Symmetric Hilbert spaces arising from species of structures

Abstract

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.