Wednesday, 27 January 2016
Wednesday, 20 January 2016
Monday, 11 January 2016
Studying gravitation theory, one conventionally requires that it incorporates Einstein’s General Relativity based on Relativity and Equivalence Principles reformulated in the fibre bundle terms.
Relativity Principle states that gauge symmetries of classical gravitation theory are general covariant transformations. Fibre bundles possessing general covariant transformations constitute the category of so called natural bundles. Let π: Y → X be a smooth fibre bundle. Any automorphism (Φ, f) of Y, by definition, is projected as π◦Φ = f ◦π onto a diffeomorphism f of its base X. The converse need not be true. A fibre bundle Y → X is called the natural bundle if there exists a group monomorphism of a group Diff(X) of diffeomorphisms of X to a group Aut(Y) of bundle automorphisms of Y → X. This functorial lift of Diff(X) to Aut(Y) are called general covariant transformations of Y.
Let us consider one-parameter groups of general covariant transformations and their infinitesimal generators. These are defined as the functorial lift T(u) of vector fields u on a base X onto Y so that the corresponding map T: V(X) → V(Y) of the Lie algebra V(X) of vector fields on X to the Lie algebra V(Y) of vector fields on a natural bundle Y is the Lie algebra morphism, i. e.,
The tangent bundle TX of X exemplifies a natural bundle. Any diffeomorphism f of X gives rise to the tangent automorphisms Tf of TX which is a general covariant transformation of TX. The associated principal bundle is a fibre bundle LX of linear frames in tangent spaces to X. It also is a natural bundle. Moreover, all fibre bundles associated to LX are natural bundles. For instance, tensor bundles are natural bundles.
Following Relativity Principle, one thus should develop gravitation theory as a field theory on natural bundles.
WikipediA: General covariant transformations.
G. Sardanashvily: Classical gauge gravitation theory, Int. J. Geom. Methods Mod. Phys. 8 (2011) 1869-1895 (#); arXiv: 1110.1176
Saturday, 2 January 2016
Our new article: G. Sardanashvily, A. Zamyatin, “Deformation quantization on jet manifolds” in arXiv:1512.06047
Abstract. Deformation quantization conventionally is described in terms of multidifferential operators. Jet manifold technique is well-known provide the adequate formulation of theory of differential operators. We extended this formulation to the multidifferential ones, and consider their infinite order jet prolongation. The infinite order jet manifold is endowed with the canonical flat connection that provides the covariant formula of a deformation star-product.
2 Deformation quantization of Poisson manifolds
- 2.1 Gerstenhaber’s deformation of algebras
- 2.1.1 Formal deformation
- 2.1.2 Deformation of rings
- 2.2 Star-product
- 2.3 Kontsevich’s deformation quantization
- 2.3.1 Differential graded Lie algebras
- 2.3.2 L^∞-algebras
- 2.3.3 Formality theorem
3 Deformation quantization on jet manifolds
- 3.1 Multidifferential operators on C^∞(X)
- 3.2 Deformations of C^∞(X)
- 3.3 Jet prolongation of multidifferential operators
- 3.4 Star-product in a covariant form
- 4.1 Fibre bundles
- 4.2 Differential forms and multivector fields
- 4.3 First order jet manifolds
- 4.4 Higher and infinite order jets
- 4.5 Hochschild cohomology
- 4.6 Chevalley–Eilenberg cohomology of Lie algebras