Continued from Part 1

On Einstein’s side there followed two years of intense theoretical and experimental work probing the subtleties of the new quantum mechanics of multi-particle systems. With advice from instein, the Berlin experimentalist and master of the technology of coincidence counting, Walther Bothe, pursued a kind of proto-Bell experimental program, investigating a variety of different correlation phenomena (see, for example, Bothe 1926). On the theoretical front, Einstein’s efforts ulminated in May of 1927–a few months before the first of the famous clashes with Bohr at the Solvay meeting–in a failed attempt to provide his own hidden variables model of Schrödinger’s new ave mechanics. 

Ironically, Einstein’s model failed precisely because it included the very entanglement Einstein was hoping to avoid. Though it reached the proof stage, Einstein’s lecture to the Berlin Academy was never published because Einstein could not find a way around the entanglement. He explained in a “Note Added in Proof”:

I have found that the schema does not satisfy a general requirement that must be imposed on a general law of motion for systems.

Consider, in particular, a system Σ that consists of two energetically independent subsystems, Σ₁ and Σ₂; this means that the potential energy as well as the kinetic energy is additively composed of two parts, the first of which contains quantities referring only to Σ₁ the second quantities referring only to Σ₂. It is then well known that
Ψ = Ψ₁ ∙ Ψ₂ , wher Ψ₁ depends only on the coordinates of Σ₁, Ψ₂ only on the coordinates of Σ₂. In this case we must demand that the motions of the composite system be combinations of possible motions of the subsystems.

The indicated scheme [Einstein’s hidden variables model] does not satisfy this requirement. In particular, let μ be an index belonging to a coordinate of Σ₁, υ an index belonging to a coordinate of Σ₂. Then Ψ_μυ does not vanish.

(Einstein 1927a)

Here, then, we have Einstein on the eve of the 1927 Solvay meeting. For twenty-two years he had known that the full story of the quantum would involve some fundamental compromise with the classical notion of the mutual independence of interacting systems. In 1924, Bose showed him that this failure of mutual independence was not incidental but an essential feature of the quantum realm, the deep fact underlying the Planck formula. In 1925, Schrödinger developed the wave mechanical formalism that relates the new quantum statistics to the symmetry properties of the two particle wave function. 

In the spring of 1927, Einstein’s attempt at a hidden variables model of wave mechanics is abandoned because it, too, evinces the very entanglement whose presence in wave mechanics Einstein was seeking to evade. It was only a few months earlier, in a letter to Max Born of 4 December 1926, that Einstein had complained that God “does not play dice” with the world (Born 1969, 129-130).    But that entanglement, not indeterminacy, was the chief source of Einstein’s misgivings about quantum mechanics should now be clear. Indeterminacy was but a symptom; entanglement was the underlying disease.

Bohr, too, had been thinking about what we now call entanglement for a long time by the fall of 1927. Since not long after Carl Ramsauer discovered in 1920 the counterintuitive fact that the scattering cross-section of electrons passing through certain noble gases goes to zero as the electron velocity is reduced (see Ramsauer 1920, 1921a, 1921b), Bohr suspected that a variety of nonclassical phenomena would find their explanation in violations of classical assumptions about the mutual independence of interacting systems, or, in Bohr’s own idiom, in failures of “ordinary spacetime description.” 

It was hard to imagine how slower electrons could more readily penetrate the gas if the scattering of the electron by, say, an argon atom were modeled as a classical hard-sphere collision. Clinton Davisson and Lester Germer’s later discovery of interference effects in electron diffraction from nickel crystals (Davisson and Germer 1927), usually remembered as the first experimental proof of the wave nature of the electron, in fact just reinforced an already well-founded impression of the highly non-classical character of electron scattering.

Worries about the failure of “ordinary space-time description” were among the main reasons for Bohr’s well-known skepticism about the photon hypothesis, for remember that Einstein’s pre-1924, pre-Bose photons were wrongly assumed to be mutually independent, corpuscle-like carriers of field energy behaving entropically like the mutually independent molecules of a Boltzmann gas. But even Bohr was slow to grasp the fundamental character of entanglement. He was helped by the disproof of the Bohr-Kramers-Slater (BKS) theory in 1924.

A lineal descendent of Einstein’s 1909 idea of wave-particle duality, the BKS theory sought to explain phenomena such as the absorption and emission of radiation by associating with each elementary structure, such as valence electrons, a virtual field with component probability amplitudes corresponding to each possible transition that the system could undergo. But it was still a “classical” theory in the sense that it explained interactions–as in the emission and absorption of a photon by two electromagnetically interacting atoms–as the result of the conjoint but still separate doings of the virtual fields associated with each interacting system. This is why, though energy and momentum were conserved on average, the BKS theory predicted violations of energy and momentum conservation in individual atomic events, because the separate actions of the two virtual fields implied the statistical independence of, say, photon emission by one atom and subsequent photon absorption by the other atom.

