We present a generalization of the quantum mechanical formalism to a class of propagators that is closely related to the generalized Lorentzian path integrals we proposed in an earlier publication [R. A. Treumann, Phys. Rev. E 57, 5150 (1998)] introducing a (real) control parameter k > 0. In this formulation the SchrÎdinger and Heisenberg formalisms assume a de¢nite form when introducing a new k-Hamilton operator that is a functional of the quantum Hamiltonian operator. For k 3 I the usual quantum mechanics is recovered. This approach though apparently successful for constant time su¡ers from violating the semi-group property of quantum operators. Currently we do not see how to cure this de¢ciency.We nevertheless present the idea to the public in the hope that a way may be found to handle this problem in the time domain. The formalism is formally applied to the harmonic oscillator. We de¢ne new creation and destruction operators, ¢nd their commutators, and calculate the eigenvalues of the new k-Hamilton operator. These eigenvalues turn out to become time-dependent, whatever this means. In addition the energy levels are not anymore equally spaced. It is believed that the theory may be of use in application to quantum chaos and quantum turbulence. Physica Scripta 66 # Physica Scripta 2002 418 Rudolf A. Treumann