The Ontic Probability Interpretation of Quantum Theory Part I: The Meaning of Einstein's Incompleteness Claim Felix Alba-Juez Felix Alba-Juez, Publisher Saint George, Utah (USA) February 5, 2020 https://felixalbajuez.com ABSTRACT Ignited by Einstein and Bohr a century ago, the philosophical struggle about Reality is yet unfinished, with no signs of a swift resolution. Despite vast technological progress fueled by the iconic EPR paper (EPR)1,2,3, the intricate link between ontic and epistemic aspects of Quantum Theory (QT) has greatly hindered our grip on Reality and further progress in physical theory. Fallacies concealed by tortuous logical negations made EPR comprehension much harder than it could have been had Einstein written it himself in German. It is plagued with preconceptions about what a physical property is, the 'Uncertainty Principle', and the Principle of Locality. Numerous interpretations of QT vis à vis Reality exist and are keenly disputed4,5,6,7,8,9,10,11. This is the first of a series of articles arguing for a physical interpretation called 'The Ontic Probability Interpretation' (TOPI). A gradual explanation of TOPI is given intertwined with a meticulous logico-philosophical scrutiny of EPR. Part I focuses on the meaning of Einstein's 'Incompleteness' claim. A conceptual confusion, a preconception about Reality, and a flawed dichotomy are shown to be severe obstacles for the EPR argument to succeed. Part II analyzes Einstein's 'Incompleteness/Nonlocality Dilemma'12. Future articles will further explain TOPI, demonstrating its soundness and potential for nurturing theoretical progress. List of Acronyms (Part I) 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 1. Introduction As a realist, Einstein wrote: "there is something like the 'real state' of a physical system, which independent of any observation or measurement exists objectively and which can in principle be described by means of physical terms". However, probability-wise, Einstein was a subjectivist -blaming the stochastic makeup of QT on its incompleteness. But more than chance as Nature's modus operandi, he obstinately detested its "spooky action at a distance" -blaming again such "telepathy" predicted by QT on its incompleteness13. Poorly understood even today, EPR1 and Bohr's response2 were published on May 15 and October 15, 1935 with identical titles: "Can Quantum Mechanics Description of Physical Reality be Considered Complete?". Prior to his formal response, Bohr had sent a letter3 to Nature. EPR 2 discussed thought experiments where the position and momentum of two correlated 'particles' were predicted by QT and 'measured'. I put 'particles', and 'measured' in quotes because: (a) quantum objects are neither particles nor waves; and (b) most physical interactions are not measurements. John Bell advised for the word 'measurement' to "be banned altogether in quantum mechanics"14. Most physicists and philosophers did not listen. 2. Elements of a Physical Theory Against the Logical Positivism in vogue at the time, EPR states: EPR1: Any serious consideration of a physical theory must take into account the distinction between the objective reality, which is independent of any theory, and the physical concepts with which the theory operates. These concepts are intended to correspond with the objective reality, and by means of these concepts we picture this reality to ourselves. A factual theory is an explanatory/predictive logico-mathematical formalism whose ultimate referent is Reality; ergo, it must be put to the empirical test. A theory consists of Ontology, Foundation, Structure, Interpretation, and Evidence. The Ontology includes the presumed real entities plus known facts about their properties and behavior. The Foundation comprises: a) abstract entities/attributes; and b) unexplained explainers: principles, postulates, hypotheses, etc. The Structure entails: a) non-factual formalisms (e.g. Logic, Calculus, Geometry); b) other factual theories (e.g. Space/Time, Relativity, Electromagnetism); and c) laws and theorems about the abstract entities. The elusive Interpretation attempts to grasp Reality by proposing semantic rules via which the abstract entities/attributes represent the real ones. The Evidence incorporates the empirical support the theory possesses to claim its verisimilitude. Measurements and observers are necessary for the Evidence but are not, and must not be, part of the theory. 