Annalen der Physik, 10 January 2011 Quantum mechanical EPRBA covariance and classical probability J.F. Geurdes∗1 1 First address Received 11 January 2011 Key words EPR, Bell theorem, Locality Contrary to Bell's theorem it is demonstrated that with the use of classical probability theory the quantum correlation can be approximated. Hence, one may not conclude from experiment that all local hidden variable theories are ruled out by a violation of inequality result. Copyright line will be provided by the publisher 1 Introduction In the debate of the foundations of quantum theory (QM), Bell's theorem [1] is an important milestone. Based on Einstein's criticism of completeness [2], Bell formulated an expression for the correlation between distant spin measurements such as described by Bohm [3]. In Bell's expression, hidden variables to restore locality and causality to the theory (LHV's) are introduced through a probability (mass) density function and through their influence upon the elementary measurement functions in the two separate wings (denoted by the Aand the B-wing) of the experiment. Many experiments and theoretical developments arose from Bell's original paper. The most important experiment was performed by Aspect [4]. Aspect's results were interpreted as a confirmation of the completeness of quantum mechanics. From that point, QM was considered a non-local theory. In a previous paper [5] the present author argued that there was insufficient ground for this conclusion. The author would like to point that merely an appeal to the simplicity and apparent logical truth of Bell's theorem is advanced to support the exclusive non-locality interpretation. Beyond that, there is no proof that experimental results must exclusively be interpreted in this manner. According to the author the inequalities of Bell were given the status of theorem without sufficient supportive evidence save simplicity. In this paper it will be demonstrated that such a conclusive supportive argument of Bell's theorem does not exist because an approximate classical model is possible. The principles of the proposed model can also be expressed in numerical terms. This enables numerical simulation of the experiments on a computer. Let us shortly describe a typical idealized Bell experiment. In such an experiment, from a single source, two particles with opposite spin are sent into opposite directions. For instance, we could think of a positron and an electron arising from para-Positronium that are drawn apart by dipole radiation. Subsequently, in the respective wings of the experimental set-up, the spin of the individual particle is measured with a SternGerlach magnet. The measurements are found to be correlated with the, unitary, parameter vectors of the magnets, a = (a1, a2, a3),b = (b1, b2, b3). It is well-known that the QM correlation for singlet state electron and positron is equal to, PQM (a,b) = −(a * b) = − 3∑ k=1 akbk. (1) ∗ Corresponding author: e-mail: han.geurdes@gmail.com, Phone: +00 999 999 999, Fax: +00 999 999 999 Copyright line will be provided by the publisher 4 J.F. Geurdes: EPRBA Bell's theorem states that local hidden variables cannot recover the quantum correlation, −(a *b), because all local hidden distributions run into inequalities similar to the one stated below. Vectors a,b, c and d are unitary. |PLHV (a,b)− PLHV (a,d)|+ |PLHV (c,b) + PLHV (c,d)| ≤ 2 (2) 2 Preliminary remarks In order to study locality and causality in a quantitative manner, Bell wrote the following general expression for PLHV (a,b) based upon general, probability theoretical, assumptions about the classical distribution of hidden substance that was supposed to explain the quantum correlation between A and B measurements of spin. We have PLHV (a,b) = ∫ dλρ(λ)A(a, λ)B(b, λ) (3) Here, λ represents the hidden variables. We are allowed to take as many LHVs, λm, with, m = 1, 2, 3, ..... as we like. Moreover, ρ(λ) represents the