For over forty years Albert Einstein was concerned about "nonlocal" instantaneous actions between widely separated objects before he coined the famous description as spooky action at a distance (Spukhafte Fernwerken in German) in a letter to Max Born in May 1947, a few years before his death. We look at eleven critical years of Einstein's work on nonlocality.

Few histories point out that it was Einstein who over three decades invented (or discovered) nonlocality and entanglement, as well as the ontological chance in quantum mechanics that is the real basis of the acausality that Heisenberg years later saw in his uncertainty principle.

In accordance with the assumption to be considered here, the energy of a light ray spreading out from a point source is not continuously distributed over an increasing space but consists of a finite number of energy quanta which are localized at points in space, which move without dividing, and which can only be produced and absorbed as complete units.

We therefore arrive at the conclusion: the greater the energy density and the wavelength of a radiation, the more useful do the theoretical principles we have employed turn out to be; for small wavelengths and small radiation densities, however, these principles fail us completely.

Thermodynamically, radiation behaves like gas particles. Light cannot be spread out continuously in all directions if the energy is absorbed as a unit that ejects a photoelectron in the photoelectric effect.

[W]e further conclude that: Monochromatic radiation of low density (within the range of validity of Wien's radiation formula) behaves thermodynamically as though it consisted of a number of independent energy quanta.

("A Heuristic Viewpoint on the Production and Transformation of Light," English translation - American Journal of Physics, 33, 5, 367 )

An electron to which kinetic energy has been imparted in the interior of the body will have lost some of this energy by the time it reaches the surface. Furthermore, we shall assume that in leaving the body each electron must perform an amount of work P characteristic of the substance...

Why did Bohr not see in 1913, or Einstein point out to him, that when a jumping electron in an atom absorbs or emits energy, the energy is a single light quantum particle? He surely knew that was what was happening in the Bohr atom!

If each energy quantum of the incident light, independently of everything else, delivers its energy to electrons, then the velocity distribution of the ejected electrons will be independent of the intensity of the incident light; on the other hand the number of electrons leaving the body will, if other conditions are kept constant, be proportional to the intensity of the incident light...

In the foregoing it has been assumed that the energy of at least some of the quanta of the incident light is delivered completely to individual electrons

("A Heuristic Viewpoint on the Production and Transformation of Light," English translation - American Journal of Physics, 373 )

If the energy travels as a spherical light wave radiated into space in all directions, how can it instantaneously collect itself together to be absorbed into a single electron. Einstein already in 1905 saw something nonlocal about the photon and that there is both a wave aspect and a particle aspect to electromagnetic radiation. He will make those aspects more clear and in 1909 describe the wave-particle relationship more clearly than it is usually presented today, with all the confusion about whether photons and electrons are waves or particles or both.

Einstein greatly expanded his light-quantum hypothesis in a presentation at the Salzburg conference in September, 1909. He argued that the interaction of radiation and matter involves elementary processes that are not "invertible," a deep insight into the irreversibility of natural processes. While incoming spherical waves of radiation are mathematically possible, they are not practically achievable. Nature appears to be asymmetric in time. He speculates that the continuous electromagnetic field might be made up of large numbers of light quanta - singular points in a field that superimpose collectively to display the wavelike behavior.

Although he could not formulate a mathematical theory that does justice to both the continuous oscillatory waves and the discrete particle pictures, Einstein argued that they could be "fused" and made compatible. This was over a decade before Erwin Schrödinger's wave mechanics and Werner Heisenberg's quantum mechanics. And because gases behave statistically, he knows that the connection between the wave and particles may involve probabilistic behavior.

When light was shown to exhibit interference and diffraction, it seemed almost certain that light should be considered a wave.

The greatest advance in theoretical optics since the introduction of the oscillation theory was Maxwell's brilliant discovery that light can be understood as an electromagnetic process...One became used to treating electric and magnetic fields as fundamental concepts that did not require a mechanical interpretation.

This path leads to the so-called relativity theory. I only wish to bring in one of its consequences, for it brings with it certain modifications of the fundamental ideas of physics. It turns out that the inertial mass of an object decreases by L / c² when that object emits radiation of energy L...the inertial mass of an object is diminished by the emission of light.

Now Einstein looks for symmetry and equivalent treatment for interchangeable matter and energy.

The energy given up was part of the mass of the object. One can further conclude that every absorption or release of energy brings with it an increase or decrease in the mass of the object under consideration. Energy and mass seem to be just as equivalent as heat and mechanical energy.

Relativity theory has changed our views on light. Light is conceived not as a manifestation of the state of some hypothetical medium, but rather as an independent entity like matter. Moreover, this theory shares with the corpuscular theory of light the unusual property that light carries inertial mass from the emitting to the absorbing object. Relativity theory does not alter our conception of radiation's structure; in particular, it does not affect the distribution of energy in radiation-filled space.

Einstein is about to tell us that the distribution of energy in radiation-filled space may be similar in some respects to the distribution of particles in matter-filled space!

Nevertheless, with respect to this question, I believe that we stand at the beginning of a development of the greatest importance that cannot yet be surveyed. The statements that follow are largely my personal opinion, or the results of considerations that have not yet been checked enough by others. If I present them here in spite of their uncertainty, the reason is not an excessive faith in my own views, but rather the hope to induce one or another of you to deal with the questions considered.

In the kinetic theory of molecules, for every process in which only a few elementary particles participate (e.g., molecular collisions), the inverse process also exists. But that is not the case for the elementary processes of radiation.

Incoming spherical waves (the advanced potential considered by Wheeler and Feynman in 1945) are never observed in nature. Radiation is irreversible, one of the arrows of time

According to our prevailing theory, an oscillating ion generates a spherical wave that propagates outwards. The inverse process does not exist as an elementary process. A converging spherical wave is mathematically possible, to be sure; but to approach its realization requires a vast number of emitting entities. The elementary process of emission is not invertible. In this, I believe, our oscillation theory does not hit the mark. Newton's emission theory of light seems to contain more truth with respect to this point than the oscillation theory since, first of all, the energy given to a light particle is not scattered over infinite space, but remains available for an elementary process of absorption.

Consider the laws governing the production of secondary cathode radiation by X-rays. If primary cathode rays impinge on a metal plate P1, they produce X-rays. If these X-rays impinge on a second metal plate P2, cathode rays are again produced whose speed is of the same order as that of the primary cathode rays.

As far as we know today, the speed of the secondary cathode rays depends neither on the distance between P1 and P2, nor on the intensity of the primary cathode rays, but rather entirely on the speed of the primary cathode rays. Let's assume that this is strictly true. What would happen if we reduced the intensity of the primary cathode rays or the size of P1 on which they fall, so that the impact of an electron of the primary cathode rays can be considered an isolated process?

In his remarks after the talk, Johannes Stark confirmed that he had observed a single X-ray that traveled as far as ten meters and ejected a similar energy electron from P2.

