Abstract Einstein’s article on the Einstein-Podolsky-Rosen paradox is the most cited of his works, but not many know that it was not fully representative of the way he thought about the incompleteness of the quantum formalism. Indeed, his main worry was not Heisenberg’s uncertainty principle, which he accepted, but the experimental non-separability of spatially separate systems. The same problem was also recognized, years later, by one of us, as part of an axiomatic analysis of the quantum formalism, which revealed an unexpected structural limitation of the quantum formalism in Hilbert space, preventing the description of separate systems. As we will explain, this limitation does not manifest at the level of the states, but of the projectors describing the properties, in the sense that there are not enough properties in the formalism to describe separate systems. The question remains whether separability is a possibility at the fundamental level and if a formalism should integrate it into its mathematical structure, as a possibility. To aid our intuition, we offer a reflection based on a powerful analogy between physical systems and human conceptual entities, as the question of separability also arises for the latter. This article is part of the theme issue ‘Newton, Principia, Newton Geneva Edition (17th–19th) and modern Newtonian mechanics: heritage, past & present’.
Abstract Einstein’s article on the Einstein-Podolsky-Rosen paradox is the most cited of his works, but not many know that it was not fully representative of the way he thought about the incompleteness of the quantum formalism. Indeed, his main worry was not Heisenberg’s uncertainty principle, which he accepted, but the experimental non-separability of spatially separate systems. The same problem was also recognized, years later, by one of us, as part of an axiomatic analysis of the quantum formalism, which revealed an unexpected structural limitation of the quantum formalism in Hilbert space, preventing the description of separate systems. As we will explain, this limitation does not manifest at the level of the states, but of the projectors describing the properties, in the sense that there are not enough properties in the formalism to describe separate systems. The question remains whether separability is a possibility at the fundamental level and if a formalism should integrate it into its mathematical structure, as a possibility. To aid our intuition, we offer a reflection based on a powerful analogy between physical systems and human conceptual entities, as the question of separability also arises for the latter. This article is part of the theme issue ‘Newton, Principia, Newton Geneva Edition (17th–19th) and modern Newtonian mechanics: heritage, past & present’.
Einstein's article on the EPR paradox is the most cited of his works, but not many know that it was not fully representative of the way he thought about the incompleteness of the quantum formalism. Indeed, his main worry was not Heisenberg's uncertainty principle, which he accepted, but the experimental non-separability of spatially separate systems. The same problem was also recognized, years later, by one of us, as part of an axiomatic analysis of the quantum formalism, which revealed an unexpected structural limitation of the quantum formalism in Hilbert space, preventing the description of separate systems. As we will explain, this limitation does not manifest at the level of the states, but of the projectors describing the properties, in the sense that there are not enough properties in the formalism to describe separate systems. The question remains whether separability is a possibility at the fundamental level and if a formalism should integrate it into its mathematical structure, as a possibility. To aid our intuition, we offer a reflection based on a powerful analogy between physical systems and human conceptual entities, as the question of separability also arises for the latter.
Two problems with the Einstein-Podolsky-Rosen argument originate from the difficult-to-explain continual dismissal of Minkowski’s demonstration that the concept of spacetime represents a real four-dimensional world and is not just a mathematical space. The uncritical employment of the traditional three-dimensional language to the description of this argument leads (e.g., in its Bohm's version) to those problems: (1) due to the collapse of the common wavefunction of the constituents 1 and 2 of a disintegrated system, caused by the measurement of 1's spin (event E1), and due the conservation of the total spin, 2's spin is \textit{instantaneously} determined (event E2), which in three-dimensional language inescapably means that E2 is \textit{caused} by E1 (using merely a different word -- `correlation' -- does not explain anything in three-dimensional language); (2) events E1 and E2 are simultaneous \textit{only} for an observer A, who was at rest with respect to the disintegrated system, and therefore the total spin is not conserved for arbitrarily large periods of time for all observers in relative motion with respect to A and there exists a class of observers B for whom event E2 occurs \textit{before} E1, which demonstrates the inadequacy of the fundamental implicit assumption that the physical world is three-dimensional. When the reality of spacetime is explicitly recognized, (1) the paradoxes of the non-conservation of the total spin and that E2 happens before E1 for observers B do not arise (because there is no `before' and `after' in spacetime) and the statement that there is a correlation between E1 and E2 makes a perfect sense, and (2) a spacetime model of the quantum object can then be considered (by also taking into account the non-fully-realized experimental fact that the quantum object \textit{actually} exists everywhere where its wavefunction is different from zero) in order to address the fundamental incompleteness of quantum mechanics -- that it deals not with the quantum object itself, but only with its states. The paper ends with a discussion of a prediction by the proposed spacetime model of the electron that can be tested experimentally probably by the quadratic Stark effect.