3. Elements of 'The Ontic Probability Interpretation' (TOPI) An interpretation endows the Foundation, Structure, and Evidence with physical meaning, thereby characterizing the Ontology. Numerous interpretations/formulations of QT exist and are widely disputed4,5,6,7,8,9,10,11. Like Bunge15, I will refer to the abstract/real entities of QT as 'quantons'. Per TOPI, an abstract quanton interacts with its abstract milieu and has: a) a current abstract state that corresponds to the real quanton's state attained from the last interaction with its physical milieu; b) current abstract attributes that parallel physical properties of the real quanton in its current real state; and c) a probability distribution (PD) for the transition to its next abstract state/attributes, which is the predicted ontic PD for the real quanton to transition to its next real state/properties. Being a PD, the next state/property values are defined but undetermined. There are attributes a quanton does not possess (e.g. size, shape), i.e. they are not defined at all; and others that are defined only for some states (like the azimuthal angle is defined/undefined off/on the polar axis). Quantons are not punctiform objects. A property which is defined/undefined for the current state can be undefined/defined for the next state. If, for any state/property, the PD is as narrow as to effectively assign determinate values for all next states/properties, the theory is classically deterministic; otherwise, it is stochastic. QT is partly stochastic and partly causal, to which I call 'quantically deterministic'. Classical determinism is a degenerate type of quantic determinism. TOPI is applicable to Classical Physics16,17,18. A composite quanton can be in product-states, for which all sub-quantons are isolated; and in entangled states, for which the sub-quantons' states are undefined per se. The same current state is expressible via different linear combinations of next eigenstates (different bases for the State- 3 Space). Quanton and milieu jointly determine the basis, with the PD for the next state/properties encrypted in the resulting linear combination18. Per TOPI, QT claims neither explicative nor predictive power between current and next states. Discrete and continuous systems are covered by QT/TOPI. A 'Physical Interaction' (PI) between a quanton and its milieu is, generally, reciprocal, i.e. both change states. A PI implemented by us to acquire knowledge will be called a 'Gauge Interaction' (GI); GIs were called 'measurements' by QT pioneers and, ignoring Bell's advice, they still are by most researchers. If a GI is such that the milieu (the 'measurer') changes state and the quanton (the 'measured') does not, I call it a 'True Measurement' (TM). From a strictly physical viewpoint, the anthropic GIs and TMs occur all the time without human intervention. Only some properties may be experimentally accessible, creating the empirical Evidence. The operationalist believes a physical property has no meaning but the one given by its measurement protocol. This is not true because we must understand the real property before we can conceive a gauging technique and build and/or select the proper instrumentation.18 3.1. Heisenberg's Inequalities vis à vis TOPI QT predicts probabilities, not values. Under TOPI, probability is not epistemic but ontic15,16,17,18,19,20,21. Heisenberg's inequalities have had more misinterpretations than any other formula in history. Per QT, given two properties with noncommutative operators P1 and P2 ([P1, P2] = P1P2 − P2P1 ≠ 0), and depending on the quanton's current state, only one of the properties may have a single value (SD → 0) while the other is undefined. As for the next state and properties, only their PDs are univocally defined. Thus, for any common current state, it is impossible to jointly assign determinate current/next values to both properties. Per TOPI, it is the probability distribution for the values, not the values themselves, that constitutes the physical property of a quanton/milieu system and, hence, no single GI can characterize the property. Inequalities (1) express the so-called 'Uncertainty Principle' for generic properties P1 and P2 and for momentum P and position Q: ∆P1∆P2 ≥ (1 2⁄ )|〈[P1, P2]〉| ⇒ ∆P∆Q ≥ ħ 2⁄ (1) Under TOPI, these inequalities neither express a 'principle' nor involve 'uncertainty', not even 'indeterminacy'. They do not entail 'measurements' either. They constitute a theorem of QT relating the SDs of two random variables for the next state. The narrower one PD is, the broader is the other. This is only true when the quanton current state is the same for both properties.18 4. Correctness/Completeness/Elements of Reality EPR asserts how to judge the correctness of a physical theory: EPR2: The correctness of the theory is judged by the degree of agreement between the conclusions of the theory and human experience. This experience, which alone enables us to make inferences about reality, in physics takes the form of experiment and measurement. A theory is correct because none of its central predictions has been empirically nullified. Prima facie, EPR appears to recognize the correctness of QT. A correct theory may be incomplete because it does not predict aspects of Reality (facts) we expected it to predict. Despite being its leitmotif, EPR does not define completeness, proposing only a necessary condition: EPR3: Whatever the meaning assigned to the term complete, the following requirement for a complete theory seems to be a necessary one: every element of the physical reality must have a counterpart in the physical theory. 