If the above is really true then, because of the independence of the secondary cathode rays' speed on the primary cathode rays' intensity, we must assume that an electron impinging on P1 will either cause no electrons to be produced at P2, or else a secondary emission of an electron whose speed is of the same order as that of the initial electron impinging on P1. In other words, the elementary process of radiation seems to occur in such a way that it does not scatter the energy of the primary electron in a spherical wave propagating in every direction, as the oscillation theory demands.

That energy is possibly available "somewhere else" is the key idea of nonlocality that Einstein will present in 1927 at the Solvay conference

Rather, at least a large part of this energy seems to be available at some place on P2, or somewhere else. The elementary process of the emission of radiation appears to be directional. Moreover, one has the impression that the production of X-rays at P1 and the production of secondary cathode rays at P2 are essentially inverse processes.

Therefore, the constitution of radiation seems to be different from what our oscillation theory predicts. The theory of thermal radiation has given important clues about this, mostly by the theory on which Planck based his radiation formula...

Planck's theory leads to the following conjecture. If it is really true that a radiative resonator can only assume energy values that are multiples of hν, the obvious assumption is that the emission and absorption of light occurs only in these energy quantities. On the basis of this hypothesis, the light-quanta hypothesis, the questions raised above about the emission and absorption of light can be answered. As far as we know, the quantitative consequences of this light-quanta hypothesis are confirmed. This provokes the following question. Is it not thinkable that Planck's radiation formula is correct, but that another derivation could be found that does not rest on such a seemingly monstrous assumption as Planck's theory? Is it not possible to replace the light-quanta hypothesis with another assumption, with which one could do justice to known phenomena? If it is necessary to modify the theory's elements, couldn't one keep the propagation laws intact, and only change the conceptions of the elementary processes of emission and absorption?

As far as I know, no mathematical theory has been advanced that does justice to both its oscillatory structure and its quantum structure...

Anyway, this conception seems to me the most natural: that the manifestation of light's electromagnetic waves is constrained at singularity points, like the manifestation of electrostatic fields in the theory of the electron. It cannot be ruled out that, in such a theory, the entire energy of the electromagnetic field could be viewed as localized in these singularities, just like the old theory of action-at-a-distance. I imagine to myself, each such singular point surrounded by a field that has essentially the same character as a plane wave, and whose amplitude decreases with the distance between the singular points. If many such singularities are separated by a distance small with respect to the dimensions of the field of one singular point, their fields will be superimposed, and will form in their totality an oscillating field that is only slightly different from the oscillating field in our present electromagnetic theory of light. Of course, it need not be emphasized that such a picture is worthless unless it leads to an exact theory. I only wished to illustrate that the two structural properties of radiation according to Planck's formula (oscillation structure and quantum structure) should not be considered incompatible with one another.

("On the Development of Our Views Concerning the Nature and Constitution of Radiation," original in German, "Entwicklung unserer Anschauungen über das Wesen und die Konstitution der Strahlung," Physikalische Zeitschrift, 10, 817-825, 1909 )

In his two papers on quantum mechanics in 1916-17, Einstein's discovery of ontological chance is the most important contribution to physics and philosophy. But his insight into the asymmetry of the emission and absorption processes may be used to discover the origin of irreversibility and an explanation for Boltzmann's hypothesis of "molecular disorder."

What we might call Einstein's "radiation asymmetry" was introduced with these words,

When a molecule absorbs or emits the energy ε in the form of radiation during the transition between quantum theoretically possible states, then this elementary process can be viewed either as a completely or partially directed one in space, or also as a symmetrical (nondirected) one. It turns out that we arrive at a theory that is free of contradictions, only if we interpret those elementary processes as completely directed processes.

("On the Quantum Theory of Radiation", p.65)

The elementary process of the emission and absorption of radiation is asymmetric, because the process is directed, as Einstein had explicitly noted first in 1909, and we know he had seen as early as 1905. The apparent isotropy of the emission of radiation is only what Einstein called "pseudo-isotropy" (Pseudoisotropie), a consequence of time averages over large numbers of events. Einstein often substitutes time averages for space averages, or averages over the possible states of a system in statistical mechanics.

a quantum theory free from contradictions can only be obtained if the emission process, just as absorption, is assumed to be directional. In that case, for each elementary emission process Z_m->Z_n a momentum of magnitude (ε_m—ε_n)/c is transferred to the molecule. If the latter is isotropic, we shall have to assume that all directions of emission are equally probable.

If the molecule is not isotropic, we arrive at the same statement if the orientation changes with time in accordance with the laws of chance. Moreover, such an assumption will also have to be made about the statistical laws for absorption, (B) and (B'). Otherwise the constants B^m_n and Bⁿ_m would have to depend on the direction, and this can be avoided by making the assumption of isotropy or pseudo-isotropy (using time averages).

("On the Quantum Theory of Radiation, pp.67-68)

Now the principle of microscopic reversibility is a fundamental assumption of statistical mechanics. It underlies the principle of "detailed balancing," which is critical to the understanding of chemical reactions. In thermodynamic equilibrium, the number of forward reactions is exactly balanced by the number of reverse reactions. But microscopic reversibility, while true in the sense of averages over time, should not be confused with the reversibility of individual collisions between molecules.

The equations of classical dynamics are reversible in time. And the deterministic Schrödinger equation of motion in quantum mechanics is also time reversible. Irreversibility thus depends on the "projection" of a superposition of states into a single state, the so-called "collapse" of the wave function.

In 1924, Einstein received an amazing very short paper from India by Satyendra Nath Bose. Einstein must have been pleased to read the title, "Planck's Law and the Hypothesis of Light Quanta." It was more attention to Einstein's 1905 work than anyone had paid in nearly twenty years. The paper began by claiming that the "phase space" (a combination of 3-dimensional coordinate space and 3-dimensional momentum space) should be divided into small volumes of h³, the cube of Planck's constant. By counting the number of possible distributions of light quanta over these cells, Bose claimed he could calculate the entropy and all other thermodynamic properties of the radiation.

Bose easily derived the inverse exponential function, Einstein too had derived this. Maxwell and Boltzmann derived it, without the additional -1, by analogy from the Gaussian exponential tail of probability and the theory of errors.

(Planck had simply guessed this expression from Wien's law by adding the term - 1 in the denominator of Wien's a / e ^{- bν / T}).

All previous derivations of the Planck law, including Einstein's of 1916-17 (which Bose called "remarkably elegant"), used classical electromagnetic theory to derive the density of radiation, the number of "modes" or "degrees of freedom" of the radiation field,

But now Bose showed he could get this quantity with a simple statistical mechanical argument remarkably like that Maxwell used to derive his distribution of molecular velocities. Where Maxwell said that the three directions of velocities for particles are independent of one another, but of course equal to the total momentum,

Bose just used Einstein's relation for the momentum of a photon,

and he wrote

This led him to calculate a frequency interval in phase space as

which he simply divided by h³, multiplied by 2 to account for two polarization degrees of freedom, and he had derived the number of cells belonging to dν,

without using classical radiation laws, a correspondence principle, or even Wien's law. His derivation was purely statistical mechanical, based only on the number of cells in phase space and the number of ways N photons can be distributed among them.

