The Geometry of Deformed Boson Algebras

Phase-space realisations of an infinite parameter family of quantum deformations of the boson algebra in which the q– and the qp–deformed algebras arise as special cases are studied. Quantum and classical models for the corresponding deformed oscillators are provided. The deformation parameters are identified with coefficients of non-linear terms in the normal forms expansion of a family of classical Hamiltonian systems. These quantum deformations are trivial in the sense that they correspond to non-unitary transformations of the Weyl algebra. They are non-trivial in the sense that the deformed commutators consistently quantise a class of non-canonical classical Poisson structures. PACS: Numbers 02.90.+p, 02.40.-k, 03.65.-w This work is dedicated to John Kennedy my former teacher who died of cancer 20th of January 1995