We propose to revisit the problem of quantisation and look at it from an entirely new angle, focussing on quantisation of dynamical systems themself, rather than of their Poisson structures. We begin with a lift of a classical dynamical system to a system on a free associative algebra with non-commutative dynamical variables and reduce the problem of quantisation to the problem of studying of two-sided quantisation ideals, i.e. the ideals of the free algebra that define the commutation relations of the dynamical variables and are invariant with respect to the non-commutative dynamics. Quantum multiplication rules in the quotient algebra over a quantisation ideal are manifestly associative and consistent with the dynamics. We found first examples of bi-quantum systems which are quantum counterparts of bi-Hamiltonian systems in the classical theory. Moreover, the new approach enables us to define and present first examples of non-deformation quantisations of dynamical systems. The new approach also sheds light on the problem of operator's ordering.
References
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