Popper’s test of Quantum Mechanics Albert Bramon1 and Rafel Escribano2 5 1 Grup de F´ısica Te`orica, Universitat Aut`onomade Barcelona, E-08193 Bellaterra 0 (Barcelona), Spain [email protected] 0 2 Grup de F´ısica Te`orica and IFAE, Universitat Aut`onomade Barcelona, E-08193 2 Bellaterra (Barcelona), Spain [email protected] n a J A test of quantum mechanics proposed by K. Popper and dealing with two- 4 particle entangled states emitted from a fixed source has been criticized by 2 several authors. Some of them claim that the test becomes inconclusive once allthe quantumaspectsofthe sourceareconsidered.Moreover,anothercriti- 1 cismstatesthatthepredictionsattributedtoquantummechanicsinPopper’s v 4 analysis are untenable. We reconsider these criticisms and show that, to a 3 large extend, the ‘falsifiability’ potential of the test remains unaffected. 1 1 0 1 Introduction 5 0 / Heisenberg’s principle is the key feature of quantum mechanics [1] and plays h thecentralroleinmanyrelevantdiscussionsconcerningcounterintuitivequan- p - tumphenomena.Thisisparticularlytruewhenappliedtocompositequantum t n systems consisting of two-particle entangled states. A well known case is the a Einstein-Podolsky-Rosen paradox which appeared in 1935 short after a pre- u liminary and lesser known proposal by KarlPopper [2]. The latter “Popper’s q test”ofquantummechanicshasbeensubsequentlyreformulatedandimproved : v byPopperhimself[2]–[5],andreconsideredbyseveralauthors[6]–[12].Alook i X at the mostrecentof these papers showsthat the controversyis still open.In the present note we attempt to clarify some points of such a basic issue. r a Following Popper [5], we consider a source S decaying symmetrically into pairs of photons or equal-mass particles. In their center-of-mass frame, the two particles of eachpair are assumed to be simultaneously emitted fromthe originandtotravelinoppositedirections.Assume,forconcreteness,thattheir trajectories are contained in the plane defined by the horizontal x-axis and theverticaly-axis.Twoverticalscreensaresymmetricallyplacedatequaldis- tancesd,left(L)andright(R)ofthesource,thusintersectingperpendicularly thehorizontalaxisatx= d.Theleftandrightscreenshaves -ands -wide L R ± slits centered aroundthe horizontalaxis at x= d and x=+d, respectively. − Some pairs of the emitted particles will then pass through the slits and will 2 Albert Bramon and Rafel Escribano trigger coincidence detectors placed far away3 and completely covering the vertical space behind each slit. The vertical y-component of the momentum oftheleft-andright-movingparticles,(k ) k and(k ) k ,canthusbe 1 y 1 2 y 2 ≡ ≡ measured with this experimental setup. The other, non-vertical components ofk andk playasecondaryroleandthediscussionthuscentersonhowthe 1 2 observeddistributionsfork andk dependontheslitwidths,s ands .As- 1 2 L R sume, for instance, that one measures the k and k distributions for particle 1 2 pairs with both members passing through equal-width slits, s = s = 2a. L R Sincethisamountstopositionmeasurementswith∆y a,fromdiffraction 1,2 ≃ theory or Heisenberg’s principle one expects ∆k 1/2a for the dispersion 1,2 ≃ of the verticalmomenta behind each slit. According to Popper’s proposal,by two suitable modifications of the previous setup one can experimentally dis- criminate between his own(propensity)interpretationofquantummechanics and Copenhagen’s orthodoxy [5]. A first possibility —case (i)— consists in considering what happens to ∆k when one runs the experiment with the same s = 2a as before but 2 L with a much wider s , s 2a. Note that one still performs a position R R ≫ measurement with ∆y a on the particle passing through the left-side slit 1 ≃ and that, because of the entanglement, the right-moving particle has then to be in a state with ∆y a when it passes through the vertical plane 2 ≃ at x = +d. According to Popper [5], quantum mechanics predicts the same ∆k 1/2a as before, whereas his own (propensity) approach predicts the 2 ≃ disappearance of such a dispersion in ∆k . Moreover —case (ii)— one can 2 then narrow