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"The theory ... noncommutative geometry ... rests on two essential points:"
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Noncommutative geometryN
Noncommutative geometry
Noncommutative geometry
Noncommutative geometry
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Noncommutative geometry (NCG) is a branch of mathematics that studies geometric ideas through noncommutative algebras. In ordinary geometry, a space can often be studied by means of a commutative algebra of functions on it; noncommutative geometry extends this viewpoint to algebras in which the product of two elements need not commute. Such algebras are treated as analogues of algebras of function
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"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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Noncommutative geometry
