Quantum Mechanics: Knocking at the Gates of Mathematical Foundations

Abstract

The Weltanschauung emerging from quantum theory clashes profoundly with our classical concepts. Quantum characteristics like superposition, entanglement, wave-particle duality, nonlocality, contextuality are difficult to reconcile with our everyday intuition. In this article I survey some aspects of quantum foundations and discuss intriguing connections with the foundations of mathematics. I. WHAT IS THE PROBLEM? (IS THERE A PROBLEM?)From its very inception quantum mechanics generated a fierce debate regarding the meaning of the mathematical formalism and the world view it provides [1][2][3][4]. The new quantum Weltanschauung is characterized, on the one hand, by novel concepts like wave-particle duality, complementarity, superposition and entanglement, and on the other by the rejection of classical ideas such as realism, locality, causality and non-contextuality. For instance, quantum correlations with no causal order [5] challenge Reichenbach's principle of common cause [6].The disquieting feeling one has at the contact with quantum theory was echoed by several physicists, including the founding fathers: "Anyone who is not shocked by quantum mechanics has not understood it" (Bohr); "I think I can safely say that nobody understands quantum mechanics" (Feynman). Consequently, there is an unsolved tension between what we predict and what we understand [7,8]. Although the predictions of quantum mechanics are by far and away unmatched (in terms of precision) by any other theory, understanding "what-allthis-means" is lacking. Briefly, we would like to have a story behind the data, to understand the meaning of the formalism.There is a wide spectrum of positions concerning the problem of quantum foundations [9], with attitudes ranging from "there is no problem, don't waste my time" to "we do have a problem and we don't know how to solve it":Actually quantum mechanics provides a complete and adequate description of the observed physical phenomena on the atomic scale. What else can one wish? [10] Quantum theory is based on a clear mathematical apparatus, has enormous significance for the natural sciences, enjoys phenomenal predictive success, and plays a critical role in modern technological developments. Yet, nearly 90 years after the theory's development, there is still no consensus in the scientific community regarding the interpretation of the theory's foundational building blocks.[11]The confusion around the meaning of the formalism and the lack of an adequate solution to the measurement problem (among others) resulted in a plethora of interpretations. Apart from the (once) dominant Copenhagen interpretationinformally known as "shut-up-and-calculate" -there are numerous others: pilot wave (de Broglie-Bohm), many-worlds Historically, the tone of the discussion was set by the BohrEinstein debate on the foundations of quantum theory [2][3][4]. Einstein lost the conceptual battle due to his persistence to understand quantum phenomena in classical terms like realism, locality and causality. In this respect Einstein was ...

Schrödinger's Cat: Where Does The Entanglement Come From?

“…The (infamous) measurement problem is masterfully discussed in John Bell's (less famous) article entitled Against 'measurement' [11]. Here we did not attempt to solve the measurement problem, which would need a (presently missing) quantum ontology [12]. Our goal was more modest: we showed that, in the usual description of the Schrödinger's cat paradox [1,2], we cannot entangle the cat and the atom via a classical channel, that is, a detector.…”