probability density of the hidden variables. From this expresion, equation 2 is derived. Because a classical explanation is sought for the quantum correlation, ρ(λ) has to follow classical probability laws. This means the intuitive clear conception of probability and the associated variable space [6]. The measurement functions, A(a, λ) and B(b, λ) project in the set {−1,+1}, agreeing with the discrete result of measuring spins. The author apologizes beforehand for not having mentioned many other important researches on Bell's theorem. His main concern in the paper was to obtain the proper mathematics. Two studies need to be mentioned however. In the first place the author would like to refer to the model of Hess and Phillip [7] and to a study of de Raedt [8]. The reader should also note that, although measurement functions and variables inside particles and measuring instruments are given, there is no physical explanation given. In this sense it is a mathematical model. Use will be made of some basic facts of physics, like energy quantization, in order to justify some choices in the model. Note that the hidden variables and their ranges will be defined in the probability density section. Moreover, it is at this stage of the model extremely difficult to provide a physical interpretation of the hidden variables. 3 Measurement functions Here, we first introduce the model's measurement functions in terms of local hidden variables. Before doing that, let us postulate the variables that reside in the particles that propagate to the respective measurement instruments, A and B. σ{AB = 3∑ k=1 {akbk sign(yk) (4) This compact 'stacked' notation will be used throughout the paper. It must be read as σA = 3∑ k=1 aksign(yk), σB = 3∑ k=1 bksign(yk) (5) The yk,with, k = 1, 2, 3 in the previous formulae are standard normal Gaussian distributed variables, N(0, 1), whereas the ak and bk arise from the two Stern-Gerlach instruments unitary parameter vectors. The stacked notation implicitly denotes the similarity between the two measurement instruments. Most of the time, only the A version of the measurement functions and probability density will be given and it is implied that B follows the same mathematical structure. In this way the somewhat cumbersome stacked Copyright line will be provided by the publisher EPRBA COVARIANCE AND CLASSICAL PROBABILITY 5 notation can be avoided in discussions of the mathematics. In any case and using the stacked notation here, it is clear that |σ{AB | ≤ √ 3 (6) An additional set of definitions is necessary to better understand the model parameters. Let us define an εA, positive 'small' and similarly an εB such that NA = 1 εA , TA = √ log(NA)√ 3− 1 (7) Sometimes it is handy to either use NA, or εA or TA in the presentation of the model. The reader should note that they are related as presented in the equation above. Basing ourselves upon the definitions of the previous sums that contain parameter vector components and variables carried by the particles and remembering the A and B structural equivalence, we may now write the first component of the measurement function A as A1 = sign[ σA√ 3 θ(tA − 1 N2A )θ( 1 N2A + 4 NA − tA) sin(αAtA)− μA] (8) thereby introducing the hidden variables, αA, tA and μA. Here, the θ function is defined as θ(x) = 1, when, x ≥ 0, while,θ(x) = 0, when, x < 0 . Perhaps, not necessary to say but a similar set for B, αB , tB and μB , is thought to exist that follow the same structure as A1, in the construction of B1. The distributions of the hidden variables or the values of hidden parameters will be given later. At the moment note that, sign(x) = 1 for,x > 0, while, sign(x) = −1 for, x ≤ 0. Moreover it is illustrative to write the following integration for a certain ξ ∈ [−1, 1] 1 2 ∫ 1 −1 sign(ξ − x)dx = 1 2 ∫ ξ −1 dx − 1 2 ∫ 1 ξ dx = 12 (ξ − (−1))− 1 2 (1− ξ) = ξ (9) Because of the inequality on |σA|, we may conclude that σA√3θ(tA − 1 N2A )θ( 1 N2A + 4NA − tA) sin(αAtA) ∈ [−1, 1]. Hence an integration like in the previous is allowed, when μA ∈ [−1, 1]. In addition, a second component of the measurement function A is to be defined as A0 = { 1, when νA ∈ [−NεA , 1) sign(φA − νA),when νA ∈ [1, NεA ] (10) Here we have introduced the hidden variable νA ∈ [−NεA , NεA ]. The NεA is derived from the previously introduced NA as NεA = NA 1− 11+(NA/εA)2 ≥ NA (11) If, φA ∈ [−NεA , NεA ], then we may write a similar integration as 1 2 ∫ NεA −NεA sign(φA − νA)dνA = 1 2 ∫ φA −NεA dνA − 1 2 ∫ NεA φA dνA = φA (12) Subsequently, the φA = φA(xA) function is given by φA =  1 xA , xA ∈ [τA, 1] (1 + T 2A)/2, xA ∈ [0, τA) 1, elsewhere (13) Copyright line will be provided by the publisher 6 J.F. Geurdes: EPRBA Here τA is defined by τA = exp[−T 2A( √ 3 − 1)] = εA = 1/NA, such as given previously. As is clear from the definition of φA, letting the hidden variable xA run the whole real axis for the moment, this function projects in the interval [1, NεA ] ⊂ [−NεA , NεA ]. This is true because, φA attains its maximum value, NA when xA = τA and NA ≤ NεA . Furthermore, for xA outside the interval [0, 1], the function φA is equal to unity. In addition because, log(NA) < ( √ 3 − 1)NA and NA ≤ NεA , when φA attains the value, (1 + T 2A)/2 for xA ∈ [0, τA), it remains inside the interval [1, NεA ]. Hence the integration over the interval,[−NεA , NεA ], in the former equation may be applied. The respective A-wing and B-wing measurement functions are defined byA = A0A1 andB = −B0B1. 4 Probability model In this section we turn the attention to the probability density, generally denoted by ρ(λ) in Bell's formulation of the correlation. For an effective presentation the vanishing part of partial densities will be described in the text below the definition. Moreover, we only present the A-wing of the probability densities associated to the measuring instruments. The B-wing is implicitly identical to the A-wing. Firstly, let us define the probability density for the xA variables ρxA(xA) = 1 1 + T 2A , for xA ∈ [−T 2A, 1] (14) while ρxA(xA) = 0, elsewhere, i.e. xA /∈ [−T 2A, 1]. As can be easily verified, ρxA(xA) is a classical density for∫ ρxA(xA)dxA = ∫ 1 −T 2A 1 1 + T 2A dxA = 1 + T 2A 1 + T 2A = 1 (15) Secondly, let us define the density of tA as ρtA(tA) = 2tAε 2 ANANεA (t2AN 2 A + ε 2 A) 2 , for tA ∈ [0, 1] (16) while ρtA(tA) = 0 for tA /∈ [0, 1]. This density is also classical because∫ ρtA(tA)dtA = ∫ 1 0 2tAε 2 ANANεA (t2AN 2 A + ε 2 A) 2 dtA. (17) From this it follows that, using, ωA = NAtA∫ ρtA(tA)dtA = NεAε 2 A NA ∫ NA 0 2ωA (ω2A + ε 2 A) 2 dωA (18) Hence, according to the definition of NεA in Eq. (11)∫ ρtA(tA)dtA = −NεAε 2 A NA ∫ NA 0 d dωA ( 1 ω2A+ε 2 A )dωA = −NεA ε 2 A NA { 1 N2A+ε 2 A − 1 ε2A } = 1. (19) Thirdly, the density of νA can be given by ρνA(νA) = 1 2NεA , for νA ∈ [−NεA , NεA ] (20) while, ρνA(νA) = 0 for νA /∈ [−NεA , NεA ]. One can easily see that ρνA(νA) is a classical density because∫ ρνA(νA)dνA = 1 2NεA ∫ NεA −NεA dνA = 1. (21) Copyright line will be provided by the publisher EPRBA COVARIANCE AND CLASSICAL PROBABILITY 7 Fourthly, the density for μA is given by ρμA(μA) = 1 2 , for μA ∈ [−1, 1] (22) while ρμA(μA) = 0 for μA /∈ [−1, 1]. This density is the well-known uniform density on [−1, 1] and is classical with∫ ρμA(μA)dνA = 1 2 ∫ 1 −1 dμA = 1. (23) For completeness the Gaussian standard normal density for three variables, denoted here with N(0, 1)(y), is written as ρy(y) = 1 (2π)3/2 exp[−1 2 |y|2] (24) here, |y|2 is the Euclidean norm of the vector y. Needless to say that N(0, 1)(y) for y ∈ R3 is classical. With the use of the previously defined A-wing densities, the total A-wing and B-wing density can be given ρ{AB = ρx{AB ρt{A B ρν{A B ρμ{A B (25) From the definition of the components it follows that ρ{AB are classical. Hence, the total density ρtot = ρyρAρB is classical too. Note for completeness that ρtot plays the role of ρ(λ) in the definition of Bell. 5 Evaluation of the model, preliminaries Before presenting the complete evaluation of all the integrals in the model we show some essential results. First, note that integration of sums containing signs of yk, with, k = 1, 2, 3, gives∫ +∞ −∞ dy1 ∫ +∞ −∞ dy2 ∫ +∞ −∞ dy3 N(0, 1)(y)sign(yk)sign(ym) = δk,m. (26) Here, δk,m is Kronecker's delta k,m = 1, 2, 3. Secondly, let us evaluate the integral over theφA(xA) function. From the definitions of the measurement functions (in particular Eq. (13)) and the probability model (in particular Eq.