Einstein immediately translated the Bose paper into German and had it published in Zeitschrift für Physik, without even telling Bose. More importantly, Einstein then went on to discuss a new quantum statistics that predicted low-temperature condensation of any particles with integer values of the spin. So called Bose-Einstein statistics were quickly shown by Dirac to lead to the quantum statistics of half-integer spin particles called Fermi-Dirac statistics. Fermions are half-integer spin particles that obey Pauli's exclusion principle so a maximum of two particles, with opposite spins, can be found in the fundamental h³ volume of phase space identified by Bose.

Einstein's 1916 work on transition probabilities predicted the stimulated emission of radiation that brought us lasers (light amplification by the stimulated emission of radiation). Now his work on quantum statistics brought us the Bose-Einstein condensation. Either work would have made their discoverer a giant in physics, but these are more often attributed to Bose, just as Einstein's quantum discoveries before the Copenhagen Interpretation are mostly forgotten by historians and today's textbooks, or attributed to others.

This may have been Einstein's last positive contribution to quantum physics. Some judge his next efforts as purely negative attempts to discredit quantum mechanics, by graphically illustrating quantum phenomena that seem logically impossible or at least in violation of fundamental theories like his relativity. But information philosophy hopes to provide explanations for Einstein's paradoxes that depend on the immaterial nature of information.

The phenomena of nonlocality, nonseparability, and entanglement may not be made intuitive by our explanations, but they can be made understandable. And they can be visualized in a way that Einstein and Schrödinger might have liked, even if they might still have found the phenomena difficult to believe. We hope even the layperson will see our animations as providing them an understanding of what quantum mechanics is doing in the microscopic world. The animations present standard quantum physics as Einstein saw it, though Schrödinger never accepted the "collapse" of the wave function and the existence of particles making quantum jumps.

Sadly, despite Einstein's two decades of pioneering work on the interaction of photons and electrons, his ideas and concerns were given little attention at this Solvay, though the conference was dedicated to electrons and photons.

The conference was dominated by papers on the new quantum theory delivered by Louis de Broglie, Niels Bohr, Max Born and Werner Heisenberg. It is best known for Einstein's after-hours suggestions to Bohr and Heisenberg probing for faults in the uncertainty principle. Accounts of these events have been told largely by the victors (there are no holes in uncertainty) but Einstein has said they often missed or ignored his important point. That point was the nonlocal behavior of a spherical light wave as it collapses to get absorbed by a single electron. This was Einstein's only contribution mentioned in the published proceedings.

Here are the notes on Einstein's original remarks at the conference and Bohr's brief response. They contain much of Einstein's 1935 EPR paper, except in 1927 only one particle is involved. Entanglement in EPR requires two identical particles.

Notice how Einstein's diagram clearly shows his concerns of over two decades about reconciling a spherical wave (his example is now an electron) and its collapse to being measured at just one point as if it is a particle. At this point in the history of quantum mechanics, wave-particle duality is seen as the debate between Schrödinger's wave mechanics and Heisenberg's particle mechanics.

MR ElNSTEIN. - Despite being conscious of the fact that I have not entered deeply enough into the essence of quantum mechanics, nevertheless I want to present here some general remarks.

One can take two positions towards the theory with respect to its postulated domain of validity, which I wish to characterise with the aid of a simple example.

Let S be a screen provided with a small opening O, and P a hemispherical photographic film of large radius. Electrons impinge on S in the direction of the arrows. Some of these go through O, and because of the smallness of O and the speed of the particles, are dispersed uniformly over the directions of the hemisphere, and act on the film.

Both ways of conceiving the theory now have the following in common. There are de Broglie waves, which impinge approximately normally on S and are diffracted at O. Behind S there are spherical waves, which reach the screen P and whose intensity at P is responsible [massgebend] for what happens at P.

We can now characterise the two points of view as follows.

The waves give the probability or possibilities for a single electron being found at different locations in an ensemble of identical experiments. A wave does not describe a cloud of electrons as Schrödinger had hoped. Einstein's ensemble theory is the correct interpretation.

1. Conception I. - The de Broglie-Schrödinger waves do not correspond to a single electron, but to a cloud of electrons extended in space. The theory gives no information about individual processes, but only about the ensemble of an infinity of elementary processes.

Quantum theory is not complete in this sense. Representing each particle as a narrow wave packet aiming at the point P is mistaken.

2. Conception II. - The theory claims to be a complete theory of individual processes. Each particle directed towards the screen, as far as can be determined by its position and speed, is described by a packet of de Broglie-Schrödinger waves of short wavelength and small angular width. This wave packet is diffracted and, after diffraction, partly reaches the film P in a state of resolution [un etat de resolution].

According to the first, purely statistical, point of view | ψ |² expresses the probability that there exists at the point considered a particular particle of the cloud, for example at a given point on the screen.

If by the same particle, Einstein means that the one individual particle has a possibility of being found at more than one (indeed many) locations on the screen. This is so, but this seems to be conception I?

According to the second, | ψ |² expresses the probability that at a given instant the same particle is present at a given point (for example on the screen). Here, the theory refers to an individual process and claims to describe everything that is governed by laws.

The second conception goes further than the first, in the sense that all the information resulting from I results also from the theory by virtue of II, but the converse is not true. It is only by virtue of II that the theory contains the consequence that the conservation laws are valid for the elementary process; it is only from II that the theory can derive the result of the experiment of Geiger and Bothe, and can explain the fact that in the Wilson [cloud] chamber the droplets stemming from an α-particle are situated very nearly on continuous lines.

Einstein is right that the one elementary process has a possibility of action elsewhere, but that could not mean producing an actual second particle. That would contradict conservation laws.

The "mechanism" of action-at-a-distance is simply the disappearance of possibilities elsewhere when a particle is actualized (localized) somewhere

But on the other hand, I have objections to make to conception II. The scattered wave directed towards P does not show any privileged direction. If | ψ |² were simply regarded as the probability that at a certain point a given particle is found at a given time, it could happen that the same elementary process produces an action in two or several places on the screen. But the interpretation, according to which | ψ |² expresses the probability that this particle is found at a given point, assumes an entirely peculiar mechanism of action at a distance, which prevents the wave continuously distributed in space from producing an action in two places on the screen.

When a particle appears - just one of the multiple nonlocal possibilities becomes actual or localized - at a specific point P , what becomes of the wave that was going off in all other directions? Its "collapse" - the instantaneous going to zero of probabilities - mistakenly appears to Einstein to violate his relativity principle.

In my opinion, one can remove this objection only in the following way, that one does not describe the process solely by the Schrödinger wave, but that at the same time one localises the particle during the propagation. I think that Mr de Broglie is right to search in this direction. If one works solely with the Schrödinger waves, interpretation II of | ψ |² implies to my mind a contradiction with the postulate of relativity.

