Realizability of a model in infinite statistics
Abstract
Following Greenberg and others, we study a space with a collection of operators a(k) satisfying the “ q-mutator relations” a(l)a † (k)a(l)=δ k,l (corresponding for q=±1 to classical Bose and Fermi statistics). We show that the n!×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 Bozejko and Speicher). We also give an explicit factorization of A n (q) as a product of matrices of the form (1-q jT)±1 with 1≦ j≦ 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- k, 2≦ k≦ n.
- Publication:
-
Communications in Mathematical Physics
- Pub Date:
- June 1992
- DOI:
- 10.1007/BF02099535
- Bibcode:
- 1992CMaPh.147..199Z