Fractional Heisenberg equation

Fractional derivative can be defined as a fractional power of derivative. The commutator (i/ℏ)[H,⋅](i/ℏ)[H,⋅], which is used in the Heisenberg equation, is a derivation on a set of observables. A derivation is a map that satisfies the Leibnitz rule. In this Letter, we consider a fractional derivative on a set of quantum observables as a fractional power of the commutator (i/ℏ)[H,⋅](i/ℏ)[H,⋅]. As a result, we obtain a fractional generalization of the Heisenberg equation. The fractional Heisenberg equation is exactly solved for the Hamiltonians of free particle and harmonic oscillator. The suggested Heisenberg equation generalize a notion of quantum Hamiltonian systems to describe quantum dissipative processes.