Comprimento mínimo em mecânica quântica via modificação da álgebra de Heisenberg

Abstract

Although gravity could not be quantized yet, there are good theoretical evidences that the unification of General Relativity and Quantum Mechanics should lead to the existence of a minimal observable length, a length scale below which the very notion of length looses meaning. Such effect proves to be worth some further analysis, for it acts as a natural regulator parameter, thus avoiding the divergences that plague Quantum Field Theories. In order to investigate some of the physical consequences of the existence of this peculiar effect, we propose to include it in the framework of Quantum Mechanics by modifying the Heisenberg algebra (i.e., the commutation relation of position and momentum operators, Xˆ and Pˆ), so that a minimum non-zero value for the uncertainty ?x emerges, which, being a limitation to the localizability of particles, acts as a minimal length. The Hilbert space of the theory must be modified accordingly. As we will see, the changes are not merely quantitative. On the contrary, our main result is that the familiar concept of “position measurement” must be reformulated, as well as other concepts related to it. We present a proposal for such reformulation.