Chapter 10 PROOFS of IMPOSSIBILITY and POSSIBILITY PROOFS

Chapter 10 PROOFS OF IMPOSSIBILITY AND POSSIBILITY PROOFS There are proofs of impossibility and possibility proofs, a situation that sometimes produced a bit of confusion in the debates on hidden-variables theories and sometimes led to something worse than confusion. So, we have to clarify it. Proofs of Impossibility von Neumann Brief Beginning of the History In his famous book, providing an axiomatization of quantum mechanics, von Neumann [26] introduced in 1932 a proof of impossibility of hidden- variables theories, the famous “von Neumann’s proof.” This were a few years after the Solvay Congress of 1927 during which de Broglie’s pilot wave was defeated. Although de Broglie had already renegated his work, and rejoined the camp of orthodoxy, the publication of the proof certainly confirmed that he had lost time with his heretical detour. Furthermore, in the words of Pinch [154], “the proof was accepted and welcomed by the physics elite.” The continuation of the story shows that the acceptance was not based on a careful examination of the proof but rather on other elements such as a psychological pressure exerted by the very high reputation of its author or the unconscious desire to definitively eliminate an ill-timed and embarrassing issue. Afterward, for a long time, the rejection of hidden variables could be made just by invoking von Neumann (the man himself more than the proof). As stated by Belinfante [1], “the truth, however, happens to be that for decades nobody spoke up against von Neumann’s arguments, and that his conclusions were quoted by some as the gospel … the authority of von Neumann’s over-generalized claim for nearly two decades stifled any progress in the search for hidden variables theories.” Belinfante also [1] remarked that “the work of von Neumann (1932) was mainly concerned with the axiomatization of the mathematical methods “CH10” — 2013/10/11 — 10:28 — page 228 — #1 Hidden101113.PDF 238 10/16/2013 4:20:19 PM Proofs of Impossibility and Possibility Proofs 229 of quantum theory. His side remarks on hidden variables were merely an unfortunate step away from the main line of reasoning.” This was more than unfortunate, however; it was erroneous, and even deeply erroneous, “because of the obviousness of inapplicability of one of von Neumann’s axioms to any realistic hidden variables theory [1].” As far as I know, the first person to challenge the validity of von Neu- mann’s proof was the philosopher Grete Hermann in 1935 [117], three years after von Neumann’s book, followed nine years after, in 1944, by another philosopher, Hans Reichenbach [134], according to Pinch [154]. As noted by Bitbol [133], and can be checked in the original reference, recently republished with enlightening complementary discussions by Léna Soler, Hermann correctly identified the weakest point of von Neu- mann’s proof, namely, the misleading use of an additivity condition of average values (which are soon to be discussed), pointed out by Bell again in 1966 [188]. Furthermore, she charged von Neumann’s proof with the accusation of circularity. Likely because they were philosophers, Hermann and Reichenbach seemingly did not have much influence on the community of physicists. Very often, they are even forgotten or dismissed. For instance, Pipkin [156] could erroneously write that it was Bell, in 1966, “who first pointed out the axiom by which von Neumann’s formulation violated the elementary principles of any realistic hidden variables theory,” a statement strongly and strangely in agreement with Belinfante [1] when he claimed, five years before that “The first to publicly pinpoint the axiom by which von Neumann’s formulation violated the elementary principles of any realistic hidden variables theory was Bell (1966).” As far as I know, the first physicist to challenge von Neumann was Bohm, twenty years after the proof was proposed, in one of his 1952 papers [30]. The best rebuttal of the early Bohm has obviously been the fact that he produced a logically consistent hidden-variables theory. This was a constructivist rebuttal, explicitly showing that what was demon- strated as being impossible was in fact possible. It definitely established that something was wrong with von Neumann’s proof; but what exactly was wrong? This is something that was compulsory to establish. The for- mal answer to this issue by Bohm was rather poorly conceived, but he correctly identified the most important point, namely, that von Neumann implicitly restricted himself, in his proof, to an excessively narrow class of hidden-variables theories. It just so happened that the early Bohm’s pilot wave did not pertain to this class. Louis de Broglie then also manifested “CH10” — 2013/10/11 — 10:28 — page 229 — #2 Hidden101113.PDF 239 10/16/2013 4:20:19 PM 230 Hidden Worlds in Quantum Physics a reluctance against von Neumann’s proof, e.g., in [62, 64]. The fact that Bohm’s rebuttal was unsatisfactory, and required more clarification, was also pointed out by Bell [188], who commented, “The analysis of Bohm seems to lack clarity, or else accuracy ….” Bell’s Rebuttal of von Neumann Accepting the idea that von Neumann’s proof has been rebutted, PEDES- TRIANS (∼∼∼∼) are allowed to skip this subsubsection. Bell [188] was able to make clear in a fairly laconic way what was unclear in Bohm’s attack against von Neumann. In essence, his identifi- cation of the reason why von Neumann failed is in agreement with what Hermann had already understood, some thirty years before. It was, however, the influence of Bell’s paper, about thirty-five years after the proof was announced to the world, a long Middle Age for hidden variables, that was the most powerful. The dragon was struck down.At that time there was a dragon still living and it had to be destroyed; this was not obvious to all scientists, as we can infer from the following quotation [188]: “The present paper … is addressed to those who do find the question interesting, and more particularly to those among them who believe that ‘the question concerning the existence of such hidden variables received an early and decisive answer in the form of von Neumann’s proof on the mathematical impossibility of such variables in quantum theory’.” The influence of Bell’s paper is acknowledged by Bitbol [133] when he stated that the very sociological turnaround of the community of physicists regarding von Neu- mann’s theorem nevertheless only took place from 1966 on, the date of the publication of Bell’s paper … Bell’s attack pointed out a faulty additivity postulate used by von Neumann that, when applied to hidden-variables theories, was “one of the weakest points in the proof,” in the words of Jammer [24]. The essential assumption used by von Neumann, the one we called above the “additivity condition on average values,” from now on called the additivity postulate, is described in the following statement: “Any linear combination of any two Hermitian operators represents an observable, and the same linear combination of expectation values is the expectation value of the combination.” In the words of Mermin [197], this was “von Neumann’s silly assumption.” The sentence quoted above actually contains two statements. The first one says that any linear combination of any two Hermitian operators represents an observable. This seems obvious, but it actually is not guaranteed “CH10” — 2013/10/11 — 10:28 — page 230 — #3 Hidden101113.PDF 240 10/16/2013 4:20:19 PM Proofs of Impossibility and Possibility Proofs 231 at all. Let us, however, forget this problem as irrelevant for our purpose and focuss on the second statement, the one we properly call the additivity postulate, saying that the same linear combination of expectation values is the expectation value of the combination. Let us consider two Hermitian operators A and B, and let us assume that the sum of the two Hermitian operators A and B is another Hermitian operator, defining an observable, which we denote by C. Let us remark that we have been here a bit more careful than von Neumann. We do not claim that C is an observable for any couple (A, B). We just consider a couple such that C is indeed an observable. We then have C = A + B (10.1) The additivity postulate tells us that C=A+B (10.2) where X is the expectation value of X. In quantum mechanics, the expectation value of X for a given wave function can be evaluated according to the following rule: ∗ X= Xdr (10.3) where ∗ is the complex conjugate of . Then, the additivity postulate is seen to be always true in quantum mechanics, since C= ∗Cdr = ∗(A + B)dr ∗ ∗ = Adr + Bdr =A+B (10.4) In particular, it is true regardless of the value of the commutator [A, B]; that is, it does not depend on whether or not the operators A and B commute. As obvious as it seems, the additivity postulate is not trivial. This can be exemplified by noting that it does not necessarily apply to eigenvalues. To see this, let us consider a counterexample taken from Jammer [24]. Let us consider the spin component of an electron in the direction along “CH10” — 2013/10/11 — 10:28 — page 231 — #4 Hidden101113.PDF 241 10/16/2013 4:20:19 PM 232 Hidden Worlds in Quantum Physics the bisector line between the x axis and the y axis. This observable is represented by the operator 1 S45◦ = √ (σx + σy) (10.5) 2 The outcome of the measurement of S45◦ for the electron, a spin-1/2 particle, may be either h/¯ 2or−h/¯ 2; that√ is, the eigenvalues are ±1, in units of h/¯ 2.

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