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Basics in Noncommutative Geometry, pp. 547560 
$100.00 

Authors: (L. Monreal and J.M. Isidro, Grupo de Modelizaci'on Interdisciplinar Intertech, Instituto Univ. de Matematica Pura y Aplicada, Univ. Politecnica de Valencia, Valencia, Spain)

Abstract: The notion of spacetime as a continuum has proved to be extremely useful. Many of the enormous advances made in physics and mathematics over the last centuries can be traced back to this concept, or to the notions derived from (and built upon) it. Soon after the birth of quantum mechanics, however, it was realised that three fundamental physical constants: the speed of light c, the quantum of action ¯h, and Newton’s constant G, could be combined into the expression pG¯h/c3, with the dimensions of length. It is customary to to call this quantity the Planck length and to denote by LP . It is widely believed that LP is to geometry what ¯h is to dynamics, namely, a fundamental quantum of length. By the same token, all lengths must be quantised in units of LP , and the notion itself of length must break down at scales comparable to, or smaller than, LP . Its value is extremely small on a macroscopic scale, 1.61×10−33 cm. Thus lengths (hence also areas, volumes, angles, etc) and ultimately spacetime itself appear to us, macroscopic observers, as continuous quantities that can one can zoom in as much as one wants. Current technology is still very far from approaching, even on a logarithmic scale, the energies required to probe such small distances. To all intents and purposes, spacetime is a continuum, at least at the energy scales attainable today. 











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