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arXiv:2410.09535

Tensorial Quantum Mechanics: Back to Heisenberg and Beyond

Christian DE RONDE, Raimundo Fernández Mouján, César Massri

Published Oct 12, 2024• 0 citations
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Abstract

In this work we discuss the establishment of Standard Quantum Mechanics (SQM) developed through Schrödinger's and Dirac's wave-vectorial reformulations of Heisenberg's original matrix mechanics. We will argue that while Heisenberg's approach was consistently developed -- taking as a standpoint the intensive patterns that were observed in the lab -- as an invariant-operational formalism, Dirac's axiomatic re-formulation was, instead, developed -- taking as a standpoint Schrödinger's wave mechanics and the methodological guide of Bohr and logical positivists -- as an essentially inconsistent "recipe" intended (but unable) to predict (binary) measurement outcomes. Leaving SQM behind and attempting to restore the consistent and coherent account of a real state of affairs, we will present a new tensorial proposal which -- taking as a standpoint Heisenberg's original approach -- will prove capable not only to extend the matrix formalism to a tensorial representation but also to account for new experimental phenomena.

DOI
10.48550/arxiv.2410.09535

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In this work we discuss the establishment of Standard Quantum Mechanics (SQM) developed through Schrödinger's and Dirac's wave-vectorial reformulations of Heisenberg's original matrix mechanics. We will argue that while Heisenberg's approach was consistently developed -- taking as a standpoint the intensive patterns that were observed in the lab -- as an invariant-operational formalism, Dirac's axiomatic re-formulation was, instead, developed -- taking as a standpoint Schrödinger's wave mechanics and the methodological guide of Bohr and logical positivists -- as an essentially inconsistent "recipe" intended (but unable) to predict (binary) measurement outcomes. Leaving SQM behind and attempting to restore the consistent and coherent account of a real state of affairs, we will present a new tensorial proposal which -- taking as a standpoint Heisenberg's original approach -- will prove capable not only to extend the matrix formalism to a tensorial representation but also to account for new experimental phenomena.
In July 1925, exactly one century ago, Werner Heisenberg created Quantum Mechanics (QM) as an invariant-operational formalism by taking as a standpoint the intensive patterns that had been already observed by experimentalists in the lab. One of the few drawbacks of this proposal was the impossibility to conceptualize it in classical terms. Attempting to restore a somewhat classical spatiotemporal representation, six months later, Erwin Schrödinger would present a formalism grounded on a wave equation in configuration space. Taking as a standpoint Schrödinger’s wave mechanics together with the methodological guide of Bohr and logical positivists, Dirac would present in 1930 an axiomatic re-formulation of the theory of quanta that would become known as the “standard” version of QM (SQM). This new version of the theory, that is still today taught in physics classrooms all around the world, would become accepted regardless of its inconsistent narrative as a “recipe” intended (but unable) to predict (binary) measurement outcomes. After pointing out some of the many problems of SQM, in this work we propose to go back to Heisenberg’s matrix mechanics (and also to his understanding of physical theories). We will argue that his matrix mechanics, as an operationally-invariant standpoint, opens the door to a tensorial enlargement of the mathematical formalism capable to account for new phenomena.
In this work we present an invariant-objective formalization of multi screen-entanglement grounded on Tensorial Quantum Mechanics (TQM) [12]. This new tensorial formulation of the theory of quanta -- basically, an extension of Heisenberg's matrix mechanics -- allows not only to escape the many problems present in the current account of multi-partite entanglement grounded on the Dirac-Von Neumann Standard formulation of Quantum Mechanics (SQM) but, more importantly, to consistently represent entanglement phenomena when considering a multiplicity of different screens and detectors.
Quantum Mechanics (QM) has faced deep controversies and debates since its origin when Werner Heisenberg proposed the first mathematical formalism capable to operationally account for what had been recently discovered as the new field of quantum phenomena. Today, even though we have reached a standardized version of QM which is taught in Universities all around the world, there is still no consensus regarding the conceptual reference of the theory and, if or if not, it can refer to something beyond measurement outcomes. In this work we will argue that the reason behind the impossibility to reach a meaningful answer to this question is strictly related to the 20th Century Bohrian-positivist re-foundation of physics which is responsible for having introduced within the theory of quanta a harmful combination of metaphysical dogmatism and naive empiricism. We will also argue that the possibility of understanding QM is at plain sight, given we return to the original framework of physics in which the meaning of understanding has always been clear.
This is a brief review of the history and development of quantum theories. Starting from the experimental findings and theoretical results which marked the crisis of the classical framework, I overview the rise of axiomatic quantum mechanics through matrix and wave mechanics. I discuss conceptual problems such as the measurement problem that led scientific realists to explore other, more satisfactory, quantum theories, as well as Bell’s theorem and quantum nonlocality, concluding with a short review of relativistic theories.
Analysis of the initial stages of the logical process followed by Louis de Broglie in establishing the electron phase wave equation in his 1924 thesis, which triggered the development of Wave Mechanics when Erwin Schrödinger formalized this concept with his vectorial wave equation. This development was soon followed by Quantum Mechanics, when Schrödinger proved that the Matrix Mechanics independently developed by Werner Heisenberg was equivalent to Wave Mechanics, with both theories leaving room for some degree of uncertainty as to the physical localization of the moving electron. This is what led Heisenberg to also formalize the Uncertainty Principle to take this situation into account. This principle was soon regarded as a fundamental axiomatic principle that seemed to make further exploration of the subatomic level of magnitude appear impossible to most researchers. We will analyze in this article the reason why the phase-wave velocity established by de Broglie generated this uncertainty in the localization of the moving electron in light of the current state of knowledge on the behavior of the electron in motion, in view of establishing the relevance of maintaining the Uncertainty Principle in the study of the subatomic level of magnitude.
In a recent paper [12], we discussed the serious inconsistency present within the operational and mathematical definition(s) of the notion of pure state. Continuing this analysis, in this work we attempt to address the role of 'purity' and 'mixtures' within two different categorical approaches to QM, namely, the topos approach originally presented by Chris Isham and Jeremy Butterfield [27, 28, 29] and the more recent logos categorical approach presented by the authors of this article [10, 11, 13]. While the first approach exposes the difficulties to produce a consistent understanding of pure states and mixtures, the latter approach presents a new scheme in which their reference is erased right from the start in favor of an intensive understanding of projection operators and quantum superpositions. This new account of the theory, grounded on an intensive interpretation of the Born rule, allows us not only to avoid the orthodox interpretation of projection operators --either as referring to definite valued properties or measurement outcomes-- but also to consider all matrices (of any rank) on equal footing. It is from this latter standpoint that we conclude that instead of distinguishing between pure and mixed states it would be recommendable --for a proper understanding of the theory of quanta-- to return to the original matrix formulation of quantum mechanics presented by Werner Heisenberg in 1925.