The quon algebra is an approach to particle statistics introduced by Greenberg to provide a theory in which the Pauli exclusion principle and Bose statistics are violated by a small amount. We generalize these models by introducing a deformation of the quon algebra generated by a collection of operators \(\mathtt {a}_i\), \(i \in {\mathbb {N}}^*\) the set of positive integers, on an infinite dimensional module satisfying the \(q_{i,j}\)-mutator relations \(\mathtt {a}_i \mathtt {a}_j^{\dag } - q_{i,j}\, \mathtt {a}_j^{\dag } \mathtt {a}_i = \delta _{i,j}\). The realizability of our model is proved by means of the Aguiar-Mahajan bilinear form on the chambers of hyperplane arrangements. We show that, for suitable values of \(q_{i,j}\), the module generated by the particle states obtained by applying combinations of \(\mathtt {a}_i\)’s and \(\mathtt {a}_i^{\dag }\)’s to a vacuum state \(|0\rangle \) is an indefinite Hilbert module. Furthermore, we recover the extended Zagier’s conjecture established independently by Meljanac et al., and by Duchamp et al.

Quon代数是Greenberg引入的一种用于粒子统计的方法，旨在提供一种理论，其中少量违反了Pauli排除原理和Bose统计。我们通过引入由正整数集运算符\（\ mathtt {a} _i \），\（i \ in {\ mathbb {N}} ^ * \）生成的Quon代数的变形来概括这些模型，在满足\（q_ {i，j} \）-变量关系\（\ mathtt {a} _i \ mathtt {a} _j ^ {\ dag}-q_ {i，j} \，\ mathtt {a} _j ^ {\ dag} \ mathtt {a} _i = \ delta _ {i，j} \）。我们的模型的可实现性是通过超平面布置腔室上的Aguiar-Mahajan双线性形式证明的。我们表明，对于\（q_ {i，j} \），由粒子状态生成的模块，该粒子状态是通过应用\（\ mathtt {a} _i \）和\（\ mathtt {a} _i ^ {\ dag} \ ）到真空状态\（| 0 \ rangle \）是不确定的Hilbert模块。此外，我们恢复了由Meljanac等人和Duchamp等人独立建立的扩展Zagier猜想。