According to a popular narrative, in 1932 von Neumann introduced a theorem that intended to be a proof of the impossibility of hidden variables in quantum mechanics. However, the narrative goes, Bell later spotted a flaw that allegedly shows its irrelevance. Bell’s widely accepted criticism has been challenged by Bub and Dieks: they claim that the proof shows that viable hidden variables theories cannot be theories in Hilbert space. Bub’s and Dieks’ reassessment has been in turn challenged by Mermin and (...) Schack. Hereby I critically assess their reply, with the aim of bringing further clarification concerning the meaning, scope and relevance of von Neumann’s theorem. I show that despite Mermin and Schack’s response, Bub’s and Dieks’ reassessment is quite correct, and that this reading gets strongly reinforced when we carefully consider the connection between von Neumann’s proof and Gleason’s theorem. (shrink)

It is a widespread belief that the Kochen-Specker theorem imposes a contextuality constraint on the ontology of beables in quantum hidden variables theories. On the other hand, after Bell’s influential critique, the importance of von Neumann’s wrongly called ‘impossibility proof’ has been severely questioned. However, Max Jammer, Jeffrey Bub and Dennis Dieks have proposed insightful reassessments of von Neumann’s theorem: what it really shows is that hidden variables theories cannot represent their beables by means of Hermitian operators in Hilbert space. (...) Hereby I show that i) the very same constraint can be derived from Gleason’s theorem, and that ii) if we consider the import of von Neumann’s and Gleason’s theorems, the relevance of the Kochen-Specker theorem for hidden variables theories gets substantially weakened: it does not force them to be contextual in any interesting sense of the term. (shrink)

We show that the respective oversights in the von Neumann's general theorem against all hidden variable theories and Bell's theorem against their local-realistic counterparts are homologous. When latter oversight is rectified, the bounds on the CHSH correlator work out to be ±2√2 instead of ±2.

Grete Hermann’s essay “Die naturphilosophischen Grundlagen der Quantenmechanik” has received much deserved scholarly attention in recent years. In this paper, I follow the lead of Elise Crull who sees in Hermann’s work the general outlines of a neo-Kantian interpretation of quantum theory. In full support of this view, I focus on Hermann’s central claim that limited spatio-temporal, and even analogically causal, representations of events exist within an overall relational structure of entangled quantum mechanical states that defy any unified spatio-temporal description. (...) In my view, Hermann also advances an important transcendental argument that perspectival spatio-temporal representations of nature have their foundations in general relational networks that are not spatio-temporal. The key point is that the adoption of a perspectival system within the general network induces the representation but only for that context. These ideas are consistent with a perspectival subject–object principle in Kant and also with Weyl’s work on Lie groups and their representations. (shrink)

The apparent dichotomy between quantum jumps on the one hand, and continuous time evolution according to wave equations on the other hand, provided a challenge to Bohr’s proposal of quantum jumps in atoms. Furthermore, Schrödinger’s time-dependent equation also seemed to require a modification of the explanation for the origin of line spectra due to the apparent possibility of superpositions of energy eigenstates for different energy levels. Indeed, Schrödinger himself proposed a quantum beat mechanism for the generation of discrete line spectra (...) from superpositions of eigenstates with different energies. However, these issues between old quantum theory and Schrödinger’s wave mechanics were correctly resolved only after the development and full implementation of photon quantization. The second quantized scattering matrix formalism reconciles quantum jumps with continuous time evolution through the identification of quantum jumps with transitions between different sectors of Fock space. The continuous evolution of quantum states is then recognized as a sum over continually evolving jump amplitudes between different sectors in Fock space. In today’s terminology, this suggests that linear combinations of scattering matrix elements are epistemic sums over ontic states. Insights from the resolution of the dichotomy between quantum jumps and continuous time evolution therefore hold important lessons for modern research both on interpretations of quantum mechanics and on the foundations of quantum computing. They demonstrate that discussions of interpretations of quantum theory necessarily need to take into account field quantization. They also demonstrate the limitations of the role of wave equations in quantum theory, and caution us that superpositions of quantum states for the formation of qubits may be more limited than usually expected. (shrink)

The publication of the EPR paper in 1935 prompted Heisenberg to draft a manuscript on the question of the completability of quantum mechanics. We give here the English translation of this manuscript with a brief introduction and bibliography.

In this paper I attempt to broaden Feyerabend scholarship by asking whether and how Feyerabend's philosophy of science, in particular his commitments to realism and pluralism about scientific theories as well as anarchism about scientific methods, is borne out in multidisciplinary research concerning the Leggett–Garg inequalities. These inequalities were derived explicitly to be a temporal analogue to Bell's inequalities: the viability of macroscopic realism is tested against the predictions of quantum mechanics by performing a series measurements on a macroscopic variable (...) of one system at different time intervals, then evaluating correlations among resultant values. In the nearly forty years since Leggett and Garg's paper, questions continue to be raised whether common tests of these inequalities reveal genuine violations or merely experimental disturbances. There is also much debate about what genuine violations in fact signify, if (as most agree) they are not straight-forwardly analogous to Bell tests. Taking this fascinating and as yet largely unsettled episode from physics as a case study, I derive three Feyerabendian lessons that demonstrate the relevance of his views even today. (shrink)

Signal causality, the prohibition of superluminal information transmission, is the fundamental property shared by quantum measurement theory and relativity, and it is the key to understanding the connection between nonlocal measurement effects and elementary interactions. To prevent those effects from transmitting information between the generating and observing process, they must be induced by the kinds of entangling interactions that constitute measurements, as implied in the Projection Postulate. They must also be nondeterministic as reflected in the Born Probability Rule. The nondeterminism (...) of entanglement-generating processes explains why the relevant types of information cannot be instantiated in elementary systems, and why the sequencing of nonlocal effects is, in principle, unobservable. This perspective suggests a simple hypothesis about nonlocal transfers of amplitude during entangling interactions, which yields straightforward experimental consequences. (shrink)

In early 1927, Pascual Jordan published his version of what came to be known as the Dirac-Jordan statistical transformation theory. Later that year and partly in response to Jordan, John von Neumann published the modern Hilbert space formalism of quantum mechanics. Central to both formalisms are expressions for conditional probabilities of finding some value for one quantity given the value of another. Beyond that Jordan and von Neumann had very different views about the appropriate formulation of problems in the new (...) theory. For Jordan, unable to let go of the analogy to classical mechanics, the solution of such problems required the identification of sets of canonically conjugate variables, i.e., p’s and q’s. Jordan ran into serious difficulties when he tried to extend his approach from quantities with fully continuous spectra to those with wholly or partly discrete spectra. For von Neumann, not constrained by the analogy to classical physics and aware of the daunting mathematical difficulties facing the approach of Jordan ), the solution of a problem in the new quantum mechanics required only the identification of a maximal set of commuting operators with simultaneous eigenstates. He had no need for p’s and q’s. Related to their disagreement about the appropriate general formalism for the new theory, Jordan and von Neumann stated the characteristic new rules for probabilities in quantum mechanics somewhat differently. Jordan was the first to state those rules in full generality, von Neumann rephrased them and then sought to derive them from more basic considerations. In this paper we reconstruct the central arguments of these 1927 papers by Jordan and von Neumann and of a paper on Jordan’s approach by Hilbert, von Neumann, and Nordheim. We highlight those elements in these papers that bring out the gradual loosening of the ties between the new quantum formalism and classical mechanics. (shrink)