“The Heisenberg Method”: Algebra, Geometry, and Probability in Quantum Mechanics

Arkady Plotnitsky

Arkady Plotnitsky

The article reconsiders quantum theory in terms of the following principle, which can be symbolically represented as QUANTUMNESS → PROBABILITY → ALGEBRA and will be referred to as the QPA principle. The principle states that the quantumness of physical phenomena, that is, the specific character of physical phenomena known as quantum, implies that our predictions concerning them are irreducibly probabilistic, even in dealing with quantum phenomena resulting from the elementary individual quantum behavior (such as that of elementary particles), which in turn implies that our theories concerning these phenomena are fundamentally algebraic, in contrast to more geometrical classical or relativistic theories, although these theories, too, have an algebraic component to them. It follows that one needs to find an algebraic scheme able make these predictions in a given quantum regime. Heisenberg was first to accomplish this in the case of quantum mechanics, as matrix mechanics, whose matrix character testified to his algebraic method, as Einstein characterized it. The article explores the implications of the Heisenberg method and of the QPA principle for quantum theory, and for the relationships between mathematics and physics there, from a nonrealist or, in terms of this article, “reality-without-realism” or RWR perspective, defining the RWR principle, thus joined to the QPA principle.

Thomas Vidick

Quantum mechanics and the theory of operator algebras have been intertwined since their origin.In the 1930s [20] von Neumann laid the foundations for the theory of (what are now known as) von Neumann algebras with the explicit goal of establishing Heisenberg's matrix mechanics on a rigorous footing (quoting from the preface, in the translation by Beyer: "The object of this book is to present the new quantum mechanics in a unified representation which, so far as it is possible and useful, is mathematically rigorous").Following the initial explorations of Murray and von Neumann, the new theory took on a life of its own, eventually leading to multiple applications unrelated to quantum mechanics, such as to free probability or noncommutative geometry.In his 1976 paper completing the classification of injective von Neumann algebras [6] Connes made a casual

Ovidiu L. Calin, Der‐Chen Chang, Peter Charles Greiner

Mohsen Razavy

Syed Muntasir Mamun

Building directly upon Volume I of this series, the present volume undertakes a rigorous mathematical dissection of the emergent von Neumann algebraic structure arising in the infinite-tensor-product (ITP) limit of the avalanche-photodiode model introduced by Van Den Bossche and Grangier (2025). We focus on the precise classification of the centre Z(M) of the type-III von Neumann algebra M generated by the limiting sector parameters. The centre is shown to be a non-trivial abelian von Neumann algebra, isomorphic to L∞ (Ω, µ) where Ω is the space of superselection sectors. We then explore natural embeddings of this centre into the framework of quantum information geometry, interpreting the sector parameters as coordinates on a statistical manifold equipped with the quantum Fisher information metric. Finally, we recast the entire construction within non-commutative probability theory, exhibiting a faithful conditional expectation onto the centre and discussing its implications for the modular automorphism group and the algebraic Heisenberg cut. All arguments remain faithful to the concrete APD dynamics while extending the original analysis into the rich territory of operator algebras, differential geometry on state spaces, and quantum probability. K 1 S Also available on Academia.edu: https://www.academia.edu/attachments/133072735/download_file?s=portfolio