Quantum Field Theory Motivated Model for the Einstein, Podolsky, Rosen (EPR) Experiment

Canadian Journal of Physics

Journal: Canadian Journal of Physics

Manuscript ID cjp-2019-0537.R2 Manuscript Type:ForPoint Reviewof View Only Date Submitted by the 12-Mar-2020 Author:

Complete List of Authors: Schatten, Kenneth; a.i. Solutions, Inc., Physics Research Jacobs, V.J.; Naval Research Laboratory

Quantum Field Theory, Electron Dressing, EPR, Einstein, Podolsky and Keyword: Rosen Experiment, Probabilistic

Is the invited manuscript for consideration in a Special Not applicable (regular submission) Issue? :

Quantum Field Theory Motivated Model for the

Einstein, Podolsky, Rosen (EPR) Experiment

For ReviewKenneth H. Schatten* Only

NASA/GSFC

Greenbelt, MD 20771

*now at: ai-Solutions, Inc.

Lanham, MD 20706

and

Verne L. Jacobs

Center for Computational Materials Science, Code 6390,

Materials Science and Technology Division,

Naval Research Laboratory,

Washington, D. C. 20375-5345

1 © The Author(s) or their Institution(s) Canadian Journal of Physics Page 2 of 11

Abstract

Bell deduced the nature and form of “nonlocal hidden variables” that could yield the unusual pattern of responses in the paradoxical Bohm (B-EPR) version of the Einstein, Podolsky, Rosen (EPR) experiment. We consider that Quantum Field Theory (QFT)allows for the existence of Bell’s hidden variables in the “dressing” of free electrons that may explain the EPR experimental results. Specifically, the dressing of free electrons can convey random vector information. Motivated by this, we use a random vector paradigm (RVP) to explore how well a Monte Carlo computer program compares with the experimental B-EPR statistics. In this algorithm, the random vector provides a unique fingerprint that allows these charged leptons to bear unique responses when their spin is read by Stern-Gerlach detectors. The program’s numerical results compare well with Bell’s experimental summary. This work offers the opportunity to consider that virtual particles play a role in understanding the origins of QM’s unique entanglement andFor probabilistic Review behaviors. Only

1. Introduction

Einstein, Podolsky and Rosen [1] (EPR) considered that non-commuting quantities (e.g. momentum and position) could be simultaneously measured by observing these properties on two particles simultaneously released from a common source. This suggested to these authors that both properties were based in local reality. The current paper focuses on David Bohm’s [2] (B-EPR) version of the EPR experiment, that was more amenable to experimental testing. In the B-EPR experiment, two spin ½ particles, an electron and a positron, are formed from an initial zero spin state in this generic EPR experiment. To conserve spin, each particle must provide a deterministic opposite spin from the other when the Stern-Gerlach(SG) detectors have a collinear orientation. Thus, if one measures spin UP (+), the other measures spin DOWN (-), and for any other opposite orientation, e.g., RIGHT, LEFT, etc. Non-collinear orientations, however, reveal differing stochastic behaviors. For example, when the detectors are oriented orthogonally, the particles’ spins are randomly distributed: half correlated (+ + or - -) and half anti-correlated (+ - or - +). Mixed statistics of differing amounts occur for other relative orientation angles. How deterministic and probabilistic properties can both exist within these charged particles is an integral component of what has been called the “EPR paradox.”

Bell [3] was the first to deduce a commonality to these puzzling experiments. From his deep insight and thoughtful interpretation of the B-EPR experiments, he showed that “non- local hidden variables” could be required to provide the unusual Stern-Gerlach detector responses. Yet, the subject still remains shrouded in mystery because his hidden variables were counter-intuitive, requiring what was called “non-local,” “action at a distance,” and even “spooky” behavior, seemingly violating relativity. Further important steps followed, one by CHSH [4], who provided a more complete account of the degree to which non-

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locality needed to be involved with generalized EPR experiments. Even now, in the 21st century, we have not really answered the question posed by EPR, as to whether quantum mechanics is “complete.” This phrasing essentially indicates that one needs to have a mechanism that determines where each particle goes. Yet despite a probabilistic view, Quantum Field Theory [5] (QFT) has shown itself to be the best physical theory we have to date, calculating atomic properties with a precision from 12-14 significant digits.