The difference between the conceptual framework of classical physics and that of quantum physics which is due to the nonseparability property of the latter theory is better understood in specific examples. This chapter uses some specific examples to test the validity, both in classical and in quantum mechanics, of two conceptual principles. First is Reality Principle . The very concept of a reality may be criticized by philosophically minded empiricists. A widespread opinion, however, is that somehow a physical reality does exist, which should not just amount to the talks we give about it. In their search for a criterion that would make it possible to recognize elements of reality, A. Einstein, B. Podolsky, and N. Rosen had to take such considerations into account. Second, Principle of the Separability of Mechanically Isolated Systems, one can easily guess that this second principle, which remains somewhat implicit in the work of Einstein and his colleagues may well be the source of insuperable difficulties.
We discuss the question of entanglement versus separability of pure quantum states in direct product Hilbert spaces, and the relevance of this issue to physics. Different types of separability may be possible, depending on the particular factorization of the Hilbert space. A given orthonormal basis set for a Hilbert space is defined to be of type (p,q) if p elements of the basis are entangled and q are separable, relative to a given bi-partite factorization of that space. We conjecture that not all basis types exist for a given Hilbert space. The phenomenon of entanglement is of central importance in the interpretation of quantum mechanics. Historically, entanglement was the focus of the famous Einstein-Podolsky-Rosen (EPR) paper [1], which suggested that standard quantum mechanics is an incomplete theory of physical reality. The central argument of the EPR paper was that more information about incompatible variables such as momentum and position could in principle be deduced about an entangled two particle quantum state than quantum mechanics permits, effectively giving information about each particle
From a phenomenological and experimental viewpoint, quantum mechanics is remarkably successful in explaining effects and processes in the microscopic world. Recently, however, a growing body of literature has revisited the theory's own formal foundations, specifically addressing issues of self-consistency and completeness. As a major historical example of these foundational challenges, the Einstein-Podolsky-Rosen (EPR) argument famously claimed that the theory is incomplete. While many valid criticisms and refutations of the EPR logic exist, they generally rely on a combination of technical results and conceptual objections to EPR's interpretive assumptions, thus transcending the plain quantum framework itself. Motivated by these trends of examining the theory's structural limits, here we revisit the EPR argument. Focusing on elements essential to the EPR reasoning, we first analyze general quantum correlations for EPR states: (i) the fact that their observables are always associated with non-commuting operators, and (ii) the nature of the correlated information obtained through measurement. From these two points alone, and relying strictly on the core rules of quantum mechanics, we demonstrate how to overturn the alleged EPR incompleteness without invoking any extraneous propositions. Consequently, we show that the standard formalism alone suffices to resolve such type of skepticism regarding the theory's physical reach. This offers, at least in a paradigmatic instance, a powerful indication of a structurally well-founded, self-consistent quantum theory.
In classical mechanics, two (or more) systems that do not interact – that do not exert any kind of force on each other – are completely separated (see Sec. 1.1), i.e. experiments performed locally on one of them cannot in any way influence experiments performed locally on the other. Quantum mechanics, on the other hand, admits that there can be a form of interdependence even in absence of physical interaction. This type of correlation goes under the name of entanglement (see Def. 5.1: p. 183). We already know that entanglement means non-factorizability of the state vector describing a system with at least two different degrees of freedom. In this chapter, we shall see that this may be the case also for systems that are very far away (space-like distant) from each other and we shall learn many interesting consequences of this state of affairs. Recent developments show that one may entangle even particles that have never interacted with each other. The starting point of the following analysis is necessarily represented by the pioneering paper of Einstein, Podolsky, and Rosen, who inaugurated an incredible series of theoretical and experimental investigations right up to the present day that have confirmed the predictions of quantum mechanics. The “irony” of this history is that Einstein, Podolsky, and Rosen, EPR for short, aimed to present a definitive proof of the inconsistency of quantum mechanics relative to classical physical theory. In fact, they initiated one of the most prolific fields of modern science.