The permutation of two identical particles does not produce two different points in multidimensional (configuration space).
Einstein and Bose discovered the new quantum statistics and indistinguishability. Dirac and Fermi extended it to electrons.
For example, interchange of the two electrons in the filled first electron shell, 1s², just produces a change of sign for the two-particle wave function.

I should also like to point out briefly two arguments which seem to me to speak against the point of view II. This [view] is essentially tied to a multi-dimensional representation (configuration space), since only this mode of representation makes possible the interpretation of | ψ |² peculiar to conception II. Now, it seems to me that objections of principle are opposed to this multi-dimensional representation. In this representation, indeed, two configurations of a system that are distinguished only by the permutation of two particles of the same species are represented by two different points (in configuration space), which is not in accord with the new results in statistics. Furthermore, the feature of forces of acting only at small spatial distances finds a less natural expression in configuration space than in the space of three or four dimensions.

(Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference, G. Bacciagaluppi and A. Valentini, 2009. p.442)

Bohr's reaction to Einstein's presentation has been preserved. He didn't understand a word! He disingenuously claims he does not know what quantum mechanics is. His response is vague and ends with his ideas on complementarity and the inability to describe a causal spacetime reality.

Does Bohr really not understand? As we have seen, Einstein has been making this general point for many years. Only recently has Bohr taken Einstein's concept of light quanta seriously.

MR BOHR. I feel myself in a very difficult position because I don't understand what precisely is the point which Einstein wants to [make]. No doubt it is my fault.

As regards general problem I feel its difficulties. I would put [the] problem in [an]other way. I do not know what quantum mechanics is. I think we are dealing with some mathematical methods which are adequate for description of our experiments. Using a rigorous wave theory we are claiming something which the theory cannot possibly give. [We must realise] that we are away from that state where we could hope of describing things on classical theories. [I] Understand [the] same view is held by Born and Heisenberg. I think that we actually just try to meet, as in all other theories, some requirements of nature, but [the} difficulty is that we must use words which remind [us] of older theories. The whole foundation for causal spacetime description is taken away by quantum theory, for it is based on [the] assumption of observations without interference. ... excluding interference means exclusion of experiment and the whole meaning of space and time observation ... because we [have] interaction [between object and measuring instrument] and thereby we put us on a quite different standpoint than we thought we could take in classical theories. If we speak of observations we play with a statistical problem There are certain features complementary to the wave pictures (existence of individuals). ...

The saying that spacetime is an abstraction might seem a philosophical triviality but nature reminds us that we are dealing with something of practical interest. Depends on how I consider theory. I may not have understood, but I think the whole thing lies [therein that the] theory is nothing else [but] a tool for meeting our requirements and I think it does.

(Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference, G. Bacciagaluppi and A. Valentini, 2009. pp.440-442)

Twenty-two years later, in his contribution to the Schilpp memorial volume on Einstein, Bohr had no better response to Einstein's 1927 concerns. But he does remember vividly and provides a picture of what Einstein drew on the blackboard.

Here is Bohr's 1949 recollection:

At the general discussion in Como, we all missed the presence of Einstein, but soon after, in October 1927, I had the opportunity to meet him in Brussels at the Fifth Physical Conference of the Solvay Institute, which was devoted to the theme "Electrons and Photons."

Note that they wanted Einstein's reaction to their work, but actually took little interest in Einstein's concern about the nonlocal implications of quantum mechanics, nor did they look at his work on electrons and photons, despite the conference title.

At the Solvay meetings, Einstein had from their beginning been a most prominent figure, and several of us came to the conference with great anticipations to learn his reaction to the latest stage of the development which, to our view, went far in clarifying the problems which he had himself from the outset elicited so ingeniously. During the discussions, where the whole subject was reviewed by contributions from many sides and where also the arguments mentioned in the preceding pages were again presented, Einstein expressed, however, a deep concern over the extent to which a causal account in space and time was abandoned in quantum mechanics.

To illustrate his attitude, Einstein referred at one of the sessions to the simple example, illustrated by Fig. 1, of a particle (electron or photon) penetrating through a hole or a narrow slit in a diaphragm placed at some distance before a photographic plate.

On account of the diffraction of the wave connected with the motion of the particle and indicated in the figure by the thin lines, it is under such conditions not possible to predict with certainty at what point the electron will arrive at the photographic plate, but only to calculate the probability that, in an experiment, the electron will be found within any given region of the plate.

Bohr's labels for points A and B are helpful. The "nonlocal" effects at point B are just that the probability of an electron being found at point B goes to zero instantly (not an "action at a distance") when an electron is localized at point A

The apparent difficulty, in this description, which Einstein felt so acutely, is the fact that, if in the experiment the electron is recorded at one point A of the plate, then it is out of the question of ever observing an effect of this electron at another point (B), although the laws of ordinary wave propagation offer no room for a correlation between two such events.

("Discussion with Einstein," by Niels Bohr, Albert Einstein, Philosopher-Scientist, P. A. Schilpp, 1949. pp.211-213)

Although Bohr seems to have missed Einstein's point completely, Werner Heisenberg at least came to explain it well. In his 1930 lectures at the University of Chicago, Heisenberg presented a critique of both particle and wave pictures, including a new example of nonlocality that Einstein had apparently developed since 1927. It includes Einstein's concern about "action-at-a-distance" that might violate his principle of relativity, and anticipates the Einstein-Podolsky-Rosen paradox. Heisenberg wrote:

In relation to these considerations, one other idealized experiment (due to Einstein) may be considered. We imagine a photon which is represented by a wave packet built up out of Maxwell waves. It will thus have a certain spatial extension and also a certain range of frequency. By reflection at a semi-transparent mirror, it is possible to decompose it into two parts, a reflected and a transmitted packet. There is then a definite probability for finding the photon either in one part or in the other part of the divided wave packet. After a sufficient time the two parts will be separated by any distance desired; now if an experiment yields the result that the photon is, say, in the reflected part of the packet, then the probability of finding the photon in the other part of the packet immediately becomes zero. The experiment at the position of the reflected packet thus exerts a kind of action (reduction of the wave packet) at the distant point occupied by the transmitted packet, and one sees that this action is propagated with a velocity greater than that of light. However, it is also obvious that this kind of action can never be utilized for the transmission of signals so that it is not in conflict with the postulates of the theory of relativity.

(The Physical Principles of the Quantum Theory, W. Heisenberg, 1930. p.39)

In 1933, shortly before he left Germany to emigrate to America, Einstein attended a lecture on quantum electrodynamics by Leon Rosenfeld. Keep in mind that Rosenfeld was perhaps the most dogged defender of the Copenhagen Interpretation. After the talk, Einstein asked Rosenfeld,

“What do you think of this situation?” Suppose two particles are set in motion towards each other with the same, very large, momentum, and they interact with each other for a very short time when they pass at known positions. Consider now an observer who gets hold of one of the particles, far away from the region of interaction, and measures its momentum: then, from the conditions of the experiment, he will obviously be able to deduce the momentum of the other particle. If, however, he chooses to measure the position of the first particle, he will be able tell where the other particle is.

