Symmetric Hilbert spaces arising from species of structures

Abstract.

Symmetric Hilbert spaces such as the bosonic and the fermionic Fock spaces over some lsquo;one particle space’${\mathcal K}$ 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 $\Gamma_F$ of the category of Hilbert spaces with contractions mapping a Hilbert space ${\mathcal K}$ to a symmetric Hilbert space $\Gamma_F({\mathcal K})$ 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. Bożejko. 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.