Solving the Einstein-Podolksy-Rosen puzzle: a possible origin of non-locality
aa r X i v : . [ qu a n t - ph ] S e p Solving the Einstein-Podolksy-Rosen puzzle: a possible origin of non-locality.
Werner A. Hofer
Department of Physics, University of LiverpoolL69 3BX Liverpool, United Kingdom
So far no mechanism is known, which could connect the two measurements in a Bell-type experi-ment with a speed beyond the speed of light, commonly considered the ultimate limit of propagationof any field-like interaction. Here, we suggest such a mechanism, based on the phase of a photonfield during its propagation. We show that two measurements, corresponding to two independentrotations of the fields, are connected, even if no signal passes from one point of measurement tothe other. The non-local connection of a photon pair is the result of its origin at a common source,where the two fields acquire a well defined phase difference. Therefore, it is not actually a non-localeffect in any conventional sense.
One of the most puzzling results in modern physics,based originally on a paper by Einstein, Podolsky, andRosen (EPR)[1], is the apparent non-locality of correla-tion measurements in quantum optics [2]. The measure-ments performed on pairs of entangled photons, begin-ning with the experiments by Alain Aspect in 1982[3],seem to prove beyond doubt that the two measurementsare not independent. The measurements are usually in-terpreted in terms of the Bell inequalities[4], which assertthat their violation, corresponding to the experimentalresults and also the theoretical predictions of quantummechanics, amounts to a non-local connection betweenthe two independent measurements[2]. In Figure 1 weshow the setup of a Bell-type experiment. Two photonswith a defined phase difference ∆ are emitted from acommon source. After traveling an integer number ofwavelengths λ of their associated field, either n , in pos-itive z -direction, or m , in negative z -direction, they aresubjected to a measurement, which is assumed to consistof a rotation of the photon fields. Rotations in geomet-ric algebra[5] are described by a multiplication with ageometric product of two vectors. Here, we assume rota-tions perpendicular to the direction of photon propaga-tion, which act on a Poynting-like vector of the electro-magnetic fields. The rotations are then described by: R ( z ) = exp ( e e ) e z π/λR ( z ) = exp − ( e e ) e z π/λ (1)where the values of z i are limited by 0 ≤ z i ≤ λ . The λ n λ - λ -m λ Source Photon 1Photon 2Rotator 2 Rotator 1 zFilter 2 Filter 1
FIG. 1: (Color online) Bell-type experiment. Two photons areemitted from a common source, and subjected at two definedpoints − mλ and nλ to separate measurements, here assumedto be a rotation of the fields and a filtering process. rotations thus cover all values from zero to a full rotationof 2 π . It is evident that the rotations are local measure-ments, i.e. the rotation at point − mλ is independentof the rotation at point nλ . Given that the geometricproducts involve a product of the three frame vectors e i , the brackets can be omitted and the triple product( e e ) e = e e e = i . The two rotations are thus: R ( ϕ ) = exp iz π/λ = e iϕ R ( ϕ ) = exp − i ( z π/λ + ∆) = e − i ( ϕ +∆) (2)where we symbolized the product z i π/λ by ϕ i . Thenormalized probability p of detecting photons after a ro-tation with angle ϕ i shall be given by the square of thereal part of the rotation, or: p ( ϕ i ) = ( ℜ ( ϕ i )) (3)The probability in this case models a filter, acting afterthe rotator. Here, the measurement depends on the phasedifference between the source of the photon pair and theend point of the rotation. The real part of the phase dif-ference is thus the square root of the detection probabil-ity. For coincidence measurements we have therefore toconsider the phase difference between the two end pointsof the rotation of both photons. The correlations betweentwo measurements with angles ϕ , ϕ are then describedby a square of the real part of the product R ( ϕ ) · R ( ϕ ).The relations between rotations and probabilities of pho-ton measurements are: p ( ϕ ) = cos ϕ p ( ϕ ) = cos ϕ p ( ϕ , ϕ ) = cos ( ϕ − ϕ − ∆) (4)It is evident that in this case the correlations cannotbe composed of a product of the individual probabili-ties. The reason for this feature is that the rotations arecomplex phases, while the individual measurements arereal values; the product of probabilities thus does