(Niels Bohr, His Life and Work as seen by His Friends and Colleagues, 1967, S. Rozental, pp.128-129)

It is most unfortunate that Einstein did not explain that measuring the momentum of the first particle allows us to deduce the momentum of the second particle because of the conservation of linear momentum.

The same conservation principle explains as Einstein says, "If, however, he chooses to measure the position of the first particle, he will be able tell where the other particle is." If Einstein had called this ability to tell "knowledge (information) at a distance," instead of "spooky action at a distance," entanglement might never have been thought "spooky" at all, just a correlation of properties.

We can diagram a simple case of Einstein’s question as follows after the particles have interacted and separate from the center. We use electrons instead of generic particles, an anachronism introduced by David Bohm in 1952.

After the particles interact at t₁, quantum mechanics describes them with a single two-particle wave function that is not the product of independent single-particle wave functions. In the case of electrons, which are indistinguishable interchangeable particles, it is not proper to say electron 1 goes this way and electron 2 that way. (Nevertheless, it is convenient to label the particles, as we do in the illustration.)

Einstein then asked Rosenfeld, “How can the final state of the second particle be influenced by a measurement performed on the first after all interaction has ceased between them?” This was the germ of the EPR paradox, and ultimately the problem of two-particle entanglement.

Why does Einstein question Rosenfeld and describe this as an “influence,” suggesting an “action-at-a-distance?”

It is only paradoxical in the context of Rosenfeld’s Copenhagen Interpretation, since the second particle is not itself measured and yet we know something about its properties, which the Copenhagen Interpretation says we cannot know without an explicit measurement..

Einstein was clearly correct to tell Rosenfeld that at a later time t₂, a measurement of one particle's position would instantly establish the position of the other particle - without measuring it. Einstein simply used conservation of linear momentum implicitly to calculate (and know) the position of the second particle.

Two years later, reacting to EPR, Schrödinger described two such particles as becoming "entangled" (verschränkt) at their first interaction, so "nonlocal" phenomena are also known as "quantum entanglement."

Although conservation laws are rarely cited as the explanation, they are the physical reason that entangled particles always produce correlated results for all properties. If the results were not always correlated, the implied violation of a fundamental conservation law would cause a much bigger controversy than entanglement itself, as puzzling as that is.

In 1934, Einstein described one way to reconcile nonlocality with a four-dimensional spacetime continuum theory. At this time, Einstein is clearly supportive of Heisenberg's uncertainty principle and the probabilistic nature of quantum theory:

The idea that the light wave (or wave function) gives the probabilities of finding a particle was seen by Einstein decades earlier, though he never published his idea of a "ghost field" (Gespensterfeld). Einstein is too modest.

The modern quantum theory, as associated with the names of de Broglie, Schrödinger, and Dirac, which of course operates with continuous functions, has overcome this difficulty by means of a daring interpretation, first given in a clear form by Max Born: - the space functions which appear in the equations make no claim to be a mathematical model of atomic objects. These functions are only supposed to determine in a mathematical way the probabilities of encountering those objects in a particular place or in a particular state of motion, if we make a measurement. This conception is logically unexceptionable, and has led to important successes...

Einstein sees that nonlocality may be unavoidable.

On the other hand, it seems to me certain that we have to give up the notion of an absolute localization of the particles in a theoretical model. This seems to me to be the correct theoretical interpretation of Heisenberg's indeterminacy relation. And yet a theory may perfectly well exist, which is in a genuine sense an atomistic one (and not merely on the basis of a particular interpretation), in which there is no localizing of the particles in a mathematical model. For example, in order to include the atomistic character of electricity, the field equations only need to involve that a three-dimensional volume of space on whose boundary the electrical density vanishes everywhere, contains a total electrical charge of an integral amount. Thus in a continuum theory, the atomistic character could be satisfactorily expressed by integral propositions without localizing the particles which constitute the atomistic system.

Only if this sort of representation of the atomistic structure be obtained could I regard the quantum problem within the framework of a continuum theory as solved.

("On the Method of Theoretical Physics," Philosophy of Science, Vol. 1, No. 2 (Apr., 1934), pp. 163-169 )

Einstein and colleagues Boris Podolsky and Nathan Rosen, proposed in 1935 a paradox (known by their initials as EPR or as the Einstein-Podolsy-Rosen paradox) to exhibit internal contradictions in the new quantum physics. They hoped to show that quantum theory could not describe certain intuitive "elements of reality" and thus was incomplete. They said that, as far as it goes, quantum mechanics is correct, just not "complete."

Einstein was correct that quantum theory is "incomplete" relative to classical physics, which has twice as many dynamical variables that can be known with arbitrary precision. But half of this information is missing in quantum physics, due to the indeterminacy principle which allows only one of each pair of non-commuting observables (for example momentum or position) to be known with arbitrary accuracy. Even more important, an individual particle, cannot be said to have a known position before a measurement, since evolution described by the unitary and deterministic Schrödinger equation provides us only probabilities.

The most that can be said is that the particle can be found anywhere the probability amplitude is non-zero. This was the core idea of Einstein's claim of "incompleteness." For Bohr to deny this and call quantum mechanics "complete" was just to play word games, which infuriated Einstein.

Einstein was also correct that indeterminacy makes quantum theory an irreducibly discontinuous and statistical theory. Its predictions and highly accurate experimental results are statistical in that they depend on an ensemble of identical experiments, not on any individual experiment. Einstein wanted physics to be a continuous field theory, in which all physical variables are completely and locally determined by the four-dimensional field of space-time in his theory of relativity.

Einstein and his colleagues Erwin Schrödinger, Max Planck, (later David Bohm), and others hoped for a return to deterministic physics, and the elimination of mysterious quantum phenomena like the superposition of states, the mysterious "collapse" of the wave function, and Schrödinger's famous cat. EPR continues to fascinate determinist philosophers of science who hope to prove that quantum indeterminacy does not exist.

But Einstein was also bothered by what is known as "nonlocality," as we saw at the 1927 Solvay conference. This mysterious phenomenon was even more clearly exhibited in EPR experiments as the apparent transfer of something physical faster than the speed of light. Einstein may have already seen this inconsistency with his relativity theory in his 1905 papers.

The 1935 EPR paper was based on a question of Einstein's about two electrons fired in opposite directions from a central source with equal velocities. He imagined them starting at t₀ some distance apart and approaching one another with high velocities. Then for a short time interval from t₁ to t₁ + Δt the particles are in contact with one another.

