Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 949associated with the principal connections on the principal bundle LX −→X 4 . Hence, there is the 1–1 correspondence between the world connectionsand the global sections of the quotient bundleC = J 1 (X 4 , LX)/GL + (4, R). (5.438)We therefore can apply the standard procedure of the Hamiltonian gaugetheory in order to describe the configuration and phase–spaces of worldconnections [Sardanashvily (1993); Sardanashvily (1994)].Also, there is the 1–1 correspondence between the world metrics g onX 4 and the global sections of the bundle Σ of pseudo–Riemannian bilinearforms in tangent spaces to X 4 . This bundle is associated with theGL 4 −principal bundle LX. The 2–fold covering of the bundle Σ is thequotient bundle LX/SO(3, 1).The total configuration space of the affine–metric gravitational variablesis the productJ 1 (X 4 , C) × J 1 (X 4 , Σ). (5.439)coordinated by (x µ , g αβ , k α βµ, g αβ α, k α βµλ). Also, the total phase–spaceΠ of the affine–metric gravity is the product of the Legendre bundles overthe above–mentioned bundles C and Σ. It has the corresponding canonicalcoordinates (x µ , g αβ , k α βµ, p α αβ , p βµλ α ).On the configuration space (5.439), the Hilbert–Einstein Lagrangiandensity of general relativity readsL HE = − 12κ gβλ F α √βαλ −gω, with (5.440)F α βνλ = k α βλν − k α βνλ + k α ενk ε βλ − k α ελk ε βν.The corresponding Legendre morphism is given by the expressionsp αβ α ◦ ̂L HE = 0,p α βνλ ◦ ̂L HE = π α βνλ = 12κ (δν αg βλ − δ α αg βν ) √ −g, (5.441)which define the constraint space of general relativity in the affine–metricvariables.Now, let us consider the following connections on the bundle C × Σ inorder to construct a complete family of Hamiltonian forms associated withthe Lagrangian density (5.440).