(14)) it follows that∫ 1 −T 2A φA(xA) 1+T 2A dxA = ∫ 0 −T 2A 1 1+T 2A dxA + ∫ τA 0 (1+T 2A)/2 1+T 2A dxA+∫ 1 τA 1 (1+T 2A)xA dxA (27) Hence, from the previous it follows that∫ 1 −T 2A φA(xA) 1+T 2A dxA = T 2A 1+T 2A + 12exp[−T 2 A( √ 3− 1)] + T 2 A( √ 3−1) 1+T 2A = √ 3 1+(1/T 2A) +O(εA). (28) Note that O(εA) is employed in the sense of Landau [9], without by necessity taken a limit. 6 Evaluation of the model, expectation value In order to evaluate the integrations in our model for Bell's expression for the correlation, let us define the expectation values. For a general proper function F in the A-wing variables, xA, tA, νA, μA and the B-wing variables xB , tB , νB , μB we define E(F ) = ∫ ρtotFdxAdtAdνAdμAdxBdtBdνBdμBdy1dy2dy3 (29) Copyright line will be provided by the publisher 8 J.F. Geurdes: EPRBA More in specific in an 'operator'-like form whith the order of integration as indicated E(F ) = ∫ +∞ −∞ dy1 ∫ +∞ −∞ dy2 ∫ +∞ −∞ dy3 ∫ 1 −T 2A dxA ∫ 1 0 dtA ∫ NεA −NεA dνA ∫ 1 −1 dμA × ∫ 1 −T 2B dxB ∫ 1 0 dtB ∫ NεB −NεB dνB ∫ 1 −1 dμB{ρtotF} (30) Hence, from the previous we may conclude that E(1) = 1 as required. In order to have a less cumbersome presentation we may rewrite the previous total expectation as E(F ) = Ey(EA(EB(F ))) (31) From Eq. (30) and the definition of the total density, ρtot we may obtain that taking the Gaussian expectation is equal to evaluate a proper function f as Ey(f) = ∫ +∞ −∞ dy1 ∫ +∞ −∞ dy2 ∫ +∞ −∞ dy3 N(0, 1)(y)f (32) Moreover, we are able to write EA(FA) for a proper FA as EA(FA) = ∫ 1 −T 2A dxA ∫ 1 0 dtA ∫ NεA −NεA dνA ∫ 1 −1 dμA ρAFA (33) and similar for B-wing expectation EB(FB). Observe that Bell's theorem does not disallow integral ordering. Moreover, there appears to be no experiment possible that may exclude the ordering presently employed. It may be noted too that PLHV (a,b) = E(AB), which justifies the title of (quantum mechanical) covariance. When E(A) = E(B) = 0 and E(A2) = E(B2) = 1 the covariance and correlation coincide. 7 Evaluation of the model, approximation Let us now inspect more closely EA(A) and note that A = A0A1. Firstly we take a closer look at the integration of A1. As already explained we may write∫ ρμAA1dμA = 1 2 ∫ +1 −1 sign[ σA√ 3 θ(tA − 1N2A )θ( 1 N2A + 4NA − tA) sin(αAtA)− μA]dμA (34) or ∫ ρμAA1dμA = σA√ 3 θ(tA − 1 N2A )θ( 1 N2A + 4 NA − tA) sin(αAtA). (35) Secondly, the integration of A0 over νA gives 1 2NεA ∫ NεA −NεA A0dνA = 1 2NεA ∫ 1 −NεA dνA + 1 2NεA ∫ NεA 1 sign(φA − νA)dνA (36) or 1 2NεA ∫ NεA −NεA A0dνA = 1 2NεA {1 +NεA + ∫ φA 1 dνA − ∫ NεA φA dνA} = φA NεA . (37) Hence, in the evaluation of EA(A) we see that the product of σA√3θ(tA− 1 N2A )θ( 1 N2A + 4NA − tA) sin(αAtA) and φANεA enters the integration over tA and xA. As was agreed in the expectation value part of the evaluation of the model the tA integral 'goes first'. Copyright line will be provided by the publisher EPRBA COVARIANCE AND CLASSICAL PROBABILITY 9 We obtain the following expression I = σA√ 3 φA NεA ∫ 1 0 2tAε 2 ANANεA (t2AN 2 A + ε 2 A) 2 θ(tA − 1 N2A )θ( 1 N2A + 4 NA − tA) sin(αAtA)dtA. (38) This equation can be rewritten, noting εANA = 1, as I = σAφAεA√ 3 ∫ 1 N2 A + 4NA 1 N2 A 2tA (t2AN 2 A + ε 2 A) 2 sin(αAtA)dtA. (39) Therefore we employ the filter θ(tA − 1N2A )θ( 1 N2A + 4NA − tA) ∈ {0, 1} in the A1 to 'catch' one of the first few violent fluctuations. If, NA 0 the integration collapses to a single term approximation