After the particles are measured and become entangled at t₁, quantum mechanics describes them with a single two-particle wave function that is not the product of two one-particle wave functions. Because electrons are indistinguishable particles, it is not proper to say electron 1 goes this way and electron 2 that way. (Nevertheless, it is convenient to label the particles, as we do in illustrations below.) It is misleading to think that specific particles have distinguishable paths.

Einstein said correctly that at a later time t₂, a measurement of one electron's position would instantly establish the position of the other electron - without measuring it explicitly. And this is correct, just as after the collision of two billiard balls, measurement of one ball tells us exactly where the other one is due to conservation of momentum. But this is not "action at a distance." It is more nearly "knowledge at a distance."

Note that Einstein used conservation of linear momentum to calculate the position of the second electron. Although conservation laws are rarely cited as the explanation, they are the physical reason that entangled particles always produce correlated results. If the results were not always correlated, the implied violation of a fundamental conservation law would be a much bigger story than mysterious entanglement itself, as interesting as that is.

This idea of something measured in one place "influencing" measurements far away challenged what Einstein thought of as "local reality." It came to be known as "nonlocality." Einstein called it "spukhaft Fernwirkung" or "spooky action at a distance." Schrödinger described the two electrons as "entangled" (verschränkt) at their first measurement. Verschränkt means something like cross-linked. It describes someone standing with arms crossed. Today EPR is the classic example of entanglement.

Einstein criticized the collapse of the wave function as "instantaneous-action-at-a-distance." This criticism resembles the criticisms of Newton's theory of gravitation. Newton's opponents charged that his theory was "action at a distance" and instantaneous. Einstein's own field theory of general relativity shows that gravitational influences travel at the speed of light and are mediated by a gravitational field that shows up as curved space-time.

For Einstein, fields like gravitation and electromagnetism are ponderable, a disturbance of the field at one place is propagated at some finite velocity to other parts of the field. But mathematical probability is not a ponderable field in this sense.

When a probability function collapses to unity in one place and zero elsewhere, nothing physical, neither matter nor energy, is moving from one place to the other. Only information changes.

In his 1936 essay for the Journal of the Franklin Institute, Einstein wrote...

Probably never before has a theory been evolved which has given a key to the interpretation and calculation of such a heterogeneous group of phenomena of experience as has the quantum theory. In spite of this, however, I believe that the theory is apt to beguile us into error in our search for a uniform basis for physics, because, in my belief, it is an incomplete representation of real things, although it is the only one which can be built out of the fundamental concepts of force and material points (quantum corrections to classical mechanics). The incompleteness of the representation is the outcome of the statistical nature (incompleteness) of the laws. I will now justify this opinion.

I ask first: How far does the ψ function describe a real condition of a mechanical system? Let us assume the ψ_r to be the periodic solutions (put in the order of increasing energy values) of the Schr­­ödinger equaτion. I shall leave open, for the time being, the question as to how far the individual ψ_r are complete descriptions of physical conditions. A system is first in the condition ψ₁ of lowest energy S₁. Then during a finite time a small disturbing force acts upon the system. At a later instant one obtains then from the Schr­­ödinger equation a ψ function of the form

ψ = Σc_rψ_r,

where the c_r are (complex) constants. If the ψ_r are "normalized," then |c_{123 to this condition a definite energy S, and, in particular, such an energy as exceeds S1 by a small amount (in any case S1 < S < S2). Such an assumption is, however, at variance with the experiments on electron impact such as have been made by J. Franck and G. Hertz, if, in addition to this, one accepts Millikan's demonstration of the discrete nature of electricity. As a matter of fact, these experiments lead to the conclusion that energy values of a state lying between the quantum values do not exist. From this it follows that our function ψ does not in any way describe a homogeneous condition of the body, but represents rather a statistical description in which the cr represent probabilities of the individual energy values. It seems to be clear, therefore, that the Born statistical interpretation of the quantum theory is the only possible one. The ψ function does not in any way describe a condition which could be that of a single system; it relates rather to many systems, to "an ensemble of systems" in the sense of statistical mechanics. If, except for certain special cases, the ψ function furnishes only statistical data concerning measurable magnitudes, the reason lies not only in the fact that the operation of measuring introduces unknown elements, which can be grasped only statistically, but because of the very fact that the ψ function does not, in any sense, describe the condition of one single system. The Schr­­ödinger equation determines the time variations which are experienced by the ensemble of systems which may exist with or without external action on the single system.}

Such an interpretation eliminates also the paradox recently demonstrated by myself and two collaborators, and which relates to the following problem.

Here Einstein presents a much clearer version of EPR than in the "paradox" paper.

Consider a mechanical system constituted of two partial systems A and B which have interaction with each other only during limited time. Let the ψ function before their interaction be given. Then the Schrödinger equation will furnish the ψ function after the interaction has taken place. Let us now determine the physical condition of the partial system A as completely as possible by measurements. Then the quantum mechanics allows us to determine the ψ function of the partial system B from the measurements made, and from the ψ function of the total system. This determination, however, gives a result which depends upon which of the determining magnitudes specifying the condition of A has been measured (for instance coördinates or momenta). Since there can be only one physical condition of B after the interaction and which can reasonably not be considered as dependent on the particular measurement we perform on the system A separated from B it may be concluded that the function is not unambiguously coördinated with the physical condition. This coördination of several ψ functions with the same physical condition of system B shows again that the function cannot be interpreted as a (complete) description of a physical condition of a unit system. Here also the coördination of the ψ function to an ensemble of systems eliminates every difficulty.⁴

The fact that quantum mechanics affords, in such a simple manner, statements concerning (apparently) discontinuous transitions from one total condition to another without actually giving a representation of the specific process, this fact is connected with another, namely the fact that the theory, in reality, does not operate with the single system, but with a totality of systems. The coefficients c_r of our first example are really altered very little under the action of the external force. With this interpretation of quantum mechanics one can understand why this theory can easily account for the fact that weak disturbing forces are able to produce alterations of any magnitude in the physical condition of a system. Such disturbing forces produce, indeed, only correspondingly small alterations of the statistical density in the ensemble of systems, and hence only infinitely weak alterations of the ψ functions, the mathematical description of which offers far less difficulty than would be involved in the mathematical representation of finite alterations experienced by part of the single systems. What happens to the single system remains, it is true, entirely unclarified by this mode of consideration; this enigmatic happening is entirely eliminated from the representation by the statistical manner of consideration.

But now I ask: Is there really any physicist who believes that we shall never get any inside view of these important alterations in the single systems, in their structure and their causal connections, and this regardless of the fact that these single happenings have been brought so close to us, thanks to the marvelous inventions of the Wilson chamber and the Geiger counter? To believe this is logically possible without contradiction; but, it is so very contrary to my scientific instinct that I cannot forego the search for a more complete conception.