The Compton-Simon and Bothe-Geiger experiments proved strict energy and momentum conservation in individual atomic events (Compton and Simon 1925a, 1925b, Bothe and Geiger 1924, 1925) and so refuted the BKS theory. But the standard histories fail to emphasize the main implication drawn by Bohr and others from this failure. What Bohr found important was precisely the proof that coupled atomic events are not thus statistically independent of one another, that strict energy and momentum conservation thus implies the failure of “ordinary space-time description.” Here is how he put the point in a letter to Geiger in April of 1925, where the immediate subject was a different experiment of Bothe’s (Bothe 1926) that yielded evidence against another aspect of the BKS theory:

I was quite prepared to learn that our proposed point of view about the independence of the quantum processes in separated atoms would turn out to be wrong. . . . Not only were Einstein’s objections very disquieting; but recently I have also felt that an explanation of collision phenomena, especially Ramsauer’s results on the penetration of slow electrons through atoms, presents difficulties to our ordinary space-time description of nature similar in kind to the those presented by the simultaneous understanding of interference phenomena and a coupling of changes of state of separated atoms by radiation. In general, I believe that these difficulties exclude the retention of the ordinary space-time description of phenomena to such an extent that, in spite of the existence of coupling, conclusions about a possible corpuscular nature of radiation lack a sufficient basis.

(Bohr to Geiger, 21 April 1925; Bohr 1984, 79)

On the same day he made a similar point in a letter to James Franck:

It is, in particular, the results of Ramsauer concerning the penetration of slow electrons through atoms that apparently do not fit in with the assumed viewpoint. In fact, these results may pose difficulties for our customary spatio-temporal description of nature that are similar in kind to a coupling of changes of state in separated atoms through radiation. But then there is no more reason to doubt such a coupling and the conservation laws generally.

(Bohr to Franck, 21 April 1925; Bohr 1984, 350)

Given the frequent complaints about the alleged opacity of Bohr’s idiolect, it is worth pausing a moment to make sure that we understand what Bohr means by the failure of the space-time mode of description. Note the manner in which Bohr associates the failure of space-time description with the failure of both corpuscular models of radiation–the photon hypothesis–and the assumed statistical independence of coupled changes of state. The associations are the same that Einstein had been making since 1905, when he argued that a corpuscular model of the radiation field means precisely the assumption of the statistical independence of two spatially separated photons occupying given cells of phase space.

How are corpuscularity and statistical independence related, in turn, to space-time description?

For both Bohr and Einstein, it was because the assumption of a difference in spatial location was thought to be sufficient for securing the mutual independence of thus differently situated systems. But corpuscular models are not alone in being thus linked to space-time description and the assumed statistical independence of coupled changes of state. Wave models can likewise be seen as providing a space-time description and undergirding the assumed statistical independence of changes of state, as long as separate wave structures are associated with spatially separated systems, as in Einstein’s early speculations about wave-particle duality and in the BKS theory. Schrödinger’s introduction of entangled n-particle wave functions written not in 3-space but in 3n-dimensional configuration space offends against space-time description because it denies the mutual independence of spatially separated systems that is a fundamental feature of a space-time description.

Bohr’s thinking about quantum foundations finally began to coalesce in the summer of 1927, stimulated by his less than wholly positive reaction to the Heisenberg indeterminacy principle (Heisenberg 1927). His new ideas, specifically the concept of complementarity, were premiered in his lecture to the Volta Congress at Como, Italy in September.

By this time Bohr had assimilated the fact that, in the quantum theory, coupled systems do not evince the mutual independence that classical physics ordains for spatially separated systems. The important new idea is that “observers,” too, are physical systems, and that when an “observed object” is coupled with an “observer” in measurement, as it must be if the measurement is to engender the object-observer correlations necessary for the measurement’s being a measurement, then both observer and object lose their mutual independence, if, that is, we treat the measurement interaction like any other physical

interaction in the quantum domain. That instrument and object form an entangled pair is the premise from which Bohr derives complementarity:

Now, the quantum postulate implies that any observation of atomic phenomena will involve an interaction with the agency of observation not to be neglected. Accordingly, an independent reality in the ordinary physical sense can neither be ascribed to the phenomena nor to the agencies of observation. . . .

This situation has far-reaching consequences. On one hand, the definition of the state of a physical system, as ordinarily understood, claims the elimination of all external disturbances. But in that case, according to the quantum postulate, any observation will be impossible, and, above all, the concepts of space and time lose their immediate sense. On the other hand, if in order to make observation possible we permit certain interactions with suitable agencies of measurement, not belonging to the system, an unambiguous definition of the state of the system is naturally no longer possible, and there can be no question of causality in the ordinary sense of the word. The very nature of the quantum theory thus forces us to regard the space-time co-ordination and the claim of causality, the union of which characterizes the classical theories, as complementary but exclusive features of the description, symbolizing the idealization of observation and definition respectively.

