Abstract

We propose to revisit the problem of guantisation and look at it from an entirely new analefocusing on the guantisation of dynamical systems themselves, rather than of their Poissonstructures. We begin with a dynamical system defined on a free associative algebra fA generatedby non-commutative dynamical variables al, ac2,..., and reduce the problem of guantisation to theproblem of studying two-sided guantisation ideals. The dynamical system defines a derivation ofthe algebra 'p : fAmfA. By definition, a two-sided ideal 'cI of 'fA is said to be alemphfquantisation ideal} for (\fA, p) if it satisfies the following two properties:1. The ideal \cl is \p-stable: p(\cI) C\cI; 2. The quotient \fA/\cI admits a basis of normallyordered monomials in the dynamical variables. The multiplication rules in the quantum algebrafA/\cI are manifestly associative and consistent with the dynamics. We found first examples ofbi-guantum systems which are quantum counterparts of bi-Hamiltonian systems in the classicatheory. Moreover, the new approach enables us to define and present first examples of non.deformation guantisations of dynamical systems, i.e. quanum systems that cannot be obtained asdeformations of a classical dynamical system with commutative variables. in order to apply thenovel approach to a classical system we need firstly lift it to a system on a free algebra preservingthe most valuable properties, such as symmetries, conservation laws, or Lax integrability. The newapproach sheds light on the long standing problem of operator's ordering. We will use the wellknown Volterra hierarchy and stationary KdV equations to illustrate the methodology.

References:

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