Operator realizations of the q-deformed Heisenberg algebra

It is shown that the q-deformed Heisenberg algebra with q being a phase (q = e(l theta), with theta real) can be reduced to Jannussis' Lie-deformed He isenberg algebra, whose realizations with differential operators are given by Jannussis et al. On the other hand, difference operators satisfying the q-deformed Heisenberg algebra are well-known. In this Letter two different methods are given to show that these two realizations are really identical. On the basis of the above discussions, a possible physical interpretation of q-deformation is given. (C) 2001 Elsevier Science B.V. All rights reserv ed.