Quantenlogische Systeme und Tensorproduktraeume
Tobias Starke
TL;DR
This work analyzes how logical descriptions of physical systems differ between classical and quantum theories, framing both within lattice-theoretic structures. It motivates Mackey’s quantum logic and the tensor-product description of joint quantum systems, contrasting it with classical Cartesian composition. The main contributions show that classical propositions form a complete, distributive Boolean lattice, while quantum propositions form a complete, orthomodular lattice lacking distributivity, and that composite quantum systems are naturally modeled by tensor products, not Cartesian products. The results underscore the role of tensor products in capturing entanglement and the correct state-space structure, with methodological groundwork for extending quantum logic to more general state spaces via advanced functional-analytic constructs.
Abstract
In this work we present an intuitive construction of the quantum logical axiomatic system provided by George Mackey. The goal of this work is a detailed discussion of the results from the paper 'Physical justification for using the tensor product to describe two quantum systems as one joint system' [1] published by Diederik Aerts and Ingrid Daubechies. This means that we want to show how certain composed physical systems from classical and quantum mechanics should be described logically. To reach this goal, we will, like in [1], discuss a special class of axiomatically defined composed physical systems. With the help of certain results from lattice and c-morphism theory (see [2] and [23]), we will present a detailed proof of the statement, that in the quantum mechanical case, a composed physical system must be described via a tensor product space.