Causality in Schwinger's Picture of Quantum Mechanics

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.

Keywords: causal categories; causal sets; causality; groupoids; incidence algebras; triangular algebras; von Neumann algebras.

Figure 1

Diagram representing a Minkowski strip space $\mathbb{M}(a,b)$ (in blue) as a subspace of Minkowski space. The causal cone $C\left(u\right)$ of an event u is marked in orange (right). Two events $x,y$ not causally related can be joined by a seesaw path $(x, z_1, z_2, z_3, z_4, z_5, z_6, z_7, z_8, z_9, z_{10}, z_{11}, y)$, consisting of causal geodesics (dark blue) contained in $\mathbb{M}(a, b)$. Note that the set of points $J^{-}\left(x\right)\cap J^{-}\left(y\right)$ in the common causal past of $x, y$, is the causal past $J^{-}\left(z\right)$ of z (in red) which is out of the Minkowski strip.