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Einstein's Boxes Einstein’s Boxes Travis Norsen∗ Marlboro College, Marlboro, Vermont 05344 (Dated: February 1, 2008) At the 1927 Solvay conference, Albert Einstein presented a thought experiment intended to demon- strate the incompleteness of the quantum mechanical description of reality. In the following years, the experiment was modified by Einstein, de Broglie, and several other commentators into a simple scenario involving the splitting in half of the wave function of a single particle in a box. This paper collects together several formulations of this thought experiment from the literature, analyzes and assesses it from the point of view of the Einstein-Bohr debates, the EPR dilemma, and Bell’s the- orem, and argues for “Einstein’s Boxes” taking its rightful place alongside similar but historically better known quantum mechanical thought experiments such as EPR and Schr¨odinger’s Cat. I. INTRODUCTION to Copenhagen such as Bohm’s non-local hidden variable theory. Because of its remarkable simplicity, the Einstein Boxes thought experiment also is well-suited as an intro- It is well known that several of quantum theory’s duction to these topics for students and other interested founders were dissatisfied with the theory as interpreted non-experts. by Niels Bohr and other members of the Copenhagen school. Before about 1928, for example, Louis de Broglie The paper is organized as follows. In Sec. II we intro- duce Einstein’s Boxes by quoting a detailed description advocated what is now called a hidden variable theory: a pilot-wave version of quantum mechanics in which parti- due to de Broglie. We compare it to the EPR argument and discuss how it fares in eluding Bohr’s rebuttal of cles follow continuous trajectories, guided by a quantum wave.1 David Bohm’s2 rediscovery and completion of the EPR. In Sec. III we present and discuss Einstein’s original version of the thought experiment as well as some of his pilot-wave theory in 1952 led de Broglie back to these ideas; Bohm’s theory has continued to inspire interest related comments regarding the boxes and their connec- tion to EPR. In Sec. IV we present a new, Bell-inspired in alternatives to the Copenhagen interpretation.3 Erwin Schr¨odinger was likewise doubtful that the quantum wave formulation of Einstein’s Boxes and relate the ideas to Bell’s celebrated theorem about hidden variable theories. function could alone constitute a complete description of physical reality. His famous “cat” thought experiment4 Section V begins with some comments of Heisenberg on was intended to demonstrate quantum theory’s incom- the thought experiment, considers an experimentally re- pleteness by magnifying the allegedly real quantum in- alized version of it, and assesses some of Heisenberg’s definiteness up to the macroscopic level where it would statements on non-locality. In Sec. VI we conclude with directly conflict with experience. a general discussion. By far the most important critic of quantum the- 6 ory, however, was Albert Einstein. Several of his early II. EPR AND DE BROGLIE’S VERSION OF thought experiments attacking the completeness doctrine THE BOXES are memorably recounted in Bohr’s reminiscence.7 Ein- stein’s most important argument against quantum the- In a 1964 book de Broglie gave a detailed statement of arXiv:quant-ph/0404016v2 8 Feb 2005 ory’s completeness is the paper (EPR) he co-authored 10 with Boris Podolsky and Nathan Rosen in 1935.8 the Einstein’s Boxes thought experiment. The purpose of this paper is to resurrect another “Suppose a particle is enclosed in a box B thought experiment which, like EPR and Schr¨odinger’s with impermeable walls. The associated wave cat, is intended to argue against the orthodox doctrine of Ψ is confined to the box and cannot leave it. quantum completeness. This thought experiment – “Ein- The usual interpretation asserts that the par- 9 stein Boxes” – is due originally to Einstein, although it ticle is “potentially” present in the whole of has also been discussed and reformulated by de Broglie, the box B, with a probability Ψ 2 at each Schr¨odinger, Heisenberg, and others. point. Let us suppose that by some| | process Given its unique simplicity, clarity, and elegance, the or other, for example, by inserting a parti- relative obscurity of this thought experiment is unjusti- tion into the box, the box B is divided into fied. Although generally aiming to establish the same two separate parts B1 and B2 and that B1 8 dilemma as that posed in the EPR paper, Einstein’s and B2 are then transported to two very dis- Boxes establishes this conclusion with a more straight- tant places, for