  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.
