Realizability of a model in infinite statistics
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Following Greenberg and others, we study a space with a collection of operatorsa(k) satisfying the “q-mutator relations”a(l)a†(k)a(l)=δ k,l (corresponding forq=±1 to classical Bose and Fermi statistics). We show that then!×n! matrixAn(q) representing the scalar products ofn-particle states is positive definite for alln ifq lies between −1 and +1, so that the commutator relations have a Hilbert space representation in this case (this has also been proved by Fivel and by Bozejko and Speicher). We also give an explicit factorization ofAn(q) as a product of matrices of the form(1−qjT)±1 with 1≦j≦n andT a permutation matrix. In particular,An(q) is singular if and only ifqM=1 for some integerM of the formk2−k, 2≦k≦n.
KeywordsNeural Network Statistical Physic Hilbert Space Complex System Nonlinear Dynamics
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