Self-distributive structures in physics

Abstract: In all of our current physical theories, it is a central feature that observables generate 1-parameter groups of transformations. For example, a Hamiltonian generates time translations, while the angular momentum observable generates rotations. In this talk, I will explain how this property is captured algebraically by the new notion of Lie quandle. The central ingredient is a version of the self-distributivity equation $x\rhd(y\rhd z)=(x\rhd y)\rhd(x\rhd z)$. I will argue that Lie quandles can be thought of as nonlinear generalizations of Lie algebras. It is intriguing that not only the observables of physical theories form a Lie quandle; the same is true for the (mixed) states, where the Lie quandle structure is given by the formation of probabilistic mixtures.

machine learningcommutative algebraalgebraic geometryalgebraic topologycombinatoricscategory theoryoperator algebrasrings and algebrasrepresentation theory

Audience: researchers in the topic

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Algebraic and Combinatorial Perspectives in the Mathematical Sciences

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