Symmetric Hilbert spaces arising from species of structures

TL;DR

This paper explores new symmetric Hilbert spaces derived from combinatorial species, developing a framework for creation and annihilation operators, and characterizing when certain commutation relations can be realized on these spaces.

Contribution

It introduces a novel construction of symmetric Hilbert spaces from species, extending symmetrization procedures and analyzing operator realizations within this framework.

Findings

A general framework for creation and annihilation operators on these spaces.

Conditions for realizing specific commutation relations on symmetric Hilbert spaces.

Connection to generalized Brownian motions of Speicher and Bożejko.

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.