of calculating the area under a recangular triangle with base, dtA = 4NA . We have σAφAεA√ 3 ∫ 1N2 A + 4NA 1 N2 A 2tA (t2AN 2 A+ε 2 A) 2 sin(αAtA)dtA ≈ σAφA 2 √ 3 2/N5A (2/N4A) 2 1 N2A 4 NA sin(α 1 N2A ). (40) with, tA = 1/N2A in the integrand leading to a height of the triangle H = 1 2N 3 A 1 N2A sin(αA 1 N2A ) and B = dtA = 4 NA its base. The well-known formula for the triangular area is 12HB = 1 2 1 2NA 4 NA sin(αA 1 N2A ). Hence, when α = πN2A/2 it follows that I ≈ σAφA√ 3 . (41) The use of the area of a triangle is justified by the form of the integrand function in the interval [ 1 N2A , 1 N2A + 4 NA ].It is an approximation that will become better and better when NA is selected larger and larger. Because, we take NA 0 it follows that 1/NmA > 0, m = 1, 2, 3, 4. Thus, for increasing NA with increasing precision an approximation of the integral with a rectangular triangle with height, H , computed in tA = 1/N2A and a base, 4/NA can be employed. In order to justify the lower and upper bound of integration, the quantum of energy, expressed in Planck's constant (for the historical discovery of the quantum of energy in the blackbody radiation theory see e.g. [10]), h can be invoked. Physically, there is overwhelming evidence that energy is quantised and that the minimum positive amount of energy, E > 0, cannot be lower than Planck's constant. This may perhaps explain the minimum stepsize of ∆t = 4/N = h. The granulation of the integral comes from the θ filter in the measurement function. The following relation between N , ignoring the A or B index, and h is supposed to be N = 1/4h. In this respect 1/N2 can be interpreted as the smallest possible 'form' that allows measurement. The reader is reminded that the whole mathematical exercise is merely there to explain (correlation between) measurements of spins. Hence, restriction based on the finitude of energy in measurement may be invoked to allow equation 40 as the optimal form of (collapsed) integration. It also suggests a relation between the t variables and energy processes in the measurement. 8 Correlation From equations 28 and 41 it is then inevitable that EA(A) ≈ σA√ 3 [ √ 3 1 + (1/T 2A) +O(εA)] (42) This previous result can be rewritten in terms of O(1/T 2A) as EA(A) ≈ σA(1 +O(1/T 2A)) (43) Copyright line will be provided by the publisher 10 J.F. Geurdes: EPRBA A similar result obtains for EB(B), thereby keeping in mind that, B = −B0B1. Hence, EB(B) ≈ −σB(1 +O(1/T 2B)). (44) Assuming that εB is small positive of the same order as εA. Because both A as well as B project in the set {−1, 1}, and we already established that E(1) = 1, it follows that E(A2) = E(B2) = 1. The Gaussian integration is included. Moreover, because of the results in 43 and 44 we now can conclude from the symmetry of the Gaussian that E(A) = E(B) ≈ 0, thereby suppressing the 1/T 2A and 1/T 2B error terms. Finally, from equation26 and combining equations 43 and 44 again, suppressing the 1/T 2A and 1/T 2 B error terms and noting that E(AB) = Ey(EA(EB(AB))) = Ey(EA(A)EB(B)) ≈ −EyσAσB we get E(AB) ≈ − ∫ +∞ −∞ dy1 ∫ +∞ −∞ dy2 ∫ +∞ −∞ dy3 N(0, 1)(y) 3∑ k=1 3∑ m=1 akbmsign(yk)sign(ym) (45) Hence E(AB) ≈ − 3∑ k=1 3∑ m=1 akbmδk,m = − 3∑ k=1 akbk (46) which is the quantum correlation between the two measurement parameters. 9 Conclusion and discussion In the previous sections the quantum correlation −(a * b) was obtained from a classical model. As can be verified, the measurement functions project in {−1, 1}. The employed densities are all positive and each density integrates to unity. Furthermore, some of the mathematics was backed with numerical analysis to optimize the integration. The selected parameters agree with the specifications of a classical model and are, hence, allowed. It should be clear that the numerical exploration is only there to justify further elaboration of the integral. The collapse of the integral to the single term with stepsize ∆t = 4/N is backed by