Generalized Uncertainty Relations in Quantum Mechanics and the Principles of Completeness in Physics

Focusing on the initial development of quantum mechanics, we will give a brief historical synopsis of the theory foundations, based on the Fourier framework and stating the philosophical conclusions inspired by that same mathematical formalism. We will then proceed, introducing an alternative way of describing the undulatory aspects of quantum entities, using local Gaussian Morlet wavelets. As we shall see, this change implies different philosophical interpretations about quantum reality and, even more, about the contemporary accepted differences between the quantum and the macroscopic realms. From these we will witness the formal and heuristic power of wavelet local analysis applied to the physical description of Nature. The ideas presented in this paper are initial standpoints of what can hopefully be expected to be a more mature and unifying physical theory, still undergoing development.