Wick order, spreadability and exchangeability for monotone commutation relations

TL;DR

This paper constructs a basis for a $*$-algebra related to monotone commutation relations and explores how spreadability and exchangeability of associated stochastic processes relate to algebraic and dynamical structures.

Contribution

It provides a basis for the algebra of monotone commutation relations and links spreadability and exchangeability of processes to algebraic actions and state symmetries.

Findings

Spreadability arises from a monoidal action inducing dissipative dynamics.

The structure of spreadable processes is characterized via spreading invariant states.

Exchangeable processes correspond to symmetric monotone states.

Abstract

We exhibit a Hamel basis for the concrete $*$-algebra $\mathfrak{M}_o$ associated to monotone commutation relations realised on the monotone Fock space, mainly composed by Wick ordered words of annihilators and creators. We apply such a result to investigate spreadability and exchangeability of the stochastic processes arising from such commutation relations. In particular, we show that spreadability comes from a monoidal action implementing a dissipative dynamics on the norm closure $C^*$-algebra $\mathfrak{M} = \overline{\mathfrak{M}_o}$. Moreover, we determine the structure of spreadable and exchangeable monotone stochastic processes using their correspondence with sp\-reading invariant and symmetric monotone states, respectively.