the left-slit width s while maintaining the same wide and fixed L s 2a, as before. According to quantum mechanics this narrowing of s R L ≫ implies larger values for ∆k 1/2a, while the opposite is expected from 2 ≃ Popper’s approach [5]. The proposal summarized in the previous two paragraphs —a crucial ex- periment, according to Popper— has however been criticized by several au- thors. All these criticisms are based on the fact that the source S cannot be exactly localized at the originand perfectly at rest, as required to argue that the two entangled final particles separate from the origin strictly with oppo- site momenta. The undecayed source itself or, better, the global two-particle final system has to obey Heisenberg’s principle and, accordingly, the vertical components of the CM-position, (y +y )/2, and total momentum, k +k , 1 2 1 2 must satisfy ∆(k + k )∆[(y +y )/2] 1/2. Once this constraint is im- 1 2 1 2 ≥ posed,the analysesofRefs.[6,7]—basedonsimple andintuitivegeometrical arguments— claim that Popper’s proposal is no longer able to discriminate betweenthe twoapproaches.Similarly,the discussionin[11,12]—basedona simplifiedwavefunctionforthetwo-particlesystemwhichobeysHeisenberg’s principle— claims that for case (ii) standard quantum mechanics predicts no 3These detectors are distributed in slightly different configurations according to the various authors. If the distance to the slits is large enough, all the various distributions become equivalent. Popper’s test of QuantumMechanics 3 increase of ∆k when narrowing the left-side slit. A claim which contradicts 2 Popper’s original analysis [5] and would make his test inconclusive as well. We now proceed to discuss that these claims —based on simple geometrical arguments and a too naive wave function— are not completely justified and that Popper’s test maintains most of its valuable ‘falsifiability’ potential. 2 Entangled two-particle state Consider the following wavefunction describing the behavior of an entangled two-particle system eik1y1 eik2y2 Ψ(y ,y ;t)= dk dk Ψ(k ,k ;t) , (1) 1 2 Z Z 1 2 1 2 √2π √2π where Ψ(k1,k2;t)= √πσ1+σ−e−4σ+1(t)2(k1+k2)2e−4σ−1(t)2(k1−k2)2 = 1 e−41(cid:16)σ+1(t)2+σ−1(t)2(cid:17)(k12+k22)e−14(cid:16)σ+1(t)2−σ−1(t)2(cid:17)2k1k2 , √πσ+σ− (2) 1 1 +i t accountsforthetimeevolutionalongtherelevant,verticaly- σ±(t)2 ≡ σ±2 m axis,andmisthemassofeachparticle.Hereandinwhatfollows,integrations extend from to + unless otherwise is stated. −∞ ∞ Note that for the global system we have chosen a Gaussian wave packet [1] with ∆(k +k ) = σ . This allows for analytical computations and at 1 2 + the decay time, t = 0, one has ∆(k + k )∆[(y +y )/2] = 1/2, which is 1 2 1 2 the minimum value compatible with Heisenberg’s principle; in this sense, our state is the closestquantumanalogto Popper’soriginalproposalwith a fixed and well localized source. NotealsothatwehavesimilarlychosenaGaussianpacketwith∆2k 1,2 | |≃ ∆(k k )=σ to describethe verticalspreadofthe finalmomenta.Admit- 1 2 − − tedly, this somehow reduces the generality of our treatment, but our Gaus- sianchoicesimplifiestheanalysisandbynomeansprecludesthediscussionof Popper’s proposal which was intended to be valid for a generic wave packet. Physically, the momentum distribution is isotropic in s-wave decays, such as positronium annihilation into two photons; restricting to particles moving in the xy-plane, the vertical components of their momenta, k , are uniformly 1,2 distributed in the range k k +k . For vertically polarized 1,2 1,2 1,2 −| | ≤ ≤ | | spin-1 states decaying into two spinless particles, as in vector-meson decays into two pseudoscalarmesons,one has k = k cosθ, where θ is the angle 1,2 1,2 between k and the x-axis, and a vertical m| ome|ntum distribution peaked 1,2 around k = 0. Initial sources of higher spin can lead to distributions with 1,2 morepronouncedpeaksaroundk =0.We cansomehowmimic this various 1,2 possibilities by a judicious choice of σ in our Gaussian packet. Note finally that one has σ σ for any realistic−value of k . + 1,2 − ≫ | | 4 Albert Bramon and Rafel