reference to the fundamental quantum of energy, h. In this sense, the integral over t is approximated with a sum with a stepsize that is minimally ∆t = 4/N . The θ filtering reduces this to a single term. This form of approximation is completely classically and the use of a rectangular triangle can justified by the shape of the integrand. The use of a finite stepsize does not violate classicality because approximations with a finite computer are considered valid (classical) approximations and not in need of a conceptual jump to a granular form of non-locality computing. Note also that with selection of the parameters, no limit is taken in the model itself. In addition, the mathematical operations leading to EA(A) ≈ σA√3 [ √ 3 1+(1/T 2A) + O(εA)] are elementary and tested many times in other theories. The problem of a possible quickly vanishing integral I for small ε is solved by employing θ filtering. For completeness, E(AB) = Ey(EA(EB(AB))) = Ey(EB(EA(AB))). Hence, because A and B are completely independent, the previous interchange of expectation values may be employed. This is also a sign of genuine locality. As can be verified, the densities plus measurement functions of the A and B wing are completely independent. The employed integration is in a definite order. Ordered integrations are allowed in Bell's theorem. If one wants to disallow them there has to be given solid reasons based on observations in physical nature without reference to non-locality of course. Finally it must be admitted that the physical nature of the local hidden parameters is unknown. The only differentiation between hidden variables at this moment is the sign of Gaussian variables in the particles themselves on the one hand and the sets related to the A and B wing on the other. In the paper only some initial steps were taken and further research will be necessary to establish a physical picture. However, it can be concluded that classical probability is not equivalent Copyright line will be provided by the publisher EPRBA COVARIANCE AND CLASSICAL PROBABILITY 11 to classical physics and Bell's manipulations on his expression of the correlation in terms of local hidden variables do not by necessity establish a theorem. We might observe from the definition of measurement functions that A ≤ 1. Now if EA(A) ≈ σA and σA as defined in equation (5) then with a properly behaving expectation operation we would have expected to see: EA(A) ≤ EA(1). However, when EA(A) ≈ σA this will be violated, because, EA(1) = 1. Note that the complete expectation E does behave as expected and, hence, the behavior of EA cannot be uncovered in statistical physics experiment. Because Born's probabilistic interpretation of the quantum theory also is in need of an ensemble of particles before anything can be said about the probability, the previous statement can be defensed. In addition, the most simple defense would be to state that the EA operation does not respect the sign inA ≤ 1 because it is not a statistical expectation that can be measured in physical reality. The model would not be able to obtain the quantum correlation from a classical probability model. Experimental tests to accomplish quantum behavior would then be largely trivial. The question is: can this curiosity of the EA operator be neutralized by calling it an error. Below we will try to demonstrate that the curious behavior of EA is neither an error nor a deviation from a classical probabilistic description. The demonstration will be done by employing a model from the theory of generalized functions. Let us inspect Hadamard's finite part function. For x 6= 0 we have Pf θ(x) x = d dx {θ(x) log |x|} . (47) Here, θ(x) = 1 when x > 0 and θ(x) = 0 when x ≤ 0. Let us employ x in the interval [−e, e]. Let us also note that Pf θ(x) x ≥ 0 for all real x, e is the base of Napier's log function ln. Let us define an Eh operator like Eh(φ(x)) = ∫ +e −e φ(x)Pf θ(x) x dx (48) with