6

**0 Years**of the first**gauge theory of gravitation**suggested by R. Utyama, "Invariant Theoretical Interpretation of Interaction",*Phys. Rev. D*,**101**(1956) 1597-1607 (Section 4)**(#)****WikipediA**: Gauge gravitation theory

6**0 Years** of
the first **gauge theory of gravitation** suggested by R. Utyama, "Invariant
Theoretical Interpretation of Interaction", *Phys. Rev. D*, **101** (1956) 1597-1607 (Section 4) **(#)**

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**

Our new
article: G.
Sardanashvily, A. Zamyatin, “Deformation quantization on jet manifolds” in **arXiv:1512.06047**

1 Introduction

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 Appendix

- 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

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