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Noncommutative differential geometry, quantum mechanics and gauge theory. (English) Zbl 0744.53042

Differential geometric methods in theoretical physics, Proc. 19th Int. Conf., Rapallo/Italy 1990, Lect. Notes Phys. 375, 13-24 (1991).
Summary: [For the entire collection see Zbl 0741.00044.]
We describe a noncommutative differential calculus, introduced in [the author, C. R. Acad. Sci., Paris, Sér. I 307, No. 8, 403-408 (1988; Zbl 0661.17012)], which generalizes the differential calculus of differential forms of E. Cartan. We show that besides the classical (commutative) situation, this differential calculus is well suited to deal with ordinary quantum mechanics. That is quantum mechanics falls in the framework of a noncommutative symplectic geometry. We then introduce the simplest corresponding gauge theories. We show that these theories describe ordinary gauge theories but with multivacua structures which provide a sort of alternative to the Higgs mechanism. Most of this lecture is based on a joint work with R. Kerner and J. Madore [J. Math. Phys. 31, No. 2, 316-322 (1990; Zbl 0704.53081); ibid., 323-330 (1990; Zbl 0704.53082); Phys. Lett. B 217, 485-488 (1989); Classical Quantum Gravity 6, No. 11, 1709-1724 (1989; Zbl 0675.53071)].

MSC:

53Z05 Applications of differential geometry to physics
81T13 Yang-Mills and other gauge theories in quantum field theory
58A30 Vector distributions (subbundles of the tangent bundles)
53C80 Applications of global differential geometry to the sciences
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