Consequences of Deformation of the Heisenberg Algebra

In this paper we will demonstrate that like the existence of a minimum measurable length, the existence of a maximum measurable momentum, also influence all quantum mechanical systems. Beyond the simple one dimensional case, the existence of a maximum momentum will induce non-local corrections to the first quantized Hamiltonian. However, these non-local corrections can be effectively treated as local corrections by using the theory of harmonic extensions of functions. We will also analyses the second quantization of this deformed first quantized theory. Finally, we will analyses the gauge symmetry corresponding to this deformed theory.