Escribano 3 Popper’s proposal: case ii) Inordertodiscussthestandardquantummechanicalpredictionfor∆k when 2 the right-side slit is wide open and the width 2a of the left-side screen is modified, we need to Fourier transform state (2) into Ψ(y1;k2;t)= √dk21πeik1y1 R 1 e−41(cid:16)σ+1(t)2+σ−1(t)2(cid:17)(k12+k22)e−14(cid:16)σ+1(t)2−σ−1(t)2(cid:17)2k1k2 ×√πσ+σ− = √2 σ+(t)σ−(t) √πσ+σ− √σ+(t)2+σ−(t)2 e−σ+(t)2+1σ−(t)2(σ+(t)2σ−(t)2y12+k22−i(σ+(t)2−σ−(t)2)y1k2). × (3) Fromtheseexpressionsitiseasytocomputetheprobabilityforobservingthe vertical position of the left-moving particle within the range a y +a 1 − ≤ ≤ allowedbytheslitincoincidencewithagivenvalue,k ,fortheverticalcompo- 2 nent of the momentum of its right-moving partner. The former measurement requires detecting the left particle behind the slit using, for instance, a single detectorplacedonthe negativex-axisfarleft ofthe slit.The measurementof the verticalcomponent,k ,ofthe right-sidemomentumis achievedthanks to 2 thedistantsetofright-sidedetectors.Thequantummechanicalpredictionfor the spread of the k distribution in these coincidence measurements is then 2 unambiguous: dk k2 +ady Ψ(y ;k ;t)2 (∆k2)2|a = R dk2 2|R+−aady Ψ1(y ;1k ;2t)2| . (4) 2| a 1 1 2 | R R− We can now consider several values of the left-slit width 2a. If this is infinitely narrow, 2a 0 and y =0, one easily finds 1 → 2 (∆k2)2|a→0 = σ+2 +4 σ−2 1+(cid:16)1σ2+2σ+++σσσ−−22(cid:17)σ2σt+22σ−2 mt22 , (5) + m2 − which decreases from 1(σ2 +σ2) at t=0 to σ+2σ−2 when t . Note that 4 + − σ+2+σ−2 →∞ theseresultsholdnotonlyfory =0butalsoforanyotherpreciselocalization 1 (2a 0) at a given y of the left-moving particle. 1 → We next increase the width of the left-side slit to a value 2a small enough to allow for an expansion of the y -Gaussian. Retaining the first three terms 1 of the expansion, the quantum mechanical prediction for the k distribution 2 turns out to be (∆k )2 =(∆k )2 (1 2a2δ) , (6) 2 a 2 a 0 | | → − where Popper’s test of QuantumMechanics 5 2 δ = 1 (σ+2 −σ−2)2 1 1−(cid:16)σ2+2σ++σσ−−2 (cid:17) σ+2σ−2 mt22 , (7) 12 σ+2 +σ2 1+σ+2σ2 mt22 1+ 2σ+σ− 2σ2σ2 t2 − − (cid:16)σ+2+σ−2 (cid:17) + −m2 which is positive for reasonablevalues of t and thus (∆k )2 decreases when 2 a | increasing the slit-width s =2a. L We finally consider the other extreme case a . From Eq. (4) one → ∞ obtains σ2σ2 (∆k2)2|a→∞ = σ2++−σ2 , (8) + − which is easily seen to be never largerthan all the preceding results, Eqs.(5) and (6). Thepredictionsquotedinthelastthreeparagraphsfullyconfirmthequan- tum mechanical analyses by Popper [3]–[5] on the dependence of ∆k on the 2 left-side slit width 2a.The larger(narrower)this width is chosen,the smaller (wider) is the k -dispersion ∆k . It is easy to see that our treatment allows 2 2 to confirma relatedanalysis by Peres [8] where the single left-side slit is sub- stituted by a double slit thus producing interference effects on the right-side in coincidence measurements. 4 Popper’s proposal: case i) LetusfinallymovetotheotherpossibilityconsideredbyPopper.Inthiscase, onehastofixthewidthofleft-sideslittos =2aandcomparethepredictions L for∆k2 whentheright-sideslithasthesamewidth,s =s =2a,withthose 2 R L from another setup with a wide open right-slit, s s = 2a. This requires R L ≫ to perform a second Fourier transform of the state we are dealing with. It then reads, σ (t)σ (t) Ψ(y1,y2;t)= + − e−41[(σ+(t)2+σ−(t)2)(y12+y22)+(σ+(t)2−σ−(t)2)2y1y2] . (9) √πσ+σ − Fromthisexpressionitiseasytocomputethespreadingoftheverticalposition ofthe right-movingparticle,∆y2 ,when it crossesthe verticalplane x=+d 2|a at time t. Simultaneously, its left-side partner passes through the window a y +a allowed by the left-slit and will be detected much later. For 1 − ≤ ≤ these left-right coincidence detections, one has dy y2 +ady Ψ(y ,y ;t)2 (∆y2)2|a = R dy2 2|R+−aady Ψ1(y ,1y ;2t)2| . (10) 2| a 1 1 2 | R R− For a 0 one obtains → 6 Albert Bramon and Rafel Escribano (∆y )2 0 , (11) 1 a 0 | → → 1+σ4 t2 1+σ4 t2 1 +m2 m2 (∆y )2 = (cid:16) (cid:17)(cid:16) − (cid:17) 0 , (12) 2 |a→0 σ2 +σ2 1+σ2σ2 t2 ≥ + + m2 − − with ∆y2 = ∆y2 = 0 only if t 0 and σ , i.e., in the case 2|a→0 1|a→0 → ± → ∞ considered by Popper of a perfectly localized source (σ ) and ignoring + →∞ the wave packet spreading with t. For finite but small a one similarly finds (∆y )2 =(∆y )2 (1+2a2δ ) , (13) 2 a 2 a 0 ′ | | → where 2 δ′ = σ+2 1+2σ−2 2 1+σ14+t2σ+2σ1−2+mt2σ24 t2 − 11++(cid:16)σσ22+2σσ++2σσ−−2t2(cid:17)  , (14)  +m2 m2 + m2   (cid:0) (cid:1)(cid:0) − (cid:1) −  is never negative. According to the results of the two last paragraphs, the spreading of the verticalright-sidemomentum,∆k ,incoincidenceeventswithaleft-sidepar- 2 ticle passing through the s =2a wide slit, is indeed affected by the physical L presence of an equal 2a-wide slit on the right side. Its presence filters a nar- rower (in y ) right-moving wave packet and this translates into a ∆k which 2 2 is larger than in the case of removing (or making s 2a for) the right-side R ≫ slit. This conclusion is in agreement with the analysis made by Short [10] of the optical experiment performed by Kim and Shih [9]. 5 Conclusions We have reconsidered Popper’s test using the standard quantum mechani- cal formalism and, consequently, using a wave packet for the source —or, equivalently, for the initial two-particle system— which satisfies Heisenberg’s principle, ∆(k +k )∆[(y +y )/2] 1/2. This contrasts with the original 1 2 1 2 ≥ Popper’s proposal involving a fixed source and therefore subjected to the criticisms raised by several authors [6, 7, 8, 10, 11, 12]. In spite of this and contrarytothe claimsofsomeofthese authors[6]–[12],wefindthatPopper’s test can be conclusive in that a narrowing of the left-side slit increases ∆k2 2 of the freely right-moving particle in coincidence detections. In other words, the qualitative behavior of ∆k2 that Popper attributes to standard quantum 2 mechanics remains valid with our improved treatment of the initial state. In agreement with a related analysis by Short [10], we find however that the other aspect ofPopper’sproposalgets modified by our analysis;namely,∆k2 2 necessarily increases when a second slit is really and symmetrically inserted on the right-side of the setup. Popper’s test of QuantumMechanics 7 Ouranalysisshowsthat,tosomeextend,Popper’stestcanindeedbecon- clusiveto discriminatebetweenhisownapproachandthe standardversionof quantum mechanics. The latter turns out to be favored by recent optical ex- periments, which are somehow related to the originalproposal[9, 13, 14, 15]. Quantum non-locality, a key question in all these discussions, is nowadays firmly established. Recent experiments tend to falsify Popper’s approach but his understanding of quantum mechanics as early as in 1934 [2] is quite re- markable. Note added: After publication of the present paper in the Proceedings of the Fundamental Physics Meeting “Alberto Galindo” we have received an improved version of Qureshi’s paper in Ref. [12]. The state discussed in this new version coincides with our Eqs. (1) and (2) once we define σ2 = 1/4Ω2 + 0 and σ2 =4σ2. − Acknowledgements This work is partly supported by the Ramon y Cajal program (R.E.), the Ministerio de Ciencia y Tecnolog´ıa and FEDER, BFM-2002-02588, and the EU, HPRN-CT-2002-00311,EURIDICE network. References 1. Galindo A,Pascual P (1978) Mec´anica Cu´antica; ed. Alhambra,Madrid 2. Popper K R (1934) Die Naturwissenschaften 22:807 3. Popper K R (1982) Quantum Theory and the Schism in Physics. Hutchinson, London 4. Popper K R (1984) Open Questions in Quantum Physics. Tarozzi G, van der Merwe A (eds) Dreidel, Dordrecht 5. Popper K R (1987) Nature328:675 and (E) Nature329:112 6. Bedford D, Selleri F(1985) Lett NuovCim 42:325 7. Collett M J, Loudon R (1987) Nature326:671 and Nature328:675 8. Peres A (2002) Stud History Philos Modern Physics 33:23 9. Kim Y H,Shih Y H (1999) FoundPhys29:1849 10. Short A K (2001) FoundPhys Lett 14:275 11. QureshiT (2004) Int J Quant Inf2:1 12. QureshiT (2004) quant-ph/0405057 13. Strekalov D V (1995) Phys RevLett 74:3600 14. Zeilinger A (1999) Rev Mod Phys71:S288 15. Dopfer B (1998) Thesis, University of Innsbruck