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Final Answers
© 2000-2023 Gérard P. Michon, Ph.D.

Connes' Geometry
Noncommutative Geometry

The use of noncommutative geometry (NCG)
as a tool for constructing particle physics models
originated in the 1990s. Matilde Marcolli (2017)

Michon

Videos :

Noncommutative geometry and particle physics (8:24) by Kevin McSherry (Radboud University, 2016年04月12日).
A rapid tour through NCG (1:02:23) by Nigel Higson (ESI, 2019年02月26日).

Videos in French :

Non-commutativité, moteur du temps (6:46) Etienne Klein (2014年05月21日).
Discours introductif aux travaux d'Alain Connes (19:01) Etienne Klein (2018).
Géométrie non-commutative (23:02, 15:15) by J-P. Luminet (2020年05月09日/11).

Videos of Alain Connes :

Non-commutative geometry (53:54) Visions in Mathematics (1999年08月26日).
Interview (1:05:59) by Stéphane Dugowson & Anatole Khélif (2014年02月05日).
Face à la réalité mathématique (7:03) Collège de France (2014).
Quanta of Geometry (1:38:01) ESI (2015年03月10日).
The Arithmetic Site (59:04, 54:14) ESI (2015年03月11日).
Quantum Emergence of Time (58:37) at IHES (2015年04月09日).
Pensée en mouvement (French, 1:55:02) Université PSL (2015年11月12日).
Géométrie non-commutative & physique (1:18:17) Guillaume Faye, IAP (2015).
Why 4 dimensions? QG & NCG (1:54:49) at IHES (2017年10月24日/27).
Entropy and the spectral action (51:09) at IHES (2017年12月24日).
Parcours d'un mathématicien (1:25:10) SAPT (2018年12月17日).
On the Fine-Structure of Space-Time (1:03:36) at IHES (2019年02月27日).

Noncommutative geometry

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On Alain Connes' Noncommutative Geometry

"It is quite conceivable that the metric relations of space in the infinitely small do not conform to the hypotheses of geometry." Bernhard Riemann (1854)

"Le point matériel était une abstraction mathématique dont nous avions pris l'habitude et à laquelle nous avions fini par attribuer une réalité physique. C'est encore une illusion que nous devons abandonner..." Elie Cartan (1931)

David Hilbert David Hilbert (2020年05月12日) Hilbert spaces (David Hilbert, 1909)

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Johnny von Neumann Johnny von Neumann Von Neumann's coat-of-arms (2020年05月03日) Von Neumann Algebras (1929) At first, rings of operators on Hilbert spaces.

When Dirac first formalized quantum theory, he posited that the ultimate state of reality was a vector belonging to an abstract ad hoc Hilbert space called the space of kets (or, equivalently, the space of the bra covectors).

However, a ket isn't directly accessible. All we can do is apply to it an operator associated to an observable physical quantity. Doing so transforms the ket into an eigenvector of that operator, whose associated eigenvalue is construed to be the result of a measurement (it's always a real quantity if we only use hermitian operators, henceforth called observables, for short).

The original motivation was to understand how hermitian operators (quantum observables) act on a system composed of several subsytems (call that entanglement if you must).

C* Algebras (Gelfand & Naimark, 1943) :

By definition, a C* algebra (pronounced "C star") is a Banach algebra (i.e., a Banach space endowed with the added structure of an algebra) on which a conjugation is an involution extending the conjugation on the scalar field (using the same postfixed star "*" notation):

X** = X (i.e., conjugation is an involution).
(k X)* = k* X* º X* k* (antilinearity).
(X + Y)* = X* + Y* (additive homomorphism).
(X Y)* = Y* X* (multiplicative antihomomorphism).

Cyclic and separating vector | Gelfand-Naimark-Segal construction (GNS)

(2020年05月08日) Compact operators

A linear operator between normed spaces is continuous iff it's bounded.

A compact operator is a linear operator for which the image of any bounded subset is precompact (i.e., its closure is compact).

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Bounded operator | Compact operator | Finite-rank operator
Spectrum | Spectral theory of compact operators | Spectral theory of normal C*-algebras

Jacques Dixmier Jacques Dixmier (2020年05月15日) Dixmier Trace (1966)

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Connes' Dictionary

Space X Algebra A

Real Variable x^m Self-adjoint Operator

Infinitesimal form dx Compact Operator e

Integral ò e = Coefficient of
Log L in Tr_L(e)

Infinitesimal displacement
(g_mn dx^m dxⁿ )^½ Fermion propagator D^-1

Dixmier trace | Dixmier mapping | Jacques Dixmier (1924-)

[画像: Minoru Tomita ] Minoru Tomita (2020年05月03日) Tomita-Takesaki theory (1967) Introduced by Minoru Tomita (1924-2015).

Tomita (1924-2015) had been hearing-impaired since the age of 2 and his theory remained obscure until it was exposed in a 1970 book based on lecture notes compiled by his student Masamichi Takesaki (1933-).

Masamichi Takesaki
Masamichi Takesaki

By sheer luck, young Alain Connes (still utterly ignorant of the subject) bought a copy form the Princeton bookstore to occupy a five-day train journey to a conference in Seattle. Not knowing that Takesaki would be one of the speakers. Connes chose to attend all the lectures given by Takesaki... That launched his career with work leading to his Fields medal in 1982.