Symmetric Hilbert spaces arising from species of structures

Symmetric Hilbert spaces such as the bosonic and the fermionic Fock spaces over some `one particle space’ $\text{K}$ 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 ${\text{G}}_F$ of the category of Hilbert spaces with contractions mapping a Hilbert space $\text{K}$ to a symmetric Hilbert space ${\text{G}}_F(\text{K})$ 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 $a_i{a}_j^{\ast }-a_j^{\ast }{a}_i=f(N){\delta }_{ij}$ with $N{a}_i^{\ast }-a_i^{\ast }N=a_i^{\ast }$ admits a realization on a symmetric Hilbert space whenever $f$ has a power series with infinite radius of convergence and positive coefficients.