Note about the quantum of area in a noncommutative space

Abstract

In this note we show that in a two-dimensional non-commutative space the area operator is quantized, this outcome is compared with the result obtained by Loop Quantum Gravity methods.PACS numbers: 02.40. Gh, 03.65.Bz One of the most relevant results of quantum mechanics is that physical quantities as the energy and the angular momenta, can have a discrete spectra. In fact, this theory begins with the Planck idea about the quantum nature of light. In contrast, general relativity modifies our understanding of gravity not as a force but as geometry. In recent years, there are several attempts to reconcile the ideas of quantum mechanics and general relativity. This implies in principle to change the geometrical properties of the space-time at the Planck length scales [1]. One of this attempt is the loop quantum gravity [2]. A remarkable result of this theory, is that geometrical operators as area or volume have discrete spectra at the Planck length order. For instance, one can define an operator for the area which has the spectrumwhere the {j i } have integers and half-integers values, l p = (hG/c) 1/2 = 10 −33 cm is the Planck length and γ is a constant analogous to the Immirzi 1

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