notaccount for the full connection of rotations in three di-mensions. It is also evident that the correlations describethe conditional probabilities in quantum mechanics, eventhough the individual rotations are fully local. From thisperspective it seems difficult to reconcile the frameworkwith the local derivations by John Bell[4], which, as wellknown, assume that locality implies such a separation oflocal probability values, and that the product of theselocal probabilities is equal to the conditional probabilityor the correlation probability. For rotations and complexphases such a derivation seems not justified.The framework can be generalized to three and morerotations. Assuming that we have two rotators on eitherside, positioned at integer values of the photon wave-length, the conditional probability for four individualmeasurements with rotators ϕ to ϕ , where ϕ and ϕ are in positive z -direction while ϕ and ϕ are in negative z -direction, is equal to: p ( ϕ , ϕ , ϕ , ϕ ) = cos ( ϕ + ϕ − ϕ − ϕ − ∆) (5)To appreciate the novelty of the approach it is illuminat-ing to cite Alain Aspect’s review paper in 1999 [2]: ”Theviolation of Bell’s inequality, with strict relativistic sepa-ration between the chosen measurements, means that itis impossible to maintain the image ’´a la Einstein’ wherecorrelations are explained by common properties deter-mined at the common source and subsequently carriedalong by each photon. We must conclude that an entan-gled EPR photon pair is a non-separable object; that is,it is impossible to assign individual local properties (lo-cal physical reality) to each photon. In some sense, bothphotons keep in contact through space and time.”Here, we found that the ”common property ... carriedalong by each photon” is a complex phase, which willbe altered in a rotator. The actual normalized count p does not reveal the full physical situation; it is thereforenecessary to take the correlated normalized count for theproduct of two complex rotations and not, as assumedin the derivation of Bell’s inequalities, the product of thetwo separate normalized counts. The additional informa-tion about the imaginary component of the phase is notrevealed in the local counts, even though it is present inthe local rotations. It seems thus that the difference be-tween the physical situation (a rotation of the fields), andthe actual measurement result (a count of photons afterrotation), has not been appreciated to date. Rotationsin three dimensional space, formalized within the frame-work of geometric algebra, are the key to understandingspin properties of electrons, as recently established[6].Based on this analysis, it seems that they are equallykey to understanding polarizations of photons and elec-tromagnetic fields.Experimentally, the measurements are performed ona pair of down-converted photons[3], which are sep-arated, subjected to a polarizer which can be inter-preted as a combination of a rotator plus a filter, andthen measured either in a spin-up or spin-down stateat the detectors. Experiments are usually interpretedin terms of the Clauser-Horne-Shimony-Holt-inequalities(CHSH)[7], which are based on normalized expectationvalues E ( ϕ , ϕ ), derived from coincidence counts of pho-ton spins at the two points of measurement. Within thepresent context it is actually unnecessary to define ex- actly, what spin-up and spin-down means in a measure-ment; it suffices to assume that they will be subject tothe same relation between rotational angles and detectionprobability. For a phase-difference ∆ = 0 the normalizeddetection rates for spin-up and spin-down photons will be(we denote coincidences by a capital C, as is standard inthe literature, and also use the convention that a coinci-dence is the measurement of equal spin for both, spin-up(+) and spin-down (-) components): C ++ = C −− = cos ( ϕ − ϕ ) C + − = C − + = 1 − cos ( ϕ − ϕ ) (6)Then we obtain the same correlations of polarizations asin Aspect’s first experiments[3], namely: E ( ϕ , ϕ ) = 2 cos ( ϕ − ϕ ) − ϕ − ϕ ) (7)Correlations at different pairs of angles ϕ , ϕ canbe combined to a sum S , which, according to Bell’sderivation[4], should not be larger than two for any localmodel. Within the present model we obtain, in accor-dance with experimental results and also with predictionsof quantum mechanics: S ( ϕ , ϕ ′ , ϕ , ϕ ′ ) = E ( ϕ , ϕ ) − E ( ϕ , ϕ ′ ) + (8)+ E ( ϕ ′ , ϕ ) + E ( ϕ ′ , ϕ ′ ) = 2 √ ϕ = 0 , ϕ ′ = 45 , ϕ = 22 . , ϕ ′ = 67 .
