Noncommutative geometry

mathematical theory in which noncommutative algebraic structures are interpreted as rings of functions on an abstract space

In mathematics, noncommutative geometry is a geometric theory that establishes correspondences between noncommutative algebras and generalized geometric spaces that have local representations involving noncommutative algebras of functions or generalized functions.

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  • The theory ... noncommutative geometry ... rests on two essential points:
  1. The existence of many natural spaces for which the classical set-theoretic tools of analysis, such as measure theory, topology, calculus, and metric ideas lose their pertinences, but which correspond very naturally to a noncommutative algebra. ...
  2. The extension of the classical tools, such as measure theory, topology, differential calculus and Riemannian geometry, to the noncommutative situation. ...
  • The use of noncommutative geometry (NCG) as a tool for constructing particle physics models originated in the 1990s ... The main idea can be heuristically regarded as similar to the idea of "extra dimensions" in String Theory, except for the fact that the nature and scope of these extra dimensions is quite different. In the NGC model one considers an "almost commutative geometry", which is a product (or locally a product in a more refined and more recent version ...) of a four-dimensional spacetime manifold and a space of inner degrees of freedom, which is a "finite" noncommutative space, whose ring of functions is a sum of matrix algebras. According to the choice of this finite geometry, one obtains different possible particle contents for the resulting physics model. The physical content is expressed through an action functional, the spectral action ..., which is defined for more general noncommutative spaces, in terms of the spectrum of a Dirac operator.

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