Fortschritte der Physik / Progress of Physics: Band 28, Heft 12 1980 [Reprint 2021 ed.] 9783112523001, 9783112522998

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FORTSCHRITTE DER PHYSIK HERAUSGEGEBEN IM AUFTRAGE DER PHYSIKALISCHEN GESELLSCHAFT DER DEUTSCHEN DEMOKRATISCHEN REPUBLIK VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

H E F T 12 • 1980 • BAND 28

A K A D E M I E

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EVP 1 0 , - M 31728

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B E R L I N ISSN 0015 - 8 2 0 8

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Fortschritte der Physik 28, 6 3 3 - 6 5 7 (1980)

The Einstein Podolsky Rosen Paradox and Local Hidden—Variables Theories1) H . PAUL

Akademie

der Wissenschaften DDR-1199

der DDR, Zentralinstitut fur Optik und Berlin-Adlershof, Rudower Chaussee 6, German Democratic Republic

Spektroskopie,

Abstract Einstein's severe critique of quantum mechanics took its ultimate form in the so-called Einstein Podolsky Rosen (EPR) paradox. The latter remained a subject of mere dispute, until a remarkable progress was initiated in 1964 by Bell who showed that quantum mechanical correlations specific for an E P R type experiment cannot be reproduced in any local hidden—variables theory. Afterwards, realistic versions of E P R experiments have been conceived and, in fact, carried out. The objective of the present paper is to give a review of this recent exciting development, both theoretical and experimental, which reveals one of the most striking features of quantum theory, the "quantum interconnectedness of distant systems".

Contents 1. 2. 3. 4. 5. 6. 7. 8. 9.

Introduction The E P R argument Bell's theorem Theoretical analysis of cascade-photon experiments Stochastic approach Experimental results Implication for semiclassical radiation theory Positronium annihilation experiments Discussion

References

-

633 634 637 640 648 650 651 652 654 657

1. Introduction I t is well known that Albert Einstein, though having given strong impulses on the development of quantum mechanics by his fundamental papers on photons, the specific heat of solids and non-classical (Bose-Einstein) statistics, ultimately could not put up with the intrinsically indeterministic nature of this revolutionary theory. " G o d does not play Extended version of a lecture delivered on the " V I I Summer School in Quantum Optics", 16—22 September 1979, Wiezyca (Poland) 44

Zeitschrift „Fortschritte der Physik", Hefl 12

634

H . PAUL

dice" was the conviction he adhered to until his death. His most profound critique of quantum mechanics which he advanced, jointly with PODOLSKY and ROSEN, in 1935 [1] has become famous as the Einstein Podolsky Rosen (hereafter abbreviated as E P R ) paradox. In that paper serious arguments were presented which conclusively, as it seemed, demonstrated the quantum mechanical description to be incomplete. Although BOHR [2] soon rejected Einstein's criticism, the problem has hitherto remained a subject of discussion for theorists. In 1964 the situation changed drastically, however, when BELL [3] showed for a Gedankenexperiment of the E P R type that any hidden—variables theory which satisfies an additional, nevertheless very natural condition named locality, is in conflict with quantum mechanics, leading to certain predictions which contradict those made by quantum mechanics. The next important step ahead is due to CLAUSER and coworkers [4] who conceived a practical device suitable for an experimental study of the E P R paradox. Generalizing BELL'S theorem [3], they thus provided the basis which allowed to settle the problem by an actual experiment. In the following years several authors performed experiments along these lines, the first to do so being FREEDMAJST and CLAUSER [5]. The results fully confirmed the quantum mechanical predictions, thus ruling out local hidden—variables theories with a high degree of certainty. The goal of the present paper is to give a concise review of the progress, both theoretical and experimental, which has been initiated by the pioneering work of BELL [3]. (For further details the reader is referred to a recent review article [6].) In sec. 2 the E P R argument will be presented in some detail. Sec. 3 is devoted to the derivation of Bell's fundamental theorem. In sec. 4 the feasible version of an E P R experiment proposed by CLAUSER et al. [4] is described, and analyzed theoretically, both in terms of quantum mechanics and deterministic local hidden—variables theories. Sec. 5 contains an analysis based on a stochastic rather than a deterministic concept. Sec. 6 gives an account of experimental work on 2-photon cascade processes in atoms which provides evidence against local hidden—variables theories. In sec. 7 an important implication of those experiments for the semiclassical theory of radiation is discussed. In sec. 8 different E P R type experiments in which the positronium decay is studied, will be briefly reported. Finally, in sec. 9 the general physical relevance of the quantum mechanical correlations becoming manifest in E P R type experiments will be emphasized.

2. The E P R argument In the form advocated by BOHM [7] the E P R Gedankenexperiment can be described as follows. A pair of spin one-half particles are initially prepared to form a molecule with zero spin. Then the molecule is made to dissociate without changing the total spin, the two particles propagating in different, say opposite directions. The point is that, on the one hand, a spatial separation of the particles occurs which, in principle, may become infinitely wide, while, on the other hand, specific quantum mechanical correlations between the spins of the particles impressed on them in the initial state, persist to be present. Those correlations become manifest, when the spin components of the two particles with respect to, say the z-direction, cr.,1 and cr,11, are measured by means of a Stern Gerlach apparatus. Owing to the fact that the total spin is zero, there are only two possible results a) ff*1 =

and

, t j (j = I, II) stand for the maximum and minimum transmittance, respectively, of the filters. (Ideal filters are characterized by eM = 1 and em = 0.) It turns out that in practical cases the correction due to filter inefficiency is the dominant one. Tor an analysis of the implications of deterministic hidden—variables theories for realistic measurements we must go back to the inequality ( 4 . 5 ) . Following CLATJSEE et al. [4], we introduce a third expectation value E(ß', ß) = J A(ß', A) B(ß, X) e(A) dX.

(4.19)

Evidently, eqs. (4.3) imply that either B{ß, 1) = +A(ß', X) or Biß, X) = —A(ß', X) for any X. Hence we may divide the space of the hidden variables A, for given values of ß and ß', into two subspaces A+ and A_ which correspond to the two afore-mentioned cases. This allows us to write the quantity (4.19) as E{ß', ß) = / e(A) dX - I e(X) dX. A+

(4.20)

A-

Making use of the normalization property (3.2), we find E(ß', ß) = 1 - 2fe(X)dX. A-

(4.21)

The Einstein Podolsky Rosen Paradox and Local Hidden—Variables Theories

645

This auxiliary relation enables us, in fact, to give the integral on the right-hand side of (4.5) a suitable form. F r o m the definitions of A+ and / L it follows t h a t this integral can be represented as / B(ß, X) B(y, X) q{X) dX = f A(ß', 2) B(y, A

X) EW

dX -

A+

/ A(ß',

X)

B(y,

X)

dX

A-

= / A(ß', X) B(y, 1) q(X) dX - 2 / A(ß', X) B(y, X)