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<p>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 propose <i>Lie quandles</i> 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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10773_2025_5909_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="204" /> </InlineMediaObject> <EquationSource Format="TEX">\(x \triangleright (y \triangleright z) = (x \triangleright y) \triangleright (x \triangleright z)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>▹</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>▹</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>▹</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>▹</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>▹</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. 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.</p>

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Self-distributive Structures in Physics

  • 期刊论文

  • 发表日期：

    2025年3月7日

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  • Tobias Fritz

摘要

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 propose 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 \(x \triangleright (y \triangleright z) = (x \triangleright y) \triangleright (x \triangleright z)\) x ▹ ( y ▹ z ) = ( x ▹ y ) ▹ ( x ▹ 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.