities before observation and the collapse of the wave function. That has just been left slowly twisting in the wind by subsequent physics.

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