φ(x) an arbitrary 'well behaved' function. In this case we may write for Eh(1) the following expression Eh(1) = ∫ +e −e Pf θ(x) x dx = ∫ +e −e d dx {θ(x) log |x|} dx = [θ(x) log |x|]x=+ex=−e (49) Hence, Eh(1) = 1. Let us subsequently note that θ(1 − x)θ(1 + x) = 1 when −1 < x < 1 and zero elsewhere. Subsequently, θ(1 − x)θ(1 + x) ≥ xθ(1 − x)θ(1 + x), because for ∀ : x ∈ (−1, 1) we have 1 > x and ∀ : x /∈ (−1, 1) the inequality collapses to 0 = 0. With similar reasoning in mind as for EA one would expect Eh{θ(1− x)θ(1 + x)} ≥ Eh{xθ(1− x)θ(1 + x)}. If we then inspect the h-expectation of θ(1 − x)θ(1 + x), it is easy to see using equation (49) that Eh{θ(1− x)θ(1 + x)} = 0. For, xθ(1− x)θ(1 + x) we first note that Eh{xθ(1− x)θ(1 + x)} = ∫ +e −e xθ(1− x)θ(1 + x) d dx {θ(x) log |x|} dx (50) This then gives Eh{xθ(1− x)θ(1 + x)} = [xθ(x) log |x|]x=+1x=−1 − ∫ +1 −1 {θ(x) log |x|} dx (51) or Eh{xθ(1− x)θ(1 + x)} = − lim ε→0+ [x log |x|]x=1x=ε + lim ε→0+ ∫ +1 ε x d dx log |x|dx (52) Hence, Eh{xθ(1 − x)θ(1 + x)} = 1, despite, Eh{θ(1 − x)θ(1 + x)} = 0 and θ(1 − x)θ(1 + x) ≥ xθ(1 − x)θ(1 + x). Hence, Eh does not respect ≤ but is based on a valid density ρX(x) = Pf θ(x)x for x ∈ [−e, e]. The fact that ρX can play the role of a density will be explained below. Copyright line will be provided by the publisher 12 J.F. Geurdes: EPRBA If we associate a σ field F = {∅,Ω} with Ω = [−e, e], to the density, ρX(x) = Pf θ(x)x , ∀ : x ∈ Ω, then it can be verified that the measure, P (X) = 1, when X = Ω ∈ F and P (X) = 0 when X = ∅ ∈ F. This can formally be obtained from the computation in equation (49) and coincides with Eh(1) = 1 forX = Ω. It is true because, P (X) =def ∫ x∈(X∈F) d dx {θ(x) log |x|} dx (53) For completeness, the field F is closed under complement, under union and under intersection and is therefore a genuine algebra[11] hence, in this case, σ field. Therefore the measure P (X), X ∈ F is a classical probability measure The EA operator in the correlation model shows a similar behavior as Eh. There is nothing wrong with employing this in a statistical theory as long as the complete expectation, i.e. the E of the model does behave properly, i.e. A ≤ 1 then E(A) ≤ E(1). Note also that the use of illustrative arguments to a feature of a larger model is common practice in developing a theory. In conclusion, because the EA behavior cannot be excluded in experiments and it does not violate the 'classicality' of the density, an explanation based on local hidden variables has been found for results obtained in experiments such as Aspect's. Note that classical probability theory also has been employed to explain Hardy's paradox [12]. In addition it has been demonstrated that the mathematics that Bell used did not exclude the possibility of LHV models [13] References [1] J.S. Bell, Physics 1, (1964), 195-200. [2] A. Einstein, N. Rosen, and B. Podolsky, Phys. Rev. 47, (1935), 777-780. [3] J. A. Wheeler, and W. H. Zurek, Quantum theory and measurement (Princeton Univ Press, Princeton, 1983) 356369. [4] A. Aspect, J. Dalibard, and G. Roger, Phys. Rev. Lett. 49, (1982) 1804-1806. [5] J. F. Geurdes, Int. J. Theor. Phys., Grp. Theor. and Nonl. Optics 12, (2007), 215-223. [6] A. M. Mood, F. A. Graybill, and D. C. Boes, Introduction to the theory of Statistics, (Mac Graw-Hill, N.Y. 1974). [7] K. Hess, and W. Phillip, Proc. Nat. Acad. Sci. USA 98, (2001) 14224-14227. [8] K. De Raedt, K. Keimpema, H. De Raedt, K. Michielsen, and S. Miyashita, Euro. Phys. J. B 53, (2006) 139-142. [9] E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, (B. G. Teubner, Berlin,1909) 20-40. [10] J. Mehra, The historical development of quantum theory, (Springer-Verlag, New York, 1982). [11] J. Rosenthal, A first look at Rigorous probability theory, (World Scientific, Singapore, 2006). [12] J. F. Geurdes, paper submitted (2010). [13] J.F. Geurdes, Adv. Std. in Theor. Phys. 4(20), (2010) 945-949. Copyright line will be provided by the publisher