To set the stage for how we employ QFT to allow the electrons and positrons (e&p) used in the EPR experiment, we need to examine their QFT structure. Eq.(1) shows the difference between the bare electron and the QFT free electron,[5] the latter being a “dressed” electron, after Lindgren.[6] This equation indicates that a free electron (solid line at left) can be represented as the sum of a bare electron, dashed line, plus its dressing, consisting of the sum of all possible virtual photons, etc., that the electron can withdraw from the vacuum, corresponding to the remainderFor Reviewof the equation. Each Only sub-graph in Eq. (1) represents an integral equation in QFT.

Previously, we developed a random vector model [7] following Bell’s methodology that provided the QM statistics. This was a mathematical construct that showing that Bell’s methodology worked, however, these electrons lacked any physical attribute beyond the tightly bound electron containing 3 sets of 2x2 Pauli spin-matrices that appeared insufficient to allow sufficient hidden variables to be encoded onto these electrons, and result in the Bell statistics. A richer source of encoded information is needed to provide the B-EPR experimental results, as required by Bell’s non-local statistics. More recently, Tommasini [8] argued for a statistical interpretation of Quantum Field Theory (QFT) that would support reconciliation of locality and causality. On the subject of the statistical interpretation of the electrons used in the B-EPR experiment, we should keep in mind that e&p belong to the class of particles called leptons and also are Fermions, meaning they obey Fermi-Dirac statistics, namely unlike Bosons, no two can exist in the same quantum state.

In this paper, we develop a QFT paradigm wherein free electron dressings contain virtual particles and examine their ability, within the EPR experiment, to provide the Bell statistics. Virtual photons play an important role in physics; for example their exchange between an electron and the nucleus is the force that holds atoms together. The use of QFT here is solely limited to providing motivation for a random vector paradigm (RVP) to free electrons

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and positrons (e&p). This random vector paradigm simply describes free e&p to contain virtual particles in their dressings, represented with vector properties. This RVP can serve as Bell’s hidden variables in the B-EPR experiment. This shall enable the development of a Monte Carlo computer program for testing the B-EPR statistics with. In effect, the virtual particles, most commonly virtual photons here, are embedded onto free electrons. They may then be viewed as carriers of information that allow these leptons to have unique, deterministic responses when viewed by the Stern-Gerlach Detectors (SGDs) within these B- EPR experiments. Each particle’s “deterministic response” would be coupled to their mirror twin particle taking part at a distant location of the B-EPR experiment.

An electron’s dressing in QFT is needed by free e&p, for without this, these charged particles would radiate energy into space. The dressing shields the electron by inhibiting it from radiating energy. Without this, the electron might not have the energy to escape its bound state. Rather, itFor would lose Review energy by Bremsstrahlung Only radiation into space, and hence would be unable to transition from its bound state. It is important to point out that, owing to conservation laws and the requirement of zero radiative emission from a free electron, whatever conserved properties the dressing retains (e.g. spin, energy,, etc.), the free e&p also possesses. Hence the particular properties gathered from the vacuum acquired as the bound leptons becomes free, remain attached to these free particles. This remains so, generally, until the free particles interact with matter.

This paper offers a more detailed account of free electron properties within the B-EPR experiment than is typically considered. Let us briefly describe how Stern-Gerlach Detectors (SGDs) work. Electrons, for example, are attached to neutral silver atoms to create a beam of neutral particles that pass through the SGD. The SGDs have a strong magnetic field gradient in a direction perpendicular to the beam and detector axis. This field gradient provides a force in the direction of the field gradient, dependent upon the electron’s spin in that direction. This separates the particles along the field gradient direction by amounts dependent upon the spin component or components in that direction. Since electrons have spin 1/2 the beam splits up into two separate beams, spin UP and spin DOWN. In our RVP, we employ a single random vector (RV) to model each electron’s dressing. For this paper, our usage allows free electrons a greater degree of determinism than the 3 sets of 2x2 Pauli spin matrices provides for tightly bound electrons. In the B-EPR experiment, free electrons have their spin gauged by Stern-Gerlach(SG) detectors [9].

