The Homotopy Operator and Conservation Laws in Fractional Quantum Mechanics

Continuity equations are local conservation laws that describe the evolution of physically important quantities, such as mass, energy, and momentum, in terms of fluxes. For fractional evolution equations (FEEs), partial differential equations that are first order in time and have fractional derivatives in space, the fractional homotopy and correction operators allow the derivation of conservation laws. Conservation laws for FEEs are quasi-continuity equations of the form d_tρ = d_xj + k where the source term, k, directly manifests the non-locality of space fractional derivatives and shows that fractional derivatives force local conservation laws to become global. This is a direct result from the fractional homotopy and correction operators which give explicit form to the quasi-continuity equation for a given density. We applied these methods to describe probability and momentum transport for the space fractional Schrodinger equation which is important in quantum information processing because it describes the non-locality of connected systems. These examples reveal artifacts of the generalization of quantum mechanics to fractional quantum mechanics.