4 Being EPR3 just necessary, only incompleteness can be proven. To do so, an element in the Ontology must have no counterpart in the Foundation, viz we must identify a fact the theory cannot predict. EPR admits it is us who identify the ontic entities/properties ("elements of the physical reality") which we expect the theory to describe/explain. Thus, completeness relates to both Reality (facts) and our expectations, which could be rooted in prejudices and/or a priori philosophical views. No unexplained explainer in the Foundation (not even a principle) belongs to the Ontology: if predictions defy our prejudices, experiment must rule. EPR agrees: EPR4: The elements of the physical reality cannot be determined by a priori philosophical considerations, but must be found by an appeal to results of experiments and measurements. To identify an 'element of physical reality', EPR proposes 'The Reality Criterion' (TRC): EPR5: If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity... Regarded not as a necessary, but merely as a sufficient condition of reality, this criterion is in agreement with classical as well as quantum-mechanical ideas of reality. 4.1. The Conceptual Confusion (TCC) Palpably against EPR4, EPR5 says that for a property to be real, it is enough that we can predict its value "with certainty" and "without in any way disturbing" the system. It is against EPR4 because measurement is absent. Besides, a mere prediction cannot disturb anything physical, and the only way to know how certain our prediction was is to accurately measure the property so... I surmise EPR forgot to include 'when measured' after "the value of a physical quantity". But here is the striking EPR confusion: the text in parentheses shows that EPR5 conflated three distinct concepts: (a) the prediction certainty (predicted vs. real); (b) the measurement accuracy (measured vs. real); and (c) the probability for a property to assume one of its values. It is vital to understand that it is (c) what QT is all about, not (a); and that (b) is outside QT, serving only to test its correctness. Predicting something with a probability is not the same as predicting a probability for something. Predicting a value "with probability equal to unity" amounts to a perfect prediction (predicted = real), and it is utterly different to predicting 'a probability equal to unity for a value'. Whether correct or not, if the theory is classically deterministic, all predicted probabilities are equal to unity. Instead, for a state/property, QT predicts a PD over the next states/property values, i.e. they are all random variables. Predicting a probability less than one can be as accurate (vis à vis Reality) as predicting a probability equal to one. I call this muddle 'The Conceptual Confusion' (TCC). It could be cogently argued that TCC invalidates EPR arguments and conclusions at the outset. That would be unfair -considering the enormous technological and philosophical impact EPR has had. QT predicts a unity probability only when the quanton is in an eigenstate of the property's operator. Only then an ideal GI delivers the value the property had pre-GI, i.e. the GI is an ideal TM. But a real TM, if repeated, never delivers a single value but a distribution of them -for classical and quantic systems. We use a single value because the error-distribution is exceptionally narrow. However, most GIs are not TMs, i.e. the initial state is not an eigenstate, with QT predicting a broad PD for the next state/properties. Ergo, estimating the prediction accuracy ("certainty") requires comparing two PDs: predicted vs. real, with the latter assessed 5 by measurement. In sum, EPR confuses the nil SD of the predicted PD for a property (when the system is in an eigenstate) with the prediction and measurement accuracies for its single value. How do we then interpret EPR5? It cannot be literally, i.e. per (a), because QT does not predict the certainty of its predictions. Clearly, it must be (c) for prediction plus (b) for Reality. Thus, from now on, TRC means EPR5 so interpreted and, if I refer to (a) to rephrase EPR rationale, I will use quotes, viz "with certainty". Only doing so can we be fair to EPR, despite TCC. With this caveat, and a negligible measurement error, TRC implies that if a 'particle' is in a momentum eigenstate, the 'momentum is real' and if it is in a position eigenstate, the 'position is real'. Otherwise, TRC is mute. Under TRC, Reality might oddly depend on the 'particle' state. 5. The Reality Preconception 1 (TRP1) EPR verbalizes Heisenberg's Inequalities using the operationalist language: EPR6: A definite value of the coordinate [position], for a particle in the state given by Eq. (2) [an eigenstate of the momentum operator], is thus not predictable, but may be obtained only by a direct measurement. Such a measurement however disturbs the particle and thus alters its state. After the coordinate is determined, the particle will no longer be in the state given by Eq. (2). The usual conclusion from this in quantum mechanics is that when the momentum of a particle is known, its coordinate has no physical reality. Per QT, because position and momentum operators do not commute, the momentum eigenstate is not a position eigenstate; hence, position in such a state