Abstract
A new extension of a fractality concept in quantum physics has been developed. The path integrals over the Lévy paths are defined and fractional quantum and statistical mechanics have been developed via new fractional path integrals approach. A fractional generalization of the Schrödinger equation has been found. The new relation between the energy and the momentum of non-relativistic fractional quantum-mechanical particle has been established. We have derived a free particle quantum-mechanical kernel using Fox's H-function. The equation for the fractional plane wave function has been obtained. As a physical application of the developed fQM we have proposed a new fractional approach to the QCD problem of quarkonium. A fractional generalization of the motion equation for the density matrix has been found. The density matrix of a free particle has been expressed in term of the Fox's H-function. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum and statistical mechanics.
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