The Ontic Probability Interpretation of Quantum Theory Part II: Einstein's Incompleteness/Nonlocality Dilemma Felix Alba-Juez Felix Alba-Juez, Publisher Saint George, Utah (USA) February 5, 2020 https://felixalbajuez.com ABSTRACT After pinpointing a conceptual confusion (TCC), a Reality preconception (TRP1), and a fallacious dichotomy (TFD)1, the famous EPR/EPRB2,3,4,5,6 argument for correlated 'particles' is studied in the light of the Ontic Probability Interpretation (TOPI) of Quantum Theory (QT). Another Reality preconception (TRP2) is identified, showing that EPR used and ignored QT predictions in a single paralogism. Employing TFD and TRP2, EPR unveiled a contradiction veiled in its premises. By removing nonlocality from QT's Ontology1 by fiat, EPR preordained its incompleteness. The Petitio Principii fallacy was at work from the outset. Einstein surmised the solution to his incompleteness/nonlocality dilemma in 19497, but never abandoned his philosophical stance8. It is concluded that there are no definitions of Reality: we have to accept that Reality may not conform to our prejudices and, if an otherwise successful theory predicts what we do not believe in, no gedankenexperiment will help because our biases may slither through. Only actual experiments could assist in solving Einstein's dilemma, as proven in the last 50 years. Notwithstanding, EPR is one of the most influential papers in history and has immensely sparked both conceptual and technological progress. Future articles will further explain TOPI, demonstrating its soundness and potential for nurturing theoretical advance. List of Acronyms (Part I and Part II) QT Quantum Theory EPR The Einstein/Podolsky/Rosen Paper TOPI The Ontic Probability Interpretation PD Probability Distribution PI Physical Interaction GI Gauge Interaction TM True Measurement TRC The Reality Criterion TCC The Conceptual Confusion SD Standard Deviation of a PD TRP1 The Reality Preconception 1 TFD The Fallacious Dichotomy EPRB EPR-Bohm Gedankenexperiment TRP2 The Reality Preconception 2 1. Introduction In Part I we saw that, because -in most casesa 'measurement' (GI) disturbs the state and TRC was mute regarding the property's 'reality', EPR needed to conceive how to predict the result of any 'measurement' "with certainty" and "without in any way disturbing the system". In TOPI lingo1: how to make a GI work as a TM. Directly interacting with the local one of two entangled space-like-separated systems, EPR claimed QT predicts the property value for the remote one "with certainty" and, per the Principle of Locality, "without being disturbed". EPR further argued that the remote property must have existed all along, not somehow created by the 'measurement', as many QT pioneers had avowed would have been the case had the remote system been directly 'measured'. Ergo, it had to be an "element of reality".2 2 2. The Iconic EPR Gedankenexperiment EPR describes its iconic thought experiment: EPR8: For this purpose let us suppose that we have two systems, I and II, which we permit to interact from the time t=0 to t=T, after which time we suppose that there is no longer any interaction between the two parts. We suppose further that the states of the two systems before t=0 were known. We can then calculate with the help of Schrödinger's equation the state of the combined system I+II at any subsequent time; in particular, for any t>T. Let us designate the corresponding wave function by ψ. We cannot, however, calculate the state in which either one of the two systems is left after the interaction. This, according to quantum mechanics, can be done only with the help of further measurements, by a process known as the reduction of the wave packet. We will refer to subsystems I and II as SSI and SSII. Knowing their states prior to interaction, Schrodinger's Equation predicts the future composite state ψ but not their individual states. This is because the equation governs the temporal evolution of the composite system in its state-space (a tensor product), not of the subsystems in their own state-spaces9. However, EPR believes the subsystems' states must have definite values that can be determined "only with the help of further measurements". EPR imaginarily 'measures' momentum PI or position QI of SSI. To predict the 'measurement' of PI, the wavefunction of the composite system is expanded in terms of the momentum eigenfunctions of SSI. But because SSI is entangled with SSII due to momentum conservation, the coefficients of the expansion are functions of the momentum eigenfunctions of SSII. Thus, after the 'measurement', SSI adopts one of its momentum eigenstates and SSII (despite being afar) adopts one of its own momentum eigenstates. Mutatis mutandis for 'measuring' QI. Obviously, the system imagined by EPR is not the dull aggregate of SSI and SSII: QT predicts that, even if they are space-like separated, a 'measurement' in SSI not only may affect its state but may also affect the state of SSII. The 'spooky action at a distance' between two 'particles', so despised by Einstein, was obviously at work. A decade before, and specifically at the Solvay 1927, he had denounced the 'one-particle nonlocality' as a sign of QT's incompleteness.10 2.1. EPRB Gedankenexperiment vis à vis TOPI In 1951, David Bohm reconceived the EPR gedankenexperiment via two spin-1⁄2 qubits in the singlet state5,6. There is a homology between EPR and Bohm (EPRB) setups. The philosophical problem is easier to clutch for the latter, so we will reinterpret EPR's assertions per EPRB. The homology is set with two bijections: a) P ↔ S ?? ⁄ (Spin along ??); and b) Q ↔ S ??