The God has created a man in order that he creates that the God fails to do

Sunday, 31 July 2011

Geometry in quantum theory IV: Modern geometries

Contemporary quantum models appeal to a number of new algebraic structures and the associated geometric techniques. Let us mention the following ones.

(i) Supergeometry of graded manifolds and different types of superrmanifolds (Archive).

(ii) Non-commutative geometry and non-commutative field theory (Archive).

(iii) Hopf algebras, including quantum groups, and, in particular, two types of quantum bundles.

(iv) Formalism of groupoids and Lie groupoids, including quantization via groupoids.

(v) Finally, one of the main point of the formality theorem in deformation quantization is that, for any algebra A over a field of characteristic zero, its Hochschild cochain complex and its Hochschild cohomology are algebras over the same operad. This observation has been the starting point of 'operad renaissance'. Monoidal categories provide numerous examples of algebras for operads. Furthermore, homotopy monoidal categories lead to the notion of a homotopy monoidal algebra for an operad. In a general setting, one considers homotopy algebras and weakened algebraic structures where, e.g., a product operation is associative up to homotopy. At the same time, the formality theorem is also applied to quantization of several algebraic geometric structures such as algebraic varieties.

G.Giachetta, L.Msngiarotti, G.Sardanashvily, Geometric and Algebraic Topological Methods in Quantum Mechanics (WS, 2005)
G.Sardanashvily, G.Giachetta, What is geometry in quantum theory, arXiv: hep-th/0401080

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