Generalized quantum spins, coherent states, and Lieb inequalities
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A mathematical generalization of the concept of quantum spin is constructed in which the role of the symmetry groupO3 is replaced byO v (ν=2,3,4, ...). The notion of spin direction is replaced by a point on the manifold of oriented planes in ℝ v . The theory of coherent states is developed, and it is shown that the natural generalizations of Lieb's formulae connecting quantum spins and classical configuration space hold true. This leads to the Lieb inequalities  and with it to the limit theorems as the quantum spinl approaches infinity. The critical step in the proofs is the validity of the appropriate generalization of the Wigner-Eckart theorem.
KeywordsNeural Network Manifold Statistical Physic Complex System Nonlinear Dynamics
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