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Speed dependent polarization correlations in QED and entanglement

Exact computations of polarizations correlations probabilities are carried out in QED, to the leading order, for initially polarized as well as unpolarized particles. Quite generally they are found to be speed dependent and are in clear violation of Bells inequality of Local Hidden Variables (LHV) theories. This dynamical analysis shows how speed dependent entangled states are generated. These computations, based on QED are expected to lead to new experiments on polarization correlations monitoring speed in the light of Bells theorem. The paper provides a full QED treatment of the dynamics of entanglement.


INTRODUCTION
We carry out exact computations of joint probabilities of particle polarizations correlations in QED, to the leading order, for initially polarized and unpolarized particles. The interesting lesson we have learnt from such studies is that the mere fact that particles emerging from a process have non-zero speeds to reach detectors implies, in general, that their polarizations correlations probabilities depend on speed [1]. The present extended, and needless to say, dynamical analysis shows that this is true, in general. This is unlike formal arguments based simply on combining angular momenta. As a byproduct of this work, we obtain clear violations with Bell's inequality (cf. [2]- [4]) of LHV theories. We will also see how QED generates speed dependent entangled states.
Several experiments have been perfomed in recent years (cf. [4]- [8]) on particles' polarizations correlations. And, it is expected that the novel properties recorded here by explicit calculations following directly from field theory, which is based on the principle of relativity and quantum theory, will lead to new experiments on polarization correlations monitoring speed in the light of Bell's Theorem. We hope that theses computations will be also useful in such areas of physics as quantum teleportation and quantum information in general.
The relevant quantity of interest here in testing Bell's inequality of LHV [2] theories is, in a standard notation, as is computed from QED. Here a 1 , a 2 (a ′ 1 , a ′ 2 ) specify directions along which the polarizations of two particles are measured, with p 12 (a 1 , a 2 )/p 12 (∞, ∞) denoting the joint probability, and p 12 (a 1 , ∞)/p 12 (∞, ∞), p 12 (∞, a 2 )/p 12 (∞, ∞) denoting the probabilities when the polarization of only one of the particles is measured. [p 12 (∞, ∞) is normalization factor.] The corresponding probabilities as computed from QED will be denoted by with χ 1 , χ 2 denoting angles the polarization vectors make with certain axes spelled out in the bulk of the paper. To show that QED is in violation with Bell's inequality of LHV, it is sufficient to find one set of angles χ 1 , χ 2 , χ ′ 1 , χ ′ 2 and speed β, such that S, as computed in QED, leads to a value of S with S > 0 or S < −1. In this work, it is implicitly assumed that the polarization parameters in the particle states are directly observable and may be used for Bell-type measurements as discussed.
The need of a relativistic treatment based on explicit quantum field dynamical calculations in testing Bell-like inequalities is critically important. An intriguing and very recent reference [9], which appeared after the submission of our paper for publication, discusses the role of relativity in quantum information, in general, and traces the historical development of its role, and most importantly, in the light of our present investigations, emphasizes the need of quantum field theory as necessary for a consistent description of interactions. Most earlier analyses dealing with relativistic aspects, relevant to information theory and Bell-like tests are kinematical of nature or deal with basic general properties of local operators associated with bounded regions of spacetime setting limits on measurements and localizability of quantum systems. These probabilities are well documented in some of the recent monographs [10]- [12] on the subject. Notable important other recent references on such general aspects which are, however, non-dynamical of nature are [13]- [17], and a paper by Czachor [18] indicating how a possible decrease in violation of Bell's inequalities may occur. In the present work, we are interested in dynamical aspects and related uniquely determined probabilities (intensities) of correlations based on QED, as a fully relativistic quantum field theory (i.e., encompassing quantum theory and relativity) that meet the verdict of experiments. QED is a non-speculative theory and as Feynman [19] puts it, it is the most precise theory we have in fundamental physics. The closest investigation to our own is that of Ref. [20], a reference we encountered after the submission of our paper for publication, which considers spin-spin interactions, in a QED setting, for non-relativistic electrons and, unfortunately, does not compute their polarizations correlations which are much relevant experimentally. In the present paper, exact fully relativistic QED, computations, to the leading order, of polarizations correlations are explicitly carried out for initially polarized and unpolarized particles. The importance of also considering unpolarized spin stems from the fact that we discover the existence of non-trivial correlations, in the outcome of the processes, even for such mixed states (since one averages over spin) and not only for pure states arising from polarized spins, leading, in particular, in both cases to speed dependent probabilities. The main results of our paper are given in (2.20), (2.22), (2.23), (2.41)-(2.43), (3.10), (3.12)- (3.19). All of these probabilities lead to a violation of Bell's inequality of LHV theories. As the computations are based on the fully relativistic QED, it is of some urgency that relevant experiments are carried out by monitoring speed.