′ ⁄ (Spin along ??′). For the spin operators to be noncommutative, ?? and ??′ are to be not anti-collinear. The homology is valid for both sites: SSI↔Qubit-I and SSII↔Qubit-II. Figure 1 drafts EPRB setup. Created in the joint singlet state, the two qubits travel in spacetime and, if the system remains isolated as a whole, its state is unaffected by their spatial separation. The singlet state is: |singlet⟩ = √2 2⁄ sin(θ 2⁄ ){|sI1sII1⟩ − |sI2sII2⟩} + √2 2⁄ cos(θ 2⁄ ){|sI1sII2⟩ − |sI2sII1⟩} (1) θ = 0 ⇒ |singlet⟩ = (√2 2⁄ )|sI1sII2⟩ − (√2 2⁄ )|sI2sII1⟩ Where θ is the angle between the fields of two Stern-Gerlach magnets for potential GIs with the qubits. The other four ket-symbols stand for the four eigenvectors spanning the composite StateSpace. Neither space nor time appears in the state, so any physical interaction between the qubits 3 is not of the dynamic type. Despite depending on θ, the real state is unique; it is the basis (eigenvectors) used to express the singlet state that changes for each potential value of θ. From Equation 1 (top), the probability for Qubit-I or Qubit-II to transition (upon a GI) to any one of their two eigenstates is 50% irrespective of θ, i.e. the local spins are perfectly random for any magnet orientation. Yet, their product shows rich probabilistic patterns, e.g. deterministic for θ=0, and perfectly random for θ=90o.10 Thus, prior to any GI on a site, the qubit state is undefined. Per TOPI, it is false to say (as EPR8 does) that SSI and SSII states can only be determined with "the help of further measurements". The fact that a single GI delivers a single value does not mean such a value was predetermined; and, even if it was, it may not have been the same because a GI may or may not be a TM.1 Figure 1: Two Spin-1/2 Qubits in the Singlet State – A GI with Qubit-I occurs First Figure 1 assumes a GI occurs first to Qubit-I depicting the two cases: (a) GI-I delivers a spin aligned to the magnetic field (|sI1⟩); or (b) GI-I delivers a spin anti-collinear to the field (|sI2⟩). All arrows at the field's entrance are dotted conveying that the pre-GI Qubit-I state is undefined. However, as soon as GI-I occurs, Qubit-II detangles from Qubit-I and (from Equation 1) its probability to transition (upon any GI-II) to one of its eigenstates (|sII1⟩ or |sII2⟩) is not 50% anymore but given by: Pr(|sII1⟩ |sI1⟩⁄ ) = Pr(|sII2⟩ |sI2⟩⁄ ) = sin 2(θ 2⁄ ) (2) 4 Pr(|sII2⟩ |sI1⟩⁄ ) = Pr(|sII1⟩ |sI2⟩⁄ ) = cos 2(θ 2⁄ ) But a glimpse at Equations 2 shows they correspond to the behavior of an isolated qubit when the angle between its direction and the magnetic field is θ + π. This proves that Qubit-II, while detangling, adopted a spin anti-collinear to that of Qubit-I, namely |sII2⟩ for (a) and |sII1⟩ for (b). And this state adoption by Qubit-II occurred whether it will ever undergo a GI or not, as indicated by the only solid arrow before reaching site II. It also follows that the angle between the two spin states adopted by Qubit-II upon two GIs on Qubit-I is the same as the angle formed by the respective magnetic fields on site I. Hence, if -remaining isolated after detanglingQubit-II ever interacted with a field collinear to the one on site I (θ = 0), its pre-GI state for cases (a) and (b) would be an eigenstate, and its post-GI state would be the same: GI-II would be a TM.1 It is crucial to realize that Qubit-II is undisturbed by GI-II not because both qubits are space-like separated but because its pre-GI state is an eigenstate whenever Magnet-II is aligned to the direction Magnet-I had when GI-I occurred. The 'spooky' effect occurs (per QT/TOPI) as a result of GI-I happening; GI-II may or may not ever occur. Going back to the homologous EPR setup, θ = 0 means that the same property (P or Q) is 'measured' on sites I and II, with their homologous spins on each site being not anti-collinear and forming the same angle in both sites. With TCC proviso1, this is what EPR needed: pre-GI-II and post-GI-II results for P or Q agree and are predicted "with certainty" from the respective results of GI-I. The GIs on site I are not TMs, so TRC does not apply; the GIs on site II are TMs, so TRC applies1. It is also vital to understand that the state adopted by SSII when 'measuring' P on site I is different from that adopted when 'measuring' Q on site I. 3. The Reality Preconception 2 Firmly believing in the Principle of Locality, EPR affirms: EPR9: We see therefore that, as a consequence of two different measurements performed upon the first system, the second system may be left in states with two different wave functions. On the other hand, since at the time of measurement the two systems no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system. This is, of course, merely a statement of what is meant by the absence of an interaction between the two systems. Thus, it is possible to assign two different wave functions to the same reality (the second system after the interaction with the first). Intriguingly, instead of relying on Relativity Theory (RT), EPR enforces locality by the very assumption of 'no interaction'. Per RT, the only way for two space-like events to be correlated is through a common cause in their past11. This is valid, of course, if RT itself is complete, i.e. if every possible "Element of Reality" has been included in its Ontology and represented in its Foundation1, a topic to be argued in future articles. Despite the distinct states predicted by QT, EPR9 decrees that SSII is in the same real state. I call this 'The Reality Preconception 2' (TRP2), which violates the EPR4 dictum1. Kemble12 quickly argued that the EPR argument was not sound, but his counterargument was not sound either. Furry13 pointed out the incompatibility of EPR premises with QT. Bohm5,6 suggested TRP2. Jammer8 recounts in detail all early criticisms to the EPR argument. This is the logical trap behind TRP2: if it is true that SSI and SSII do not interact: (a) the composite state must be a product-state10,1; (b) the expansion coefficients for SSI cannot depend on SSII eigenfunctions; and c) EPR9's first sentence is not true, rendering its last sentence invalid. EPR9 relies on a cross-influence among wavefunctions that contradicts its own 'no interaction' 5 assumption. It might not be the type of interaction EPR would philosophically approve, but the mere existence of cross-relations implies an interaction. EPR used and ignored QT's controversial prediction in the same paralogism. Furthermore, TRP2 uncovers an incompleteness nuance: first, 'incomplete' meant that the same abstract state represented more than one real state ('real' by virtue of TRP1) so that a "counterpart in the physical theory" for many a 'real' state was missing1. Now EPR asserts: the same real state ('real' by virtue of TRP2) corresponds to more than one abstract state. EPR9 dogmatically removes nonlocality from the Ontology and, inevitably, preordains not only QT's incompleteness but also its incorrectness. 4. Misusing 'The Fallacious Dichotomy' Before resorting to TFD1 as the coup de grâce, EPR states: EPR10: Thus, by measuring either A [PI] or B [QI] we are in a position to predict with certainty, and without in any way disturbing the second system, either the value of the quantity P [PII] or the value of the quantity Q [QII]. In accordance with our criterion of reality, in the first case we must consider the quantity P as being an element of reality, in the second case the quantity Q is an element of reality. But, as we have seen, both wavefunctions belong to the same reality. Based on TCC1, TRP11, and TRC1, EPR concludes: in one case, it is PII the one which is real (single value); in the other, it is QII the real one. But in both cases, because what happened to SSI did not (by virtue of TRP2) affect SSII at all, PII and QII must have existed all along, i.e. they had, in EPR's language, a "simultaneous reality". Finally, using TFD1, via a reductio ad absurdum argument, EPR reaches a contradiction that, alas, already existed in its premises: EPR11: Previously we proved that either (1) the quantum-mechanical description of reality given by the wave function is not complete or (2) when the operators corresponding to two physical quantities do not commute the two quantities cannot have simultaneous reality. Starting then with the assumption that the wave function does give a complete description of the physical reality, we arrived at the conclusion that two physical quantities, with non-commuting operators, can have simultaneous reality. Thus the negation of (1) leads to the negation of the only other alternative (2). We are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete. Despite QT predicting the opposite, EPR insists the 'real' state of SSII has not changed and applies the 'Uncertainty Principle' incorrectly1. The purpose is to assert that PII and QII do not have "simultaneous reality" (one of them has a PD), while they must have it because QT predicts them "with certainty" and "without disturbance". EPR contends that such a logical contradiction (both alternatives false) can only imply that QT is incomplete. Instead of contrasting QT predictions with TRP2, EPR could have humbly admitted that perhaps QT was revealing something new about Nature's modus operandi and, ergo, a shift from thought experiments to real ones was mandatory to confirm/refute those predictions. Such refutation would prove that QT was incorrect (untrue), not merely incomplete. Technology was not ready for such a feat -not even in 1964, when John Bell14 started his seminal work (future article). The truth is that EPR did not start "with the assumption that the wave function does give a complete description of the physical reality". Otherwise, EPR would have accepted that the predicted physical reality did change in SSII when the 'measurement' in SSI took place. Had EPR fully accepted QT predictions, no "simultaneous reality" of PII and QII could have been claimed 6 and no contradiction had occurred. The contradiction EPR claimed to have unveiled was already veiled in their premises. The Petitio Principii fallacy was at work from the very outset. EPR admitted that had a stricter 'criterion