Dual Algebraic Pairs and Polynomial Lie Algebras in Quantum Physics: Foundations and Geometric Aspects

Details Dual Algebraic Pairs and Polynomial Lie Algebras in Quantum Physics: Foundations and Geometric Aspects Dual Algebraic Pairs and Polynomial Lie Algebras in Quantum Physics: Foundations and Geometric Aspects

2003-04-17

arxiv.org/abs/quant-ph/0304118v1

Physics

Quantum Physics

10 pages, LATEX; submitted to Proceedings of the Workshop "Contemporary Geometry and Related Topics" (Belgrade, May 15-21, 200

Scientific paper

We discuss some aspects and examples of applications of dual algebraic pairs $({\cal G}_1,{\cal G}_2)$ in quantum many-body physics. They arise in models whose Hamiltonians $H$ have invariance groups $G_i$. Then one can take ${\cal G}_1 = G_i$ whereas another dual partner ${\cal G}_2= g^D$ is generated by $G_i$ invariants, possesses a Lie-algebraic structure and describes dynamic symmetry of models; herewith polynomial Lie algebras $\hat g = g^D$ appear in models with essentially nonlinear Hamiltonians. Such an approach leads to a geometrization of model kinematics and dynamics.

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