is undefined while a PD is predicted for its next value under a position-GI. By stating that a definite value of the coordinate is "thus not predictable, but may be obtained only by direct measurement", EPR reveals an a priori belief in classical determinism: such a position must exist and it could have been provided by its direct 'measurement' had the previous 'measurement' of the momentum not altered the system state. When a 'particle' is in a momentum eigenstate, a momentum-GI is a TM so, per TRC, the momentum is real. As for a position-GI, being the prediction a PD, TRC is mute so it is a non sequitur to infer that if the momentum is real the position is not. EPR6 recites the Copenhagen Interpretation of QT. TRC was purposely devised as "merely" sufficient lest, having assumed QT correct, TRC would imply that the "coordinate has no physical reality" at all. EPR believed the position was real but only if it had a definite value, which is nothing but an a priori philosophical belief (violating EPR4). For Einstein, using probability amounted to confessing ignorance of the underpinning causal processes. I call this 'The Reality Preconception 1' (TRP1). 6. The Fallacious Dichotomy (TFD) Endeavoring to prove QT incomplete, EPR condenses TRC, TCC, and TRP1 into a dichotomy: EPR7: From this follows 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. For if both of them had simultaneous reality-and thus definite values-these values would enter into the complete description, according to the condition of completeness. If then the wave function provided such a complete description of reality, it would contain these values; these would then be predictable. This not being the case, we are left with the alternatives stated. The phrase "For if both of them had simultaneous reality-and thus definite values-..." is now unequivocally asserting TRP1: only attributes with definite values are real, so two 6 conjugate properties cannot be "simultaneously real" (unless QT is incomplete). EPR7 also says that the definite value of a real property must be "predictable": the "mere" sufficient character of TRC has now become also necessary. Thus, for EPR, a theory cannot be complete if, in most cases, it predicts a mere PD. EPR7 dogmatically removes probability from the Ontology and, inevitably, preordains QT's incompleteness: Petitio Principii at work. It is baffling why Reality was not so 'defined' at the outset. A plethora of convoluted logic could have been saved: QT would be incomplete simply because only rarely does it predict definite values. However, the inclusion of a priori philosophical considerations into the Ontology (against EPR4) would have been obvious. EPR7 dichotomy boils down to: either (1) the two quantities do have "simultaneous reality" (determinate values) and QT is incomplete because it does not predict them, or (2) the quantities do not have "simultaneous reality" (at least one has a PD) and QT is complete because it predicts so. EPR conflates the joint reality of two physical properties with joint predictability and measurability of single values for them. This dichotomy is fallacious because it is predicated on a priori philosophical beliefs regarding Reality. It has only analytic value (as opposed to synthetic) because QT completeness or incompleteness depends on the ad hoc definition of "simultaneous reality", not on experimental evidence. As for EPR7 phrase "..., it would contain these values; these would then be predictable", it is obviously intimating the well-known idea of 'hidden variables' which, having zero dispersion (SD = 0), would restore Classical Determinism to Physics, reaffirming TRP1. Future articles in this series will deal with hidden-variable theories and other QT interpretations/formulations. Conclusions To honor the spirit of EPR, because of the conceptual confusion (TCC), I reinterpreted its reality criterion (TRC). In violation of its own dictum for identifying the 'elements of reality' (EPR4), EPR revealed its commitment to classical determinism, associating probability only with human ignorance and, thereby, relying on a Reality preconception (TRP1). Combining TRC, TCC, and TRP1, EPR proposed a mutually exclusive disjunction (TFD), whose truth value is only analytic (not synthetic) because it depends upon an ad hoc 'definition' of Reality. Despite the above logical flaws, EPR strived to prove that option (1) in TFD was true, i.e. that the two quantities did have "simultaneous reality". But, because (in most cases) a 'measurement' (GI) disturbs the state and TRC was mute regarding the property's reality, EPR needed to conceive a way of 'measuring' without "in any way disturbing the system". In our TOPI jargon: a way of making a GI to effectively work as a TM. Such a scheme to prove QT's incompleteness was proposed by EPR, and it is dissected and proven also inadequate in Part II.12 References 1. Einstein, A., Podolsky, B. & Rosen, N. Can Quantum-Mechanical Description of Physical Reality be Considered Complete?. Physical Review, 47, 777-780 (1935). 2. Bohr, N. 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