To these considerations we should add those of another kind which also voice their plea against the idea that the methods introduced by quantum mechanics are likely to give a useful basis for the whole of physics. In the Schrödinger equation, absolute time, and also the potential energy, play a decisive role, while these two concepts have been recognized by the theory of relativity as inadmissable in principle. If one wishes to escape from this difficulty he must found the theory upon field and field laws instead of upon forces of interaction. This leads us to transpose the statistical methods of quantum mechanics to fields, that is to systems of infinitely many degrees of freedom. Although the at tempts so far made are restricted to linear equations, which, as we know from the results of the general theory of relativity, are insufficient, the complications met up to now by the very ingenious at tempts are already terrifying. They certainly will rise sky high if one wishes to obey the requirements of the general theory of relativity, the justification of which in principle nobody doubts.

To be sure, it has been pointed out that the introduct ion of a space-time continuum may be considered as contrary to nature in view of the molecular structure of everything which happens on a small scale. It is maintained that perhaps the success of the Heisenberg method points to a purely algebraical method of description of nature, that is to the elimination of continuous functions from physics. Then, however, we must also give up, by principle, the space-time continuum. It is not unimaginable that human ingenuity will some day find methods which will make it possible to proceed along such a path. At the present time, however, such a program looks like an at tempt to breathe in empty space.

There is no doubt that quantum mechanics has seized hold of a beautiful element of truth, and that it will be a test stone for any future theoretical basis, in that it must be deducible as a limiting case from that basis, just as electrostatics is deducible from the Maxwell equations of the electromagnetic field or as thermodynamics is deducible from classical mechanics. However, I do not believe that quantum mechanics will be the starting point in the search for this basis, just as, vice versa, one could not go from thermodynamics (resp. statistical mechanics) to the foundations of mechanics. In view of this situation, it seems to be entirely justifiable seriously to consider the question as to whether the basis of field physics cannot by any means be put into harmony with the facts of the quantum theory. Is this not the only basis which, consistently with today's possibility of mathemat ical expression, can be adapted to the requirements of the general theory of relativity? The belief, prevailing among the physicists of today, that such an at tempt would be hopeless, may have its root in the unjustifiable idea that such a theory should lead, as a first approximation, to the equations of classical mechanics for the motion of corpuscles, or at least to total differential equations. As a mat ter of fact up to now we have never succeeded in representing corpuscles theoretically by fields free of singularities, and we can, a priori, say nothing about the behavior of such entities. One thing, however, is certain : if a field theory results in a representation of corpuscles free of singularities, then the behavior of these corpuscles with time is determined solely by the differential equations of the field.

"Physics and reality." Journal of the Franklin Institute, 221(3), 349-382. (1936)

Reprinted in Ideas and Opinions, New York: Crown, pp.315-320. (1964)m i ffytu7i9ko

1947

The Born-Einstein Letters

I cannot make a case for my attitude in physics which you would consider at all reasonable. I admit, of course, that there is a considerable amount of validity in the statistical approach which you were the first to recognize clearly as necessary given the framework of the existing formalism.

Here Einstein characterizes the nonlinearity he has seen for decades with one of his most famous phrases.

I can not seriously believe in it because the theory cannot be reconciled with the idea that physics should accept a reality in time and space, free from spooky actions at a distance. I am, however, not yet firmly convinced that it can really be achieved with a continuous field theory, although I have discovered a possible way of doing this which seems quite reasonable. The calculation difficulties are so great that I will be biting the dust long before I myself can be fully convinced of it. But I am quite convinced that someone will come up with a theory whose objects, connected by laws, are not probabilities but considered facts, as used to be taken for granted until quite recently. I cannot, however, base this conviction on logical reasons, but only produce my little finger as witness that is, I offer no authority which would be able to command any kind of respect outside of my own hand.

The Born-Einstein Letter, Macmillan 1971, 2005 p.155

1949

Einstein Philosopher-Scientist

Before I enter upon the question of the completion of the general theory of relativity, I must take a stand with reference to the most successful physical theory of our period, viz., the statistical quantum theory which, about twenty-five years ago, took on a consistent logical form (Schrödinger, Heisenberg, Dirac, Born). This is the only theory at present which permits a unitary grasp of experiences concerning the quantum character of micro-mechanical events. This theory, on the one hand, and the theory of relativity on the other, are both considered correct in a certain sense, although their combination has resisted all efForts up to now. This is probably the reason why among contemporary theoretical physicists there exist entirely differing opinions concerning the question as to how the theoretical foundation of the physics of the future will appear. Will it be a field theory; will it be in essence a statistical theory? I shall briefly indicate my own thoughts on this point.

Physics is an attempt conceptually to grasp reality as it is thought independently of its being observed. In this sense one speaks of “physical reality.” In pre-quantum physics there was no doubt as to how this was to be understood. In Newton’s theory reality was determined by a material point in space and time; in Maxwell’s theory, by the field in space and time. In quantum mechanics it is not so easily seen. If one asks: does a Ψ-function of the quantum theory represent a real factual situation in the same sense in which this is the case of a material system of points or of an electromagnetic field, one hesitates to reply with a simple “yes” or “no”; why? What the Ψ-function (at a definite time) asserts, is this: What is the probability for finding a definite physical magnitude q (or p) in a definitely given interval, if I measure it at time t? The probability is here to be viewed as an empirically determinable, and therefore certainly as a “real” quantity which I may determine if I create the same Ψ-function very often and perform a q- measurement each time. But what about the single measured value of q? Did the respective individual system have this q-value even before the measurement? To this question there is no definite answer within the framework of the [existing] theory, since the measurement is a process which implies a finite disturbance of the system from the outside; it would therefore be thinkable that the system obtains a definite numerical value for q (or p) the measured numerical value, only through the measurement itself. For the further discussion I shall assume two physicists, A and B, who represent a different conception with reference to the real situation as described by the Ψ-function.

A. The individual system (before the measurement) has a definite value of q (or p) for all variables of the system, and more specifically, that value which is determined by a measurement of this variable. Proceeding from this conception, he will state: The Ψ-function is no exhaustive description of the real situation of the system but an incomplete description; it expresses only what we know on the basis of former measurements concerning the system.

B. The individual system (before the measurement) has no definite value of q (or p). The value of the measurement only arises in cooperation with the unique probability which is given to it in view of the Ψ-function only through the act of measurement itself. Proceeding from this conception, he will (or, at least, he may) state: the Ψ-function is an exhaustive description of the real situation of the system.

We now present to these two physicists the following instance: There is to be a system which at the time t of our observation consists of two partial systems S₁ and S₂, which at this time are spatially separated and (in the sense of the classical physics) are without significant reciprocity. The total system is to be completely described through a known Ψ-function Ψ₁₂ in the sense of quantum mechanics. All quantum theoreticians now agree upon the following: If I make a complete measurement of S₁ I get from the results of the measurement and from Ψ₁₂ an entirely definite Ψ-function of the system S₂. The character of Ψ₂ then depends upon what kind of measurement I undertake on S₁.

