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Birkhäuser Boston eBooks

Dynamical Correspondences

Erik M. Alfsen, Frederic W. Shultz

Published Jan 1, 2003• 0 citations
Publisher

Abstract

In this chapter we will discuss the notion of a “dynamical correspondence”, which is applicable in both JB and JBW contexts. This generalizes the correspondence of observables and generators of one-parameter groups of automorphisms in quantum mechanics. It is closely related to Connes’ concept of orientation [36, Definition 4.11] that he used as a key property in his characterization of the natural self-dual cones associated with von Neumann algebras (i.e., the cones P ξ ♮ of Tomita—Takesaki theory). Both can be thought of as ways to specify possible Lie structures compatible with a given Jordan structure.

DOI
10.1007/978-1-4612-0019-2_6

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In this chapter we will discuss the notion of a “dynamical correspondence”, which is applicable in both JB and JBW contexts. This generalizes the correspondence of observables and generators of one-parameter groups of automorphisms in quantum mechanics. It is closely related to Connes’ concept of orientation [36, Definition 4.11] that he used as a key property in his characterization of the natural self-dual cones associated with von Neumann algebras (i.e., the cones P ξ ♮ of Tomita—Takesaki theory). Both can be thought of as ways to specify possible Lie structures compatible with a given Jordan structure.
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Five conceptually distinct notions of symmetry in quantum theory are studied in the algebraic setting where a quantum system is characterized by a von Neumann algebra of observables and the set of normal states on the algebra. It is shown that all five symmetry notions are closely related and that the glue binding them together is the concept of a Jordan ∗-automorphism. For factor algebras a Jordan ∗-automorphism reduces either to an ∗-automorphism or a ∗-anti-automorphism. If the algebra is put in standard form then a ∗-automorphism is always unitarily implementable, whereas a ∗-anti-automorphism is always anti-unitarily implementable. However, there is no guarantee that a general von Neumann algebra admits ∗-anti-automorphisms or, if it does, that it admits order two (or involutory) ∗-anti-automorphisms). For non-factor algebras there can be genuine Jordan ∗-automorphisms that are neither ∗-automorphisms nor ∗-anti-automorphisms, and implementation is possible only through partial isometries. These developments enable generalized versions of Wigner's theorem on the implementation of transition probability preserving symmetries for von Neumann algebras. This review is largely an exercise in connecting the dots in existing mathematics and physics literature. But in the service of the philosophy of physics it is an exercise worth doing since the practitioners in this field seem largely unaware of or unappreciative of this literature and how it fits together to yield a multifaceted but unified picture of quantum symmetries. Along the way various interpretations issues worthy of further discussion are flagged.
Physical theories can be characterized in terms of their state spaces and their evolutive equations. The kinematical structure and the dynamical structure of finite dimensional quantum theory are, in light of the Choi-Jamiołkowski isomorphism, one and the same --- namely the homogeneous self-dual cones of positive semi-definite linear endomorphisms on finite dimensional complex Hilbert spaces. From the perspective of category theory, these cones are the sets of morphisms in finite dimensional quantum theory as a dagger compact closed category. Understanding the intricate geometry of these cones and charting the wider landscape for their host category is imperative for foundational physics. In Part I of this thesis, we study the shape of finite dimensional quantum theory in terms of quantum information. In Part II of this thesis, we move beyond quantum theory within the vein of Euclidean Jordan algebras. In posting this thesis on the arXiv, we hope that it might serve as a useful resource for those interested in its subjects.
An important object appearing in the framework of the Tomita–Takesaki theory is an invariant cone under the modular automorphism group of von Neumann algebras. As a result of the connection between von Neumann algebras and quantum field theory, von Neumann algebras have become increasingly important for (higher) category theory and topology. We show explicitly how an example of a class of cones discovered by Connes–Araki–Haagerup (CAH), invariant under the modular automorphism group, are related to Wishart laws and information geometry. Given its relation to 2D quantum field theory this highlights new relations between (quantum) information geometry and quantum geometry. Additionally, our results have applications to qubits. The cones and the Wishart laws defined on them carry interesting monoidal structures, revealing the possibility of using a more abstract formalism of monoidal categories. Such a language is interesting as it opens a deeper possibility for generalizations of results.
In Lecture “Lecture 10: Derivations and Generators. K.M.S. Condition. Elements of Modular Structure. Standard Form” we have discussed some aspects of quantum dynamics when it is represented by a one-parameter group of automorphisms of a C∗C^*C∗ -algebra.