Quantisation of free associative dynamical systems

Traditional quantisation theories start with classical Hamiltonian systems with variables taking values in commutative algebras and then study their non-commutative deformations, such that the commutators of observables tend to the corresponding Poisson brackets as the (Planck) constant of deformation goes to zero. I am proposing to depart from dynamical systems defined on a free associative algebra $\mathfrak{A}$. In this approach the quantisation problem is reduced to the problem of finding of a two-sided ideal $\mathfrak{J}\subset\mathfrak{A}$ satisfying two conditions: the ideal $\mathfrak{J}$ has to be invariant with respect to the dynamics of the system and to define a complete set of commutation relations in the quotient algebras $\mathfrak{A}_{\mathfrak{J}}=\mathfrak{A}\diagup\mathfrak{J}$ [AvM20].
To illustrate this approach I'll consider the quantisation problem for the Volterra family of integrable systems. In particular, I will show that odd degree symmetries of the Volterra chain admit two quantisations, one of them is a standard deformation quantisation of the Volterra chain, and another one is new and not a deformation quantisation. The periodic Volterra chain admits bi-Hamiltonian and bi-quantum structures [CMW22]. The method of quantisation based on the concept of quantisation ideals proved to be successful for quatisation of stationary Korteveg-de–Vries hierarchies [BM2021]. The Toda hierarchy also admits bi-quantum structures and non-deformation quantisation.