Now it appears to me that one may speak of the real factual situation of the partial system S₂. Of this real factual situation, we know to begin with, before the measurement of S₁, even less than we know of a system described by the Ψ-function. But on one supposition we should, in my opinion, absolutely hold fast: the real factual situation of the system S₂ is independent of what is done with the system S₁, which is spatially separated from the former. According to the type of measurement which I make of S₁ I get, however, a very different Ψ₂ for the second partial system (φ₂, φ₂², ...). Now, however, the real situation of S₂ must be independent of what happens to S₁. For the same real situation of S₂ it is possible therefore to find, according to one’s choice, different types of Ψ-function. (One can escape from this conclusion only by either assuming that the measurement of Ψ₁ ((telepathically)) changes the real situation of Ψ₂ or by denying independent real situations as such to things which are spatially separated from each other. Both alternatives appear to me entirely unacceptable.)

If now the physicists, A and B, accept this consideration as valid, then B will have to give up his position that the qp-function constitutes a complete description of a real factual situation. For in this case it would be impossible that two different types of ^-functions could be co-ordinated with the identical factual situation of S2. The statistical character of the present theory would then have to be a necessary consequence of the incompleteness of the description of the systems in quantum mechanics, and there would no longer exist any ground for the supposition that a future basis of physics must be based upon statistics.-------assun^i L It is my opinion that the contemporary quantum theory by means of certain definitely laid down basic concepts, which on ^ the whole have been taken over from classical mechanics, constitutes an optimum formulation of the connections. I believe, however, that this theory offers no useful point of departure for future development. This is the point at which my expectation departs most widely from that of contemporary physicists. They are convinced that it is impossible to account for the essential aspects of quantum phenomena (apparently discontinuous and temporally not determined changes of the situation of a system, and at the same time corpuscular and undulatory qualities of the elementary bodies of energy) by means of a theory which describes the real state of things [objects] by continuous functions of space for which differential equations are valid. They are also of the opinion that in this way one can not understand the atomic structure of matter and of radiation. They rather expect that systems of differential equations, which could come under consideration for such a theory, in any case would have no solutions which would be regular (free from singularity) everywhere in four-dimensional space. Above everything else, Ihowever, they believe that the apparently discontinuous char- jacter of elementary events can be described only by means of an f essentially statistical theory, in which the discontinuous changes < of the systems are taken into account by 'way of the continuous ' changes of the probabilities of the possible states.

All of these remarks seem to me to be quite impressive.

Albert Einstein: Philosopher-Scientist, Library of Living Philosphers, Harper, (1959) pp.81-89

Visualizing Entanglement, Nonlocality, and Nonseparability

Schrödinger said that his "Wave Mechanics" provided more "visualizability" (Anschaulichkeit) than the "damned quantum jumps" of the Copenhagen school, as he called them. He was right. We can use the wave function to visualize EPR.

But we must focus on the probability amplitude wave function of the prepared two-particle state. We must not attempt to describe the paths or locations of independent particles - at least until after some measurement has been made. We must also keep in mind the conservation laws that Einstein used to describe nonlocal behavior in the first place. Then we can see that the "mystery" of nonlocality for two particles is primarily the same mystery as the single-particle collapse of the wave function. But there is an extra mystery, one we might call an "enigma," that results from the nonseparability of identical indistinguishable particles.

As Richard Feynman said, there is only one mystery in quantum mechanics (the superposition of states, the probabilities of collapse into one state, and the consequent statistical outcomes). The only difference in two-particle entanglement and nonlocality is that two particles appear simultaneously (in their original interaction frame) when their wave function collapses.

We choose to examine a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery. We cannot make the mystery go away by "explaining" how it works. We will just tell you how it works. In telling you how it works we will have told you about the basic peculiarities of all quantum mechanics.

(The Feynman Lectures on Physics, vol III, 1-1)

In the time evolution of an entangled two-particle state according to the Schrödinger equation, we can visualize it - as we visualize the single-particle wave function - as collapsing when a measurement is made. The discontinuous "jump" is also described as the "reduction of the wave packet." This is apt in the two-particle case, where the superposition of | + - > and | - + > states is "projected" or "reduced" to one of these states, and then further reduced to the product of independent one-particle states.

In the two-particle case (instead of just one particle making an appearance), when either particle is measured we know instantly those now determinate properties of the other particle that satisfy the conservation laws, including its location equidistant from, but on the opposite side of, the source.

Animation of a two-particle wave function collapsing - click to restart

Some commentators say that nonlocality and entanglement are a "second revolution" in quantum mechanics, "the greatest mystery in physics," or "science's strangest phenomenon," and that quantum physics has been "reborn." They usually quote Erwin Schrödinger as saying

"I consider [entanglement] not as one, but as the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought."

Schrödinger knew that his two-particle wave function could not have the same simple interpretation as the single particle, which can be visualized in ordinary 3-dimensional configuration space. And he is right that entanglement exhibits a richer form of the "action-at-a-distance" and nonlocality that Einstein had already identified in the collapse of the single particle wave function.

But the main difference is that two particles acquire new properties instead of one, and they do it instantaneously (at faster than light speeds), just as in the case of a single-particle measurement, where the finite probability of appearing at various distant locations collapses to zero at the instant the particle is found somewhere.

We can enhance our visualization of what might be happening between the time two entangled electrons are emitted with opposite spins and the time one or both electrons are detected.

Quantum mechanics describes the state of the two electrons as in a linear combination of | + - > and | - + > states. We can visualize the electron moving left to be both spin up | + > and spin down | - >. And the electron moving right would be both spin down | - > and spin up | + >. We could require that when the left electron is spin up | + >, the right electron must be spin down | - >, so that total spin is always conserved.

Consider this possible animation of the experiment, which illustrates the assumption that each electron is in a linear combination of up and down spin. It imitates the superposition (or linear combination) with up and down arrows on each electron oscillating quickly.

Notice that if you move the animation frame by frame by dragging the dot in the timeline, you will see that total spin = 0 is conserved. When one electron is spin up the other is always spin down.

Since quantum mechanics says we cannot know the spin until it is measured, our best estimate is a 50/50 probability between up and down.

This is the same as assuming Schrödinger's Cat is 50/50 alive and dead. But what this means of course is simply that if we do a large number of identical experiments, the statistics for live and dead cats will be approximately 50/50%. We never observe/measure a cat that is both dead and alive!

As Einstein noted, QM tells us nothing about individual cats. Quantum mechanics is incomplete in this respect. He is correct, although Bohr and Heisenberg insisted QM is complete, because we cannot know more before we measure, and reality is created (they say) when we do measure.

Despite accepting that a particular value of an "observable" can only be known by a measurement (knowledge is an epistemological problem, Einstein asked whether the particle actually (really, ontologically) has a path and position before we measure it? His answer was yes.

Here is an animation that illustrates the unprovable assumption that the two electrons are randomly produced in a spin-up and a spin-down state, and that they remain in those states no matter how far they separate, provided neither interacts until the measurement. An interaction does what is described as decohering the two states.

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