Abstract
This paper begins the study of the relation between causality and quantum mechanics, taking advantage of the groupoidal description of quantum mechanical systems inspired by Schwinger’s picture of quantum mechanics. After identifying causal structures on groupoids with a particular class of subcategories, called causal categories accordingly, it will be shown that causal structures can be recovered from a particular class of non-selfadjoint class of algebras, known as triangular operator algebras, contained in the von Neumann algebra of the groupoid of the quantum system. As a consequence of this, Sorkin’s incidence theorem will be proved and some illustrative examples will be discussed.
Highlights
Causal Structures vs. Quantum MechanicsEntropy 2022, 24, 75. https://doi.org/The principle of causation, ”cause precedes effect” or ”every effect has a cause”, is the bedrock of modern science, and beyond it, is magic
Over Ω satisfying some natural properties. They reproduce the vast majority of various approaches to causality introduced before: from Einstein’s geometric causality to Sorkin’s causal sets theories
They will allow to place them in the context of quantum mechanical systems by using their Schwinger’s inspired groupoidal picture
Summary
The principle of causation, ”cause precedes effect” or ”every effect has a cause”, is the bedrock of modern science, and beyond it, is magic. Rovelli introducing the notion of a thermodynamic time in the description of quantum systems in a general covariant setting, the so called Connes–Rovelli thermodynamic time hypothesis [31] Such hypothesis considers that the system is described by a certain von Neumann algebra of observables and that the dynamics provided by the Tomita–Takesaki modular flow associated to a given reference state provides the natural choice for an arrow of time. An important observation regarding the full program is that, in order to incorporate both the mathematical technical tools and the physical background ideas, it is necessary to extend the theory of causal relations from its standard topological/differentiable setting to a measure theoretical one
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