POLARIZATIONS CORRELATIONS: INITIALLY POLARIZED PARTICLES
We consider the process e − e − → e − e − , in the c.m., with initially polarized electrons with one spin up, along the z-axis, and one spin down. With p 1 = γmβ(0, 1, 0) = −p 2 denoting the momenta of the initial electrons, γ = 1/ 1 − β 2 , we consider momenta of the emerging electrons with where θ is measured from the z-axis.
For the four-spinors of the initial electrons, we have (p 0 = γm) and for the final ones where the two-spinors ξ 1 , ξ 2 will be specified later.
The expression for the amplitude of the process is well known (cf. [21]) The following matrix elements are needed and are readily calculated For θ = 0, (see FIG. 1), we obtain from (2.7)-(2.15) generating the speed dependent (normalized) entangled state of the emerging electrons ρ is defined in (2.4), and the angles are measured relative to the x-axis (see FIG. 1).
The joint probability of the electrons polarizations correlations is then given by [For β → 0, one obtains a rather familiar expression P [χ 1 , If only one of the spins is measured, say, corresponding to χ 1 , we then have to form the state and similarly The probability P [χ 1 , −] may be equivalently obtained by summing P [χ 1 , χ 2 ] over the two angles for any arbitrarily chosen fixed χ 2 , i.e., as is easily checked, and similarly for P [−, For all 0 β 1, angles χ 1 , χ 2 , χ ′ 1 , χ ′ 2 are readily found leading to a violation of Bell's inequality of LHV theories. For example, for β = 0.3, 79 violating the inequality from below.

POLARIZATIONS CORRELATIONS: INITIALLY UNPOLARIZED PARTICLES
For the process e − e − → e − e − , in the c.m., with initially unpolarized spins, with momenta p 1 = γmβ(0, 1, 0) = −p 2 , we take for the final electrons and for the four-spinors A straightforward but tedious computation of the corresponding probability of occurrence with initially unpolarized electrons, (2.7) leads to which after simplification and of collecting terms reduces to where we have used the expressions for the spinors in (3.2), (3.3).
The probability of photon polarizations correlations in e + e − → 2γ with initially unpolarized e + , e − , has been given in [1] to be and a clear violation of Bell's inequality of LHV theories was obtained for all 0 β 0.2. Again, for larger values of β, alone, one cannot discriminate between LHV theories and quantum theory for this process. A violation of Bell's inequality for at least some β values, as seen, however, automatically occurs violating LHV theories.
For completeness, we mention that for the annihilation of the spin 0 pair into 2γ the following probabilities are similarly worked out: and violates Bell's inequality of LHV theories for all 0 β 1.

CONCLUSION
We have seen by explicit dynamical computations based on QED, that the polarizations correlations probabilities of particles emerging in processes depend on speed, for initially polarized as well as unpolarized particles, in general. We have also seen how QED leads directly to speed dependent entangled states. For processes with initially polarized particles (as well as for spin 0 pairs annihilation into 2γ), a clear violation of Bell's inequality of LHV theories was obtained for all speeds. This clear violation was also true for several speeds for processes with initially unpolarized particles, but the tests are more sensitive on the speed for such processes. The main results of the paper are given in (2.20), (2.22), (2.23), (2.41)-(2.43), (3.10), (3.12)- (3.19). We feel that it is a matter of some urgency that the relevant experiments are carried out by monitoring speed.