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Quantum Physics
Title: Self-distributive structures in physics
(Submitted on 21 Mar 2024 (v1), last revised 10 Apr 2024 (this version, v2))
Abstract: It is an important feature of our existing physical theories that observables generate one-parameter groups of transformations. In classical Hamiltonian mechanics and quantum mechanics, this is due to the fact that the observables form a Lie algebra, and it manifests itself in Noether's theorem. In this paper, we introduce Lie quandles as the minimal mathematical structure needed to express the idea that observables generate transformations. This is based on the notion of a quandle used most famously in knot theory, whose main defining property is the self-distributivity equation $x \triangleright (y \triangleright z) = (x \triangleright y) \triangleright (x \triangleright z)$. We argue that Lie quandles can be thought of as nonlinear generalizations of Lie algebras.
We also observe that taking convex combinations of points in vector spaces, which physically corresponds to mixing states, satisfies the same form of self-distributivity.
Submission history
From: Tobias Fritz [view email][v1] Thu, 21 Mar 2024 15:04:49 GMT (14kb)
[v2] Wed, 10 Apr 2024 10:02:02 GMT (16kb)
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