(Bohr 1928, 54-55)

These few words have given rise to much confusion in the secondary literature, even though Bohr clearly sought, in vain, to avoid one possible confusion by speaking not of the “observer” as a subjective knower but of the “agency of observation,” the point being, one presumes, to emphasize the physical nature of measurement interactions. Should Bohr also have foreseen that another misreading of these words would turn them into an endorsement of anti-realism, a crucial premise in the all-too-common, if mistaken, arguments that Bohr’s complementarity interpretation of quantum mechanics is a species of positivism? Bohr does not say, here, that there is no quantum reality. What he says is that, in virtue of what we now call entanglement, we cannot ascribe an independent reality to either the observer or the observed

This is not the place for an extended discussion of Bohr on complementarity, but note two points.(10) First, many authors, Beller included, argue that the challenge of the Einstein, Podolsky, and Rosen (EPR) paper, “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?”(herafter referred to as EPR), which employs indirect, not direct measurements, forced a dramatic revision in Bohr’s conception of complementarity. It is said that Bohr stopped talking about the complementarity between space-time description and the claims of causality introduced in the Como lecture and began to speak, instead, of complementarity as a relationship between incompatible observables, like position and momentum.

That is simply false. In the published version of the Como lecture, itself, a few pages after the just-quoted remark, Bohr wrote:

According to the quantum theory a general reciprocal relation exists between the maximum sharpness of definition of the space-time and energy-momentum vectors associated with the individuals. This circumstance may be regarded as a simple symbolical expression for the complementary nature of the space-time description and the claims of causality.

(Bohr 1928,60)

Can there be a more categorical assertion of the essential identity of the two versions of complementarity? The difference is no more than that between a conceptual and a formal characterization of one and the same notion.

Of course there is the possibility of a complementary relationship between parameters other than position and momentum or energy and time. Nothing precludes that. The mentioned instantiations of complementarity are in the foreground in 1927 for two reasons: (a) the fundamental role of position and momentum in defining the state concept of classical particle mechanics (state =_df point in phase space), and the comparably fundamental roles of time and energy as, respectively, the fundamental dynamical variable in classical mechanics and the associated conserved quantity;  (b) the fact that the state concept, the dynamics, and the conservation laws in classical particle mechanics all work together in harmony, as they must, whereas the various parts of the classical mode of description cannot be combined consistently in the quantum theory.

Bohr’s highlighting the connection between Heisenberg indeterminacy and the complementarity of space-time description and causality gives us the clue we need to understand the latter. A bit of formal hindsight helps as well. Space-time description is the easy part. A space-time description in classical particle mechanics models the behavior of a physical system by tracking the time evolution of the system in 3-space or, equivalently, the evolution of its state in phase space.

If we are dealing with interacting systems, A and B–the really crucial case–then it is the separate evolution of each in space that concerns us, the interaction handled by a potential via which one calculates the separable effects of A on B and B on A. But Bohr (and Einstein) already knew that such “pictures”–yes, literally, pictures–do not work in the quantum theory, where interacting systems are entangled.

The “claims of causality” is the harder part, at least at first glance. Bohr’s talk of “causality" is confusing, because he does not mean the univocal time evolution of physical states. What Bohr does mean is indicated by his associating the “claims of causality” with the energy-momentum vector. Energy and momentum are the fundamental conserved quantities of classical particle mechanics. A “causal” description, in Bohr’s use of the term, is a description according to which energy and momentum are conserved.

The failure of the BKS theory had taught Bohr how the two harmonious aspects of a classical mechanical description come apart in the quantum realm, for, as we just saw, violations of strict energy and momentum conservation are the famous untoward consequence of the BKS theory’s still assigning separate states (including the virtual fields) to spatially separated but interacting systems

In the quantum domain, a space-time description implies no energy and momentum conservation in individual events; the claims of causality are violated. Salvaging energy and momentum conservation in individual events requires the right, strict correlations between interacting systems: a gain in energy here means a loss of energy there, and a loss of momentum there means a gain of energy here.

As was already becoming clear to Einstein and Bohr in 1927, getting the correlations right means introducing entangled joint multi-particle states, as Schrödinger had done in wave mechanics.

The 1927 and 1930 Solvay Encounters

The 1927 Solvay meeting took place in October, one month after the Volta Congress in Como and a few months after Einstein had abandoned his attempted hidden variables model of Schrödinger’s wave mechanics. In addition to bits and pieces of contemporary correspondence, the main sources of information about the encounter there between Einstein and Bohr are the published proceedings of the meeting (Solvay 1927) and Bohr’s memoir of his discussions with Einstein, written some twenty years after the event (Bohr 1949).

