标题： 从结构种类中产生的对称Hilbert空间 显示英文标题

标题： Symmetric Hilbert spaces arising from species of structures

摘要： 对称的希尔伯特空间，如某些“单粒子空间”$\K$上的玻色子和费米子福克空间，是通过对全福克空间进行某种对称化过程形成的。 我们通过借鉴乔伊尔的组合种类概念，研究了对称化的替代方法。 任何这样的种类$F$都会生成一个从希尔伯特空间范畴（带有压缩映射）到自身的自函子$\G_F$，它将一个希尔伯特空间$\K$映射到一个具有与种类$F$相同对称性的对称希尔伯特空间$\G_F(\K)$。 发展了一个在这些空间上湮灭和产生算子的一般框架，并与 R. 斯皮彻和 M. 波泽伊科的广义布朗运动进行了比较。 作为推论，我们发现当$f$具有无限收敛半径和正系数的幂级数时，$a_ia_j^*-a_j^*a_i=f(N)\delta_{ij}$与$Na_i^*-a_i^*N=a_i^*$的对易关系在对称希尔伯特空间上有一个实现。 显示英文摘要

摘要： Symmetric Hilbert spaces such as the bosonic and the fermionic Fock spaces over some `one particle space' $\K$ are formed by certain symmetrization procedures performed on the full Fock space. We investigate alternative ways of symmetrization by building on Joyal's notion of a combinatorial species. Any such species $F$ gives rise to an endofunctor $\G_F$ of the category of Hilbert spaces with contractions mapping a Hilbert space $\K$ to a symmetric Hilbert space $\G_F(\K)$ with the same symmetry as the species $F$. A general framework for annihilation and creation operators on these spaces is developed, and compared to the generalised Brownian motions of R. Speicher and M. Bo\.zejko. As a corollary we find that the commutation relation $a_ia_j^*-a_j^*a_i=f(N)\delta_{ij}$ with $Na_i^*-a_i^*N=a_i^*$ admits a realization on a symmetric Hilbert space whenever $f$ has a power series with infinite radius of convergence and positive coefficients.