Abstract
Abstract The prototypical example of a quantum mechanical system is the spin 1/2 Hamiltonian, which describes a vast class of two level systems, including the well known Zeeman interaction between a spin 1/2 particle and a magnetic field. As a pure quantum mechanical system, it conserves probability. In the present contribution a simple extension to this problem is put forward, by taking the α-th fractional power of the spin-1/2 Hamiltonian, leading to a dissipative system. It can be used to describe a vast class of quantum mechanical problems displaying relaxation, like a spin 1/2 in the presence of a thermal bath, or the asymmetry in energy spectrum between electrons and holes in a semiconductor. The mean-field theory of a fractional magnet is explored and a fractional version of Dirac's equation in 1 + 2 dimensions is also examined.
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