Self-distributive structures in physics
Tobias Fritz
TL;DR
Observables in physics generate one-parameter transformation groups and relate to conserved quantities via Noether's theorem; the paper introduces Lie quandles as a nonlinear, minimal framework to formalize this correspondence. A Lie quandle is defined as a smooth manifold $Q$ with a family of operations $x \triangleright_t y$ satisfying Self-action, Self-distributivity, and Idempotency, with canonical quantum examples such as $Q = \{A = A^\dagger\}$ and $X \triangleright_t Y = e^{itX} Y e^{-itX}$ linking to Heisenberg-type dynamics via $\frac{d}{dt}(X \triangleright_t Y) = [X, X \triangleright_t Y]$, and with every real Lie algebra embedding as a Lie quandle via $X \triangleright_t Y = \mathrm{Ad}(e^{tX}) Y$. Nonlinear instances like the Bloch quandle on $S^2$ and Noether-quandles illustrate physics-relevant examples and connect the quandle formalism to Noether-type relations where symmetry and conservation cohere in the quandle structure. The work suggests a new geometric-algebraic lens for generalizing Hamiltonian and quantum dynamics and motivates further study of the interplay between observables, state mixtures, and self-distributive operations.
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.