by Peter Byrne
Defying the logic of classical physics, what happens to each of these entangled particles instantaneously affects its partner, even though they are separated by distances measured in light years.11 But this phenomenon is not confined to experiment; since the beginning of time particles have been interacting, exchanging energy, entangling, cooking up reality.
Entanglement was at the root of the famous “EPR paradox” posed by Einstein, Nathan Rosen, and Boris Podolsky in 1935 to challenge Bohr’s claim that quantum mechanics is a complete description of reality. In brief, EPR (as reformulated by Bohm some years later) pointed out that the instantaneous determination of the spins of a pair of entangled but spatially separated particles would violate special relativity’s prohibition against faster than light speed action or “non-locality.” The violation was supposed to occur because by measuring the up spin of one particle in the entangled pair, the spin of its (possibly superluminally) separated partner instantaneously became down: Einstein called it “spooky-action-at-a-distance.”12
As Everett knew, Schrödinger, in 1935, was of the opinion that wave functions exactly model physical reality. The inventor of wave mechanics defined entanglement as,
When two systems, of which we know the states by their respective representatives [wave functions], enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz., by endowing each of them with a representative of its own.
I would not call that one but rather the characteristic trait of quantum mechanics. By interaction the two representatives [wave functions] have become entangled…. Another way of expressing the peculiar situation is: the best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts, even though they may be entirely separated.13
In quantum mechanics the particles composing the observer continuously “entangle” with each other and the particles of the objects under observation as their respective wave functions combine. Everett was to take entanglement very seriously. It is at the heart of his universal wave function, which describes physical reality as a vast superposition of all possible quantum states and all possible worlds. In the end, thought Everett, it is not possible for an observer to stand outside a quantum state that necessarily includes himself because the whole universe is entangled with itself.
Schrödinger’s jellyfish
It was 1929. Sir Nevill Francis Mott, a British physicist, was perplexed by the results of an experiment showing the trajectory of a particle inside a Wilson cloud chamber, (a sealed container filled with super cooled water vapor). So he wrote a famous paper that made the measurement problem concrete.
Here is the story: In a cloud chamber, alpha particles emitted by the nucleus of a radioactive atom create droplets of observable condensation by interacting with water molecules in the vapor. According to Mott’s calculations, the alpha particle wave—think of it as a superposition of possible particle trajectories—spreads in all directions. It is “localized” when it interacts with a water molecule, leaving behind a droplet. A series of such droplets form a track of condensation, depicting the trajectory of the particle. So a spherically spreading wave should leave behind a droplet record that is spherical.
But this is not what happened inside the cloud chamber: only a single track appeared. Somehow, the interaction of the spherical wave with its environment selected only a few interactions to become real. Mott could not explain how or why this happened—why only one element of the superposition became real. But he decided that the spherical wave function of the alpha particle had somehow narrowed or “collapsed” itself, vanishing all possible trajectories but one.14
Mott’s conclusion bothered Schrödinger, who, like Everett, did not accept the “wave collapse” interpretation. In his thesis, Everett cited a paper in which Schrödinger insisted that experiments such as that described by Mott,
cannot [be] account[ed] for without taking the wave to be a wave, acting simultaneously throughout the region over which it spreads, not ‘perhaps here’ or ‘perhaps there,’ as the probability view would have it…. That would fail to account for the interference phenomena [i.e. the superposition principle].15
Wave functions are physically real, Schrödinger insisted, although he could not explain why only one alpha particle track emerged in the cloud chamber.
The problem is that Schrödinger’s wave equation evolves all of the possible positions of the spherical alpha wave burped from the nucleus as it interacts sequentially with molecules in the cloud chamber. Until the measurement interaction with the water molecules occurred, the wave equation described the future of each possible state of the particle. It was continuous and causal. The wave equation did not contain a mechanism for selecting out a single track in Mott’s cloud chamber to emerge to the exclusion of all other possible tracks, and yet, this happened.
In 1935, Schrödinger summarized the problem of measurement:
Any measurement suspends the law [Schrödinger equation] that otherwise governs continuous time-dependence of the ψ-function and brings about in it a quite different change, not governed by any law but rather dictated by the result of the measurement. But laws of nature differing from the usual ones cannot apply during a measurement, for objectively viewed it is a natural process like any other, and it cannot interrupt the orderly course of natural events. Since it does interrupt that of the ψ-function, the [ψ-function] … can not serve, like the classical model, as an experimentally verifiable representation of an objective reality. And yet [it does].16
In other words, the emergence of a single alpha particle track from all possible tracks for that alpha particle was inexplicable. Since Mott, several generations of physicists have searched for answers to why and how the classical world of our experience emerges from the multiplicity of possible experiences lurking inside wave functions. It is appropriate to ask, as Schrödinger did, why do we not observe macroscopic objects obeying the Schrödinger equation as their constituent parts must?
