Realizability of a model in infinite statistics.

*(English)*Zbl 0789.47042Summary: 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) |

##### Keywords:

\(q\)-mutator relations; Bose and Fermi statistics; commutator relations have a Hilbert space representation; explicit factorization
PDF
BibTeX
XML
Cite

\textit{D. Zagier}, Commun. Math. Phys. 147, No. 1, 199--210 (1992; Zbl 0789.47042)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Greenberg, O. W.: Example of infinite statistics. Phys. Rev. Lett.64, 705–708 (1990) · Zbl 1050.81571 |

[2] | Biedenharn, L. C.: J. Phys.A22, L873-L878 (1989) · Zbl 0708.17015 |

[3] | Fivel, D.: Interpolation between Fermi and Bose statistics using generalized commutators. Phys. Rev. Lett.65, 3361–3364 (1990) · Zbl 1050.81567 |

[4] | Bo\.zejko, M., Speicher, R.: Commun. Math. Phys.137, 519–531 (1991) · Zbl 0722.60033 |

[5] | Greenberg, O. W.: Q-mutators and violations of statistics. Argonne Workshop on Quantum Groups. T. Curtright, D. Fairlie, C. Zachos (eds.) Singapore: World Scientific, pp. 166–180 (1991) · Zbl 0815.17027 |

[6] | Greenberg, O. W.: Particles with small violations of Fermi or Bose statistics. Phys. Rev. D43, 4111–4120 (1991) |

[7] | Greenberg, O. W.: Interactions of particles having small violations of statistics. Physica A (to appear) |

[8] | Freund, P. G. O., Nambu, Y.: Private communication (1989) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.