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For Review Only

Figure 1. In the B-EPR experiment, a positron, p, and an electron, e, emanate from a Zero- Spin State located at X. The figure shows the particles’ inherent structure, as having a bare core (black core), surrounding which is a “dressing” (clear shell) containing one or more virtual particles (commonly a virtual photon). The two charged leptons: a positron, particle 1, and an electron, particle 2, exit their parturition site at X. Subsequently, they travel towards the Stern-Gerlach Detectors (see Feynman et al. [9]) SGD1 and SGD2. These need

not lie in a plane and are oriented at a relative angle,2 , with respect to each other. In the Random Vector Paradigm (RVP), we suggest the dressing consists of an equivalent equal and     opposite 3D random vectors that provides spin information: s1  sE  sP  s2 , creating an “entanglement” dependency.

The free electron and positron (e&p) structures contain virtual particles in their dressings. One or more virtual particles, typically photons, are drawn from the vacuum to cloak these particles as they leave any strongly bound state. A process like this has been observed, known as zitterbewegung. For the B-EPR experiment, the positron and electron leave their common zero-spin state, X, shown in Figure 1. For the purposes of the EPR experiment, when attached to a neutral atom, these particles behave similarly, allowing Stern-Gerlach Detectors (SGD) to sample their spin attributes. The only pre-condition set for the particles’ spins is that the two particles contain oppositely oriented spin vectors, owing to the conservation rule associated with their origin from a zero-spin state, namely:

  sE  sP . (2)

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We shall refer to this as the “B-EPR spin entanglement equation,” as this equation can result in quantum “entanglement.” At this point QFT’s role ends, having supplied the dressed electron paradigm. From this, we hypothesize that the sum of each particle’s dressing can be expressed as a spin vector, which we refer to as the random vector paradigm (RVP), thereby allowing these leptons to possess 3D random vector spin information, which contains a significantly greater amount of information than is available in tightly bound electrons. The B-EPR spin entanglement equation is captured geometrically in Figure1.

The RVP allows these charged leptons to transport hidden variables, such as exist, for example in a photon’s Poynting vector, although that refers to energy transport, but they are related. This dressing plays a central role for electrons when they escape tight atomic bounds as these fieldsFor inhibit radiative Review losses. Following Only the events of the two particle pairs in the B-EPR experiment, they remain “entangled” until either interacts with distant matter. Familiar examples of dis-entanglement are when an electron is captured by an atom or when its spin state is sampled by a Stern-Gerlach Detector. The 3D vector spin property of the free electron dressing is used in the numerical model we now discuss. In Section II, we discuss Bell’s analysis, wherein he brilliantly managed to summarize the inscrutable experimental findings; in Section III, we outline our Monte Carlo computer algorithm that provides a test of our “random vector paradigm” (RVP) to ascertain the extent to which this model can mimic the B-EPR experimental results; and in Section IV, we end with a discussion.

The present work cannot answer the EPR original question as to the completeness of Quantum Mechanics. It has limited goals, simply two main tasks: 1) development of a “dressed electron” model that allows random vectors to exist on free electrons, and 2) a numerical test of this mo. The next section discusses Bell’s Analysis of the B-EPR experiment, and following this, we compare Bell’s analysis with our numerical Monte Carlo Model. 2. Bell’s Analysis of the B-EPR Experimental Results

We briefly outline Bell’s analysis. He developed a “non-local, hidden variables model” that reproduced the statistical behaviors observed during the B-EPR experiments. Bell’s method utilized two quantities needed to calculate the Stern-Gerlach (SG) detectors’  responses to an electron’s spin state. The first is a 3-D random unit vector,sP , representing  the positron’s spin vector in Figure 1. The response of detector 1, DSG1 , to the positron  spin, is determined locally. Namely it is taken to be the dot (inner) product of sP with the SG1 detector’sorientation. –The responses are often expressed in a variety of ways by other authors: UP, DOWN, LEFT, RIGHT, +, -, dependent upon the authors’ predilections. The

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   s  s other detector,DSG2 , samples the second particle’s spin with E P from the B-EPR spin entanglement equation.