of reality' been adopted, concluding "simultaneous reality" would have been a non sequitur. But, after proposing a new "definition" of reality, EPR immediately labeled it as "unreasonable". There are simply no definitions of Reality: the only reasonable attitude is to accept that Reality may not conform to our prejudices and, if an otherwise very successful theory predicts something we do not believe in (nonlocality), no thought experiment will help because our prejudices may creep in. Only actual experiments could do the trick (as proven in the last 50 years). EPR finishes, stating: EPR12: While we have thus shown that the wave function does not provide a complete description of the physical reality, we left open the question of whether or not such a description exists. We believe, however, that such a theory is possible. I hope I have convincingly argued against EPR having shown that QT "does not provide a complete description of reality". 5. Einstein's Incompleteness/Nonlocality Dilemma Despite EPR's logical flaws, I am determined to be fair to Einstein, so let me reproduce what he said 13 years later15: ... when I consider the physical phenomena known to me, and especially those which are being so successfully encompassed by quantum mechanics, I still cannot find any fact anywhere which would make it appear likely that requirement [locality] will have to be abandoned. I am therefore inclined to believe that the description of quantum mechanics ... has to be regarded as an incomplete and indirect description of reality, to be replaced at some later date by a more complete and direct one. Evidently, in 1948, Einstein was open to change his mind upon learning of "any fact anywhere which would make it appear likely that..." it made sense for him to accept nonlocality. A year later, in 'Reply to Criticisms'7, he reformulated TFD1, now endowing it with synthetic value: By this way of looking at the matter, it becomes evident that the paradox [EPR] forces us to relinquish one of the following two assertions: (1) the description by means of the ψ–function is complete; (2) the real states of spatially separated objects are independent of each other. His relinquishing option (2) shows that, in 1949, he surmised the solution to his dilemma: if QT was complete then the Principle of Locality (or at least its universal supremacy) had to go. Nonetheless, Jammer8 interviewed Einstein in 1953 confirming he never abandoned his view that QT was an incomplete description of physical reality. As it is well known, in 1964, John Bell14 proved that behind the apparent Lorentz-invariance of quantum phenomena, there was a deeper level that was not Lorentz-invariant, namely that nonlocality was a real feature of Nature. He also proved that no hidden-variable theory could avoid nonlocality. Copious experimental data followed suit (next articles); I am sure that Einstein (dismayingly) would have changed his mind had he seen the evidence we have today. Even so... 20 years into the 21st century, the struggle about Reality continues. Conclusions QT/TOPI clearly shows that SSI and SSII detangle upon the first GI occurring to one of them, leaving both (now independent) systems in correlated states -irrespective of whether a GI ever happens to the other system. This will help to understand the infamous 'measurement problem' in 7 a future article. A second 'Reality Preconception' (TRP2) was identified, showing that EPR used and ignored QT's prediction of nonlocality in a single paralogism. Employing TFD1 and TRP2, EPR unveiled a contradiction already veiled in its premises. The Petitio Principii fallacy had been at work from the very outset. By 1949, Einstein lucidly surmised the solution to his incompleteness/nonlocality dilemma but never changed his mind. The lesson is that there are no definitions of Reality: we have to accept that Reality may not conform to our prejudices and, if an otherwise successful theory predicts what we do not believe in, no gedankenexperiment will help because our biases may slither through. Only actual experiments could help solving Einstein's dilemma, as proven in the last 50 years. Notwithstanding, EPR is one of the most influential papers in history and has greatly sparked both conceptual and technological progress. Future articles will further explain TOPI, demonstrating its soundness and potential for nurturing theoretical advance. References 1. Alba-Juez, F. The Ontic Probability Interpretation of Quantum Theory Part I: The Meaning of Einstein's Incompleteness Claim. https://philpapers.org/rec/ALBTOP-2 (2020). 2. Einstein, A., Podolsky, B. & Rosen, N. 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Saint George, Utah: Felix Alba-Juez, Publisher (2019). 11. Alba-Juez, F. Records of the Future Classical Entropy, Memory, and the 'Arrow of Time'. Salt Lake City: Felix Alba-Juez, Publisher (2013). 12. Kemble, E. C. The Correlation of wave functions with the states of physical systems. Physical Review, 47, 973-974 (1935). 13. Furry, W. H. Note on the quantum mechanical theory of measurement. Physical Review, 49, 393-399 (1936). 14. Bell, J. S. On the Einstein Podolsky Rosen Paradox. Physics, 1, 195-200 (1964). 15. Einstein, A. Quantum Mechanics and Reality. Dialectica, 2, 320-324 (1948).