For example, when the quantum object named Mott, himself a superposition of possible states, looked into his cloud chamber, why didn’t each Mott in the superposition of possible Motts correlate with each possible track of the alpha particle? That was, in fact, Everett’s conclusion: Mott did indeed correlate to each element included in the alpha particle wave function and he immediately split into multiple Motts, each Mott seeing and remembering a different, alpha particle track. The sum of single tracks witnessed by the sum of Motts equals the information about possible particle trajectories contained in the spherical wave function, according to Everett.
Following Schrödinger, Everett believed that Born’s probability rule was an illusion. Both theorists wanted the real world to reflect the seamless beauty of the linear wave equation, which protects the continuous individuality of each and every possibility. But Schrödinger was not prepared to follow Everett to his ultimately unsettling conclusion, one that encompassed many Motts. Schrödinger considered what would happen if we did evolve according to his equation.
In a 1952 lecture at the university of Dublin, he observed:
Nearly every result [a quantum theorist] pronounces is about the probability of this or that … happening—with usually a great many alternatives. The idea that they be not alternatives but all really happen simultaneously seems lunatic to him just impossible. He thinks that if the laws of nature took this form for, let me say, a quarter of an hour, we should find our surroundings rapidly turning into a quagmire, or sort of a featureless jelly or plasma, all contours becoming blurred, we ourselves probably becoming jelly fish. It is strange that he should believe this. For I understand he grants that unobserved nature does behave this way—namely according to the wave equation. The aforesaid alternatives come into play only when we make an observation—which need, of course, not be a scientific observation. Still it would seem that
, according to the quantum theorist, nature is prevented from rapid jellification only by our perceiving or observing it. And I wonder that he is not afraid, when he puts a ten-pound note into his drawer in the evening, he might find it dissolved in the morning, because he has not kept watching it.17
But he could not accept Bohr’s complementary approach to measurement, which he originally characterized in 1935:
Now while the new theory [Bohr’s complementarity] calls the classical model incapable of specifying all details of the mutual interrelationship of the determining parts (for which its creators intended it), it nevertheless considers the model suitable for guiding us as to just which measurements can in principle be made on the relevant object. This [is] an unscrupulous proscription against future development … born of distress.18
Unable to solve the measurement problem, or to reconcile himself with Bohr’s dualism, Schrödinger oscillated between realism (the wave function is physically real), and anti-realism (it is a mathematical ideal). Like Wigner and von Neumann, he eventually developed a “mentalist” interpretation of quantum mechanics based on the intervention of human consciousness in the measuring process.
But Everett found in Schrödinger’s pure wave mechanics a platform on which to mount his own realist interpretation, which, he claimed, could be derived directly from the quantum mechanical formalism invented by Schrödinger, without adding new assumptions and postulates. In fact, Everett’s model of quantum mechanics was simply based on following Schrödinger’s logic to wherever it might lead.
To do this, he began by tackling the notion of “wave function collapse,” which is the idea that it is acceptable to suspend the smooth and continuous evolution of the quantum world in order to accommodate our perception of single results. And here he confronted von Neumann’s most powerful and long lasting contribution to quantum theory: the wave function collapse postulate that vitiated the universality of the Schrödinger equation. The collapse theory was not a solution to the measurement problem, but it effectively swept it under the rug for decades.
11 Collapse and Complementarity
Mathematics can only prove statements of the type: if theorem A is true, then theorem B is also true; but it can never prove whether A is true or not…. No statement of physics can be proved mathematically, but … every ‘proof’ in physics consists only in deriving mathematically one physical fact from other statements about physical facts.
Philipp Frank, 19491
Physical facts cannot be derived from mathematics.
Dieter Zeh, 20082
John von Neumann’s Mathematical Foundations of Quantum Mechanics was the prevailing textbook on the subject well into the 1950s. In it, von Neumann axiomatized quantum mechanics. He presented “wave function reduction” or the “collapse” postulate as a panacea for the measurement problem. His approach was widely accepted, although it was challenged in the post-war years by some prominent theorists, including Schrödinger, Bohm, and Henry Margenau of Yale University. But it was Everett’s frontal attack on von Neumann that allowed critics of his “orthodox” interpretation of quantum mechanics some breathing space to say, yes, there is a problem with the axioms.
Axioms are unproven rules or postulates from which other rules may be derived. One of von Neumann’s axioms was that the Schrödinger equation governs how each superposed element in a quantum system dynamically changes over time as it evolves causally, deterministically. Another axiom said that the act of measurement arbitrarily and instantaneously “projects” one element of a superposition into classical reality. This axiom is known the “collapse” postulate; its discontinuity interrupts the continuity of the Schrödinger equation’s representation of reality.
