  In this paper, the deformation of the Heisenberg algebra, consistent with
both the generalized uncertainty principle and doubly special relativity, has
been analyzed. It has been observed that, though this algebra can give rise to
fractional derivative terms in the corresponding quantum mechanical
Hamiltonian, a formal meaning can be given to them by using the theory of
harmonic extensions of function. Depending on this argument, the expression of
the propagator of the path integral corresponding to the deformed Heisenberg
algebra, has been obtained. In particular, the consistent expression of the one
dimensional free particle propagator has been evaluated explicitly. With this
propagator in hand, it has been shown that, even in free particle case, normal
generalized uncertainty principle and doubly special relativity shows very much
different result.

High Energy Physics - Theory