One of the wonderful contributions that Bell [3] provided was his recognition of the dependence that exists between the two detectors’ responses. Namely, he discovered that the relative angle between the two detectors plays an important role in their relative response! We explore this later by emphasizing Bell’s form of the correlations with respect  to detector settings. Bell placed a random “correlation vector,” c , in his model with a form which accurately mimicked the EPR correlation observations. This vector determines the  temporal fraction in which the second detector responds with the same response as DSG1 , relative to the opposite response, thereby utilizing this inter-dependence of the two SGDs directly. We discuss later how Bell’s correlation form performs with our current model. For now, we just accept Bell’sFor form, Reviewand yes, it is delightfully Only puzzling!

Figure 2. This geometry shown helps describe “Bell’s [3] model” of correlations observed. He uses an angular coordinate system based upon the two Stern-Gerlach detectors (SG1

and SG2), which are at an angle  2 from each other as shown. Based upon Bell’s formula  using a random s vector that relates to how Stern-Gerlach detector 1, SG1, will respond to the particle it sees, the positron, P, and the second detector associated “correlation vector”,   c , the latter obtains the “non-local” correlations. First, if the s vector is directed more   towards the Stern-Gerlach1 detector orientation (located towardszP the direction,DSG1 )   than not, the DSG1 detector will yield a + response. The correlation vector c is non-local; it relates to the probability that the two detectors will respond in a correlated manner, namely “same” (+ + or - -) vs. anti-correlated, namely “opposite” (+ - or - +) manner. For example, when ~0o , the two detectors are aligned, and one can be sure to obtain an

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anti-correlation as SG2’s response, as the whole sphere becomes negative. Hence the  correlation vector,c , must fall into the negative “zone.”

3. Monte Carlo Computer Analysis

Now we describe our computer experiment that attempts to mimic the generic B- EPR physical experiment, using the previous assumptions that allow the Stern-Gerlach Detectors (SGD1 and SGD2) to read the vector spin information embedded within the particles’ dressings.

To compare our computer model with Bell’s summation of the experimental data, the results are placed in the same geometrical format. The difference being, that now we have random vectors, Forusing equation Review (2), the “B-EPR Only spin entanglement equation,” which now serves to guide the particles’ responses, albeit dependent upon the relative orientation of the two SGDs.

The computer code is developed along the familiar lines of traditional Monte Carlo programs. In this case, we mimic the experiment in the following manner: First pairs of numerical leptons are created with random 3D spatial vectors, but balanced via equation (2). These are then passed through the two Stern-Gerlach Detectors (SGD1 and SGD2) shown in Figure 1. Different detector settings are utilized for each separate run.These consist of 37 different detector orientations, with relative angle,2 , around the axes of travel of the positrons and electrons allowing the range to vary from 0o to 180o in5o o increments. Values of  2 beyond 180 involve the same geometry as those at , thus they have not been duplicated. For each angle, a correlation vector, cˆ , is calculated. This is allowed to be utilized, as it is solely dependent upon the detectors’ relative orientation. One can think of this equivalency: if one were to tilt a spectrograph’s grating relative to a light beam, that light beam would “see” a different relative spacing, much as X- ray diffraction, in a Bragg experiment, is affected by the crystal’s orientation, relative to the X-ray beam.

Within each run of the program it generates positron-electron zero-spin-state pairs using 3-D random Euler angle representations of the particle pairs’ spin vectors. A  subroutine in the program calculates 500 random spin vector pairs. The spin vectors,sP and  sE , are calculated. The numerical algorithm provides a spin up (+) or spin down (-) response for all angles. The responses are binned, recorded, and categorized (++, +- , -+ and - - for the two detector readings), the first and fourth (++ and - -) being positively correlated (CORR), and second and third (+- , -+) being anti-correlated (ACORR). Hence, the statistics of the program are displayed in Figure 3.

