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16 pages, 2005

Abstract:

Quantum geoometry is a relatively new branch of Physical MAthematics that incorporates geometric and algebraic geometry ideas in Quantum theories. Quantum geometry deals with quantum spaces often derived from either non-commutative C*-algebras or von Neumann algebras (W*). Quantum geoometry is a relatively new branch of Physical Mathematics that incorporates geometric and algebraic geometry ideas in Quantum theories. Quantum geometry deals with quantum spaces often derived from either non-commutative C*-algebras or von Neumann algebras (W*). http://www.matem.unam.mx/~micho/papers.html From the author's Introduction: "Quantum geometry is a new branch of mathematics. It introduces a completely new concept of space, by unifying methods of classical geometry with non-commutative C*-algebras and functional analysis, and incorporating into geometry various ideas from quantum physics. Every geometry deals with some kind of spaces. Quantum geometry deals with quantum spaces, including the classical concept of space as a very special case. In classical geometry spaces are always understandable as collections of points equipped with the appropriate additional structure (as for example a topological structure given by the collection of open sets, or a smooth structure given by the atlas). In contrast to classical geometry, quantum spaces are not interpretable in this way. In general, quantum spaces have no points at all! They exhibit non-trivial `quantum fluctuations' of geometry at all scales. A very interesting potential application of quantum geometry in physics is to provide a mathematically coherent description of the physical space-time, at all scales---in particular at the level of ultra-small distances, characterizedby the Planck length. This length is a universal physical constant, defined as a unique combination of the gravitational constant G, Planck's constant, h, and the velocity of light" (in vacuum). http://www.matem.unam.mx/~micho/papers.html

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