Could both axioms be true?
Physical reality told von Neumann that one measurement result emerges, but he could not mathematically derive this result from quantum mechanics. He explained the dilemma:
In the discussions so far, we have treated the relation of quantum mechanics to the various causal and statistical methods of describing nature. In the course of this we found a peculiar dual nature of the quantum mechanical procedure which could not be satisfactorily explained.3
The problem he pointed to is that in the objective formalism of quantum mechanics, nature proceeds causally, deterministically, but our perception of nature is subjective: macroscopic reality appears to emerge randomly, non-causally from the microcosm. Deciding in favor of a touchable (if indeterminist) reality, von Neumann continued,
It is a fundamental requirement of the scientific viewpoint—the so-called principle of psycho-physical parallelism—that it must be possible so to describe the extra-physical process of the subjective perception as if it were in reality in the physical world—i.e., to assign to its parts equivalent physical processes in the objective environment, in ordinary space.4
Adopting a dualistic approach, von Neumann decided that if both axioms were correct there are two kinds of changes in quantum mechanical states:
• The collapse postulate, which describes “the discontinuous, non-casual and instantaneously acting experiments of measurements,” or “arbitrary changes by measurement.”
• The Schrödinger equation’s “continuous and causal changes in the course of time,” which evolve the elements of a superposition as “automatic changes.”5
It is precisely this dichotomy that Everett was to attack in the first sentence of his thesis when he called for abandoning the collapse rule. There was, however, a rhyme and reason to von Neumann’s dualism. He noted that microscopic processes as guided by the Schrödinger equation are, in theory, reversible. But according to the second law of thermodynamics—that entropy increases in a closed system—a macroscopically obtained measurement of a microscopic system is not reversible (for all practical purposes). As the process of change in our macroscopic universe follows an irreversible arrow of time, von Neumann reasoned that, according to psycho-physical parallelism, collapse, as an explanation, overrules the unseen continuity expressed by the wave equation. Only, he could not prove that this was so, so he postulated it. And there were obvious flaws. For example, simply asserting that the wave function of a measured particle is reduced because a particular property of a particle somehow emerges from a superposition of possible results to a single result (without saying why) begs the question of where and at what instant does the collapse of the wave function occur?
Like Bohr, von Neumann believed that, by definition, a valid observation or measurement of a quantum system occurs outside the system (Everett claimed to put the measurement process inside the system). But, if the observer and the object observed are microscopically entangled, which von Neumann believed to be the case, where is the cut? At what point in space or time does a measurement become externalized, as entanglement with adjacent objects is constantly going on according to the superposition principle? At what precise moment does the wave function reduce into a measurement of a piece of the total system? Does the reduction occur when the Geiger counter needle registers a single measurement? But isn’t the needle in a superposition until it is measured by the eyeball of the experimenter? And what about the eyeball and, for that matter, the observer’s friend who becomes correlated to the quantum system under observation by the mere fact of entering the laboratory? Did the wave function of the experimenter who is looking at the needle registering the measurement result collapse before or after his friend looked at him? And when does the friend’s wave function collapse?
The difficulty of saying where and when the wave function collapses poses the problem of an infinite regression: how can an observer ever get outside the serially entangling system to observe it without being an intrinsic part of it?
Echoing Bohr’s arbitrary insertion of an epistemological partition between the quantum and classical realms to achieve externality, von Neumann remarked,
But in any case, no matter how far we calculate—to the [measuring device], to the retina, or into the brain, at some time
we must say: and this is perceived by the observer. That is, we must always divide the world into two parts, the one being the observed system, the other the observer. In the former we can follow up all physical processes (in principle at least) arbitrarily precisely. In the latter, this is meaningless. The boundary between the two is arbitrary to a very large extent…. That this boundary can be pushed arbitrarily deeply into the interior of the body of the actual observer is the content of the principle of the psycho-physical parallelism—but this does not change the fact that in each method of description the boundary must be put somewhere, if the method is not to proceed vacuously, i.e., if a comparison with experiment is to be possible. Indeed, experience only makes statements of this type: an observer has made a certain (subjective) observation; and never any like this: a physical quantity has a certain value.6
Unable to mathematically staunch the infinite regression—which threatened to make the whole universe quantum mechanical, thereby making external observation of a classical result impossible—von Neumann (probably influenced by Wigner) arbitrarily declared that the chain of measurement ended with the “abstract ego” of the observer. And that this arbitrary cut was philosophically justified by the principle of psycho-physical parallelism.7