  The present paper allows the sP andsE vectors to be embedded within the dressing of the two free charged leptons (the positron and electron), each carrying identical, but   mirror-image information, owing to the B-EPR spin entanglement equation: sE  sP . This,  and its relationship to the correlation vector,c , are both needed since both detectors need to respond individually and jointly in a concerted fashion to explain the intertwined B-EPR statistics. Added to this is the relationship of Bell expressing the need for “non-local hidden variables.” Indeed, the vector information in the present model is “hidden” in the dressing of free electrons, but the question of the locality remains. When the two particles are measured, they indeed are non-local to each other, however, the two originate from a zero spin state, X, and they both are local relative to X, thus supporting local reality.

This illustrates Bell’sFor wonderful Review model that manages Only to illustrate how QM weaves together a rather inscrutable tapestry of deterministic and stochastic behaviors within a single experiment, with the only varying parameter being an instrumental setting! Figure 3 shows Bell’s formulae for functional forms of sine and cosine formulae.

Figure 3. Numerical Correlations and Anti-Correlations in our B-EPR experiment (shown by the diamond and triangle points), compared with Bell’s formulae (the solid and dashed curves).The 500 value means all runs ended up with both SG detectors responding identically, when + correlated (+ + or - -) and when fully anti-correlated, they respond oppositely (+ - or - +). The 90o orthogonal angle has approximately 50-50% mix of both +

and - correlation responses for BOTH detectors. At an angle of 0o anti-correlations dominate completely; at180o positive correlations totally dominate, both with “determinism,” whereas at 90o , stochastic behavior ensues: a puzzling tapestry of responses, however, these are consistent with angular momentum conservation.

4. Discussion.

The nature of the free electron dressing appears to be a fertile ground for investigation in both theoretical and experimental physics, as well as in astronomy. The nature of virtual particles and photons play a significant role in the nature of determinism in physics as well as dark energy and massFor in cosmology. Review In physics, virtual Only photons are considered to supply the force that holds electrons to the nucleus in atoms. Virtual photons also supply the force that holds matter together, called the “Casimir Effect.” It may also be of interest in quantum computers, as electrons clearly play an important role in their development, particularly the extent to which free electrons can transport information would be dependent upon the information carrying capacity of free electrons. This paper describes a difference between the information carrying capacity of free and loosely bound electrons from that of strongly or tightly bound electrons. Although not an original goal of the present model, its success may bear fruit by offering the opportunity to examine the origin of QM probabilistic behaviors and as well as quantum entanglement.

Acknowledgments

KHS acknowledges Abner Shimony for sharing his enlightened knowledge and deep understanding of the essential aspects involved in many quantum puzzles, in particular the fascinating EPR experiment where determinism and stochastic behaviors are seemingly, inexplicably intertwined. KHS also appreciates discussions with his colleagues: Sabatino Sofia and Hans Mayr. VLJ acknowledges the financial support of the Office of Naval Research through the Basic Research Program at the Naval Research Laboratory. The authors are greatly indebted to an anonymous referee who greatly improved the paper.

References

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2. D. Bohm, Quantum Theory, (Prentice Hall, Englewood Cliffs, NJ, 1951).

3. J.S. Bell, On the Einstein Podolsky Rosen Paradox, Physics, 1 (3): 195–200,(1964). doi:10.1103, reproduced as Ch. 2 of J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press(1987).

4. J.F. Clauser; M.A. Horne; A. Shimony; R.A. Holt (CHSH) Proposed experiment to test local hidden-variable theories", Phys. Rev. L., 23 (15):880C,(1969).

5. M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Perseus Books, 175-210, (1995).For Review Only 6. I. Lindgren, et al. Analysis of the electron self-energy for tightly bound electron, Physical Review A, 58:1001-1015, (1998). 7. K. H. Schatten, Hidden Variable Model for the Bohm Einstein-Podolsky-Rosen experiment, Phys. Rev. A, 48, 1, 103-4, (1993). 8. D. Tommasini, Reality, measurement and locality in Quantum Field Theory, arXiv:hep- th/0205105v3. 9. R. F. Feynman, R. B. Leighton, and M. Sands, Feynman Lecture Notes, II 35.1-6, III 6.1-6.14. Addison-Wesley, Reading, MA,(1965).