Interesting Relations in Fock Space ⋆

Certain non-linear relations between the generators of the (qdeformed) Heisenberg algebra are found. Some of these relations are invariant under quantization and q-deformation. ⋆ Invited talk at International Conference on Complex and Hypercomplex Analysis, Mexico-city, Mexico (1994); to be published in Proceedings † On leave of absence from the Institute for Theoretical and Experimental Physics, Moscow 117259, Russia E-mail: [email protected] or [email protected] In this Talk I want to present different relations appearing in the Fock space generated by the q-deformed Heisenberg algebra. In a certain particular case, some of these relations can be summarized by the theorem: (main) THEOREM. The following differential identities hold for any natural number n and δ ∈ C ( x d dx − nx )n+1 = x d dxn+1 , ( d dx2 x− n d dx )n+1 = d dx2n+2 x , (1) ( x d dx x )n = x d dxn x , ( d dx x d dx )n = d dxn x d dxn , (2)