5, in violationof the Bell inequalities. The model thus fully accountsfor experimental values under ideal conditions (which arenearly reached in the most advanced experiments[8]), andalso for the standard predictions in quantum mechanics.The underlying reason that quantum mechanics ap-pears to be non-local is due to its formulation in termsof operators and expectation values which entail inte-grations over the whole system. A local model, basedon geometric algebra and phases, can obtain the samenumerical results, as shown here concerning EPR-typeexperiments. Moreover, while the standard model makesthe actual connection of entangled photons somewhat lessthan transparent, the model developed has the advantagethat all processes are local and transparent. There is, asshown, no connection between the two measurements ex-ceeding the speed of light. Moreover, the present modelis also a fully local model of photon entanglement. Allthat is required for entanglement, it seems, is a coherentphase between the two photons. Whether this model isthe whole answer to the problem, or only part of an even-tually fully comprehensive theory, cannot be estimated atpresent. This will to a large extent depend on subsequentexperimental tests. However, any claim that quantumentanglement is entirely incomprehensible and mysteri-ous and forces us to renounce all aspirations to modelingit in terms of standard physical concepts like realism andlocality, an opinion frequently found in the EPR litera-ture, will seem hard to defend on the basis of this work.From a statistical point of view an initial phase ϕ atthe origin for both photons does not alter the outcome.Neither the single probabilities p ( ϕ ) or p ( ϕ ), nor theconditional probability p ( ϕ , ϕ ) will be affected. In thiscontext it is interesting to note that the present analysisis based only on the field properties of photons and theirrotational features. It does not need to consider any par-ticle properties to arrive at the derived results. However,it does also not specify, whether an individual photonat a particular setting of the polarizer will actually bemeasured or not. For the macroscopic outcome such aspecification is neither necessary, nor does it form partof any theoretical model which describes the experimentsat present. It is certainly not part of quantum mechanics,which, as shown, can be replicated with a model basedon three-dimensional rotations and phases of the photonfields. But it is also not contained in Bell’s analysis ofthe original EPR problem [4], where it is never speci-fied, what will trigger the detection of individual pho-tons. Eventually, this question might be answered by adetailed analysis of the dynamical processes in the po-larizer (rotator/filter in the present model) itself, whichcontains hidden variables in the exact shape of the fieldsor the thermal fluctuations of the polarizer atoms. Sucha detailed picture is not necessary, though, to establishthe correlations which have been puzzling physicists formore than thirty years.Returning to the original EPR problem and the ques-tion whether there is an ”element of reality” in the exper-iments, which is not described by the formalism in quan-tum mechanics, it turns out that Einstein was wrong:the description in quantum mechanics via Pauli matri-ces is an equivalent way of accounting for rotations inthree dimensional space, as already emphasized [6]. Itderives from the fact that the geometric algebra of threedimensional space is equivalent to the algebra of the Paulimatrices: e i e j = δ ij + iǫ ijk e k ˆ σ i ˆ σ j = δ ij + iǫ ijk ˆ σ k (9) The formalism in quantum mechanics is thus complete.However, Einstein was also right, because the imaginarycomponent of the phase difference, which is a conse-quence of rotations in geometric algebra, is not strictlyspeaking a physical property of the system in quantummechanics, and it is not revealed in the experiments,where only the real component shows up in the photoncount. It is thus a hidden variable. But this imaginarycomponent is a - classical - geometric component of ro-tations in geometric algebra, thus it does have physicalreality. This reality has so far been denied in quantummechanics. In this sense one could say that even thoughthe framework is formally complete, its relation to phys-ical reality has not been correctly described. This seemsto account also for Bell’s derivation of his famous inequal-ities [4]: since Bell did not ascribe physical reality to theimaginary component of the phase, he did arrive only ata limited description of the situation from the viewpointof geometric algebra. Thus his inequalities can be vio-lated in a local and realistic model, even though it hasfrequently been asserted that this is not possible.On a final note we hope that this model and the clarifi-cations presented in this paper will contribute to a morerational debate about scientific issues in quantum me-chanics in the future. Irrational statements about eitherthe scope or the epistemological meaning of quantum me-chanics, which is after all only a scientific theory, seemto have been the main obstacles to scientific progress inthe past. Acknowledgements
Helpful discussions with Kon-stantin Lukin and Sheldon Goldstein are gratefully ac-knowledged. The author also acknowledges support fromthe Royal Society London. [1] A. Einstein, B. Podolsky, N. Rosen, Can Quantum-Mechanical Description of Physical Reality Be ConsideredComplete?
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