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Realizability of a model in infinite statistics. (English) Zbl 0789.47042
Summary: Following Greenberg and others, we study a space with a collection of operators $$a(k)$$ satisfying the “$$q$$-mutator relations” $$a(\ell)a^ †(k)-qa^ †(k)a(\ell)= \delta_{k,\ell}$$ (corresponding for $$q=\pm 1$$ to classical Bose and Fermi statistics). We show that the $$n!\times n!$$ matrix $$a_ n(q)$$ representing the scalar products of $$n$$- particle states is positive definite for all $$n$$ if $$q$$ 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 Bożejko and Speicher). We also give an explicit factorization of $$A_ n(q)$$ as a product of matrices of the form $$(1-q^ j T)^{\pm 1}$$ with $$1\leq j\leq n$$ and $$T$$ a permutation matrix. In particular, $$A_ n(q)$$ is singular if and only if $$q^ M=1$$ for some integer $$M$$ of the form $$k^ 2-2$$, $$2\leq k\leq n$$.

##### MSC:
 47N55 Applications of operator theory in statistical physics (MSC2000) 47B47 Commutators, derivations, elementary operators, etc. 82B05 Classical equilibrium statistical mechanics (general) 81S05 Commutation relations and statistics as related to quantum mechanics (general)
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##### References:
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