example to Paris and Tokyo. forward logical argument. Our hope is that this paper [See Fig. 1.] The particle, which has not yet will aid in re-injecting this old thought experiment into appeared, thus remains potentially present in the ongoing discussions of the significance and implica- the assembly of the two boxes and its wave tions of EPR, the completeness doctrine, and alternatives function Ψ consists of two parts, one of which, 2 gives no information about this. “We might note here how the usual interpre- tation leads to a paradox in the case of exper- iments with a negative result. Suppose that the particle is charged, and that in the box B2 in Tokyo a device has been installed which en- ables the whole of the charged particle located in the box to be drained off and in so doing to establish an observable localization. Now, if nothing is observed, this negative result will signify that the particle is not in box B2 and it is thus in box B1 in Paris. But this can reasonably signify only one thing: the parti- cle was already in Paris in box B1 prior to FIG. 1: A single particle is confined to the box B, into which the drainage experiment made in Tokyo in a barrier is inserted thus splitting the box and the particle’s box B2. Every other interpretation is absurd. wave function in two. The two half boxes (B1 and B2) are How can we imagine that the simple fact of then separated, at which point the boxes may be opened and having observed nothing in Tokyo has been their contents examined. able to promote the localization of the parti- cle at a distance of many thousands of miles away?”11 Ψ1, is located in B1 and the other, Ψ2, in B2. The wave function is thus of the form Although de Broglie did not make his reasoning ex- Ψ= c Ψ + c Ψ , where c 2 + c 2 = 1. 1 1 2 2 | 1| | 2| plicit, he gives some hints that allow us to plausibly re- “The probability laws of wave mechanics now construct the intended logical structure of this argument tell us that if an experiment is carried out in for incompleteness. The rhetorical question at the end box B1 in Paris, which will enable the pres- implies that de Broglie thinks it is impossible (“absurd”) ence of the particle to be revealed in this box, for the negative result of a measurement in Tokyo to have the probability of this experiment giving a a decisive causal effect on the contents of the box in Paris. 2 positive result is c1 , whilst the probability Likewise for a positive result: “If we show that the par- | | 2 of it giving a negative result is c2 . Accord- ticle is in box B1, it implies simply that it was already ing to the usual interpretation,| | this would there prior to localization,” and therefore, by implication, have the following significance: because the that it was already not in box B2 before the observa- particle is present in the assembly of the two tion. Generalizing, the measurement at B1 cannot have boxes prior to the observable localization, it affected the contents of the box B2, and therefore the would be immediately localized in box B1 in contents of B2 (which are known after inspection of B1) the case of a positive result in Paris. This must have been already definite before that inspection. does not seem to me to be acceptable. The According to at least one commonly held version of only reasonable interpretation appears to me the orthodox interpretation, however, it is the very act to be that prior to the observable localization of observation that disturbs the physical system in ques- in B1, we know that the particle was in one tion and brings about a definite result. Why should de of the two boxes B1 and B2, but we do not Broglie think that the act of opening B1 and examining know in which one, and the probabilities con- its contents can’t affect the contents of B2? Because B2 sidered in the usual wave mechanics are the is spatially distant. Its contents could not be affected by consequence of this partial ignorance. If we any reasonable physical mechanism by something done show that the particle is in box B1, it implies to the well-separated B1. That is evidently the point of simply that it was already there prior to lo- taking one box to Paris and the other to Tokyo before calization. Thus, we now return to the clear opening them. classical concept of probability, which springs There is thus a locality or separability assumption at from our partial ignorance of the true situa- work here. Given that B2 is physically separated from tion. But, if this point of view is accepted, B1 by a great distance, nothing we do to B1 can af- the description of the particle given by the fect the contents of B2.
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