Witnesses agree that the most interesting conversations took place off the record, at breakfast and dinner, often lasting well into the night. One would think Bohr’s memoir the better source, therefore, but for the fact that it differs so dramatically from the published record.

Bohr’s memoir and the published record agree that the one-slit diffraction experiment was of central concern. According to Bohr, the point at issue was whether or not the one-slit experiment could be manipulated to yield violations of Heisenberg indeterminacy:

The discussions . . . centered on the question of whether the quantum-mechanical description exhausted the possibilities of accounting for observable phenomena or, as Einstein maintained, the analysis could be carried further and, especially, of whether a fuller description of the phenomena could be obtained by bringing into consideration the detailed balance of energy and momentum in individual processes.

(Bohr 1949, 213)

While claiming to explain “the trend of Einstein’s arguments,” Bohr recalls his introducing his own long-since-familiar refinements, such as a movable diaphragm and a second diaphragm containing two slits, all for the purpose of countering Einstein’s efforts to prove that one can make precise determinations of both the position and the momentum of the particle traversing the apparatus.

The published record of Einstein’s one contribution to public discussion shows Einstein thinking about something very different. Einstein says that, in application to the one-slit diffraction experiment, quantum mechanics can be given either an ensemble interpretation (“Interpretation I”) or an interpretation (“Interpretation II”) according which the wave function represents the objective state of an individual quantum system.

Einstein offers three objections to “Interpretation II.” The first is that it violates relativity because, by fixing the probability that this one specific particle is located at one specific place, the second interpretation “presupposes a very particular mechanism of action at a distance which would prevent the wave continuously distributed in space from acting at two places of the screen” (Einstein 1927b, 102).

Crudely put, the idea is that if the particle has a high probability of being here, then, simultaneously (the crucial point), it has a high probability of not being there.

More interesting for and relevant to our purposes is the second objection:

[It] is essentially connected with a multidimensional representation (configuration space because only this representation makes possible the interpretation of |Ψ|² belonging to interpretation II. Now, it seems to me that there are objections of principle against this multidimensional representation. In fact, in this representation two configurations of a system which only differ by the permutation of two particles of the same kind are represented by two different points (of configuration space), which is not in agreement with the new statistical results.

(Einstein 1927b, 103)

Einstein’s objection fails because he confuses points in the two-particle configuration space with states in the state space (a Hilbert space) in which the two-particle wave functions live. But the objection is instructive, nonetheless, because it shows that Einstein was still ruminating on the consequences for quantum mechanics of the failure of classical assumptions about the mutual independence of spatially disjoint systems.

Why Bohr’s later memoir omits this objection from Einstein is not clear. The lapse of time and a lapse of memory could explain it just as well as could the hypothesis of a conscious and deliberate rewriting of history to reinforce the idea of a Copenhagen hegemony. But time lapses and memory lapses are less persuasive in explaining why Bohr’s recollection of the still more famous encounter with Einstein at the 1930 Solvay meeting directly contradicts contemporary documentary evidence

At center stage in the Einstein-Bohr encounter at the 1930 Solvay meeting was Einstein’s well-known photon box thought experiment.

A box containing a photon has an opening covered by a shutter that is activated by a timer attached to a clock inside the box by means of which we could accurately time the emission of the photon from the box. The whole box is suspended by a spring by means of which arrangement we could weigh the box both before and after the photon’s emission with whatever accuracy we desire, thus determining the photon’s energy via the mass-energy equivalence relation.

As Bohr tells the story, Einstein introduced the photon-box thought experiment for the purpose, yet again, of exhibiting violations of Heisenberg indeterminacy. Simply perform both measurements: weigh the box to fix the emitted photon’s energy and open the box to check the clock and fix the time of emission.

Bohr tells us that, at first, Einstein had him completely stumped.  He could find no flaw in Einstein’s reasoning. Only in the wee hours of the morning did it come to him. Ironically, general relativity would save quantum mechanics, specifically the general relativistic effect of a gravitational field on clock rates. A quick calculation showed Bohr that the change in the box’s mass when the photon is emitted changes, in turn, its vertical location in the earth’s gravitational field, and that the effect of the latter change on the rate of the clock in the box induces precisely the uncertainty in the clock’s rate needed to insure satisfaction of the Heisenberg indeterminacy principle (Bohr 1949, 227-228). Bohr uses general relativity against Einstein to save quantum mechanics! A wonderful story.

But is it true?

To be concluded in Part 3

Reprinted with permission from Iyyun: The Jerusalem Philosophical Quarterly,Volume 56, January 2007

ENDNOTES:

10. I discuss Bohr's complementarity interpretation of quantum mechanics in much greater detail in Howard 1979, 1994, and 2004.

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