Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 947Using (5.425), (5.426) and (5.433), we get the relations∂ β nΓ Amαµ p αλm ∂ n βλ ˜H Y M = Γ β αµS α β,r A[λα]m R m λαµ = ∂ α S α µ(r A )−Γ β µλ Sα β(r A )and we find thatt α µ√|g| = 2Sαµ − 1 2 δα µS α α, T ΓAαµ (r A ) = S α µ(r A ) − 1 2 δα µS α α(r A ),t α µ(r A ) √ |g| = T α Γ A µ(r A ) + S λ µ(r A ).Hence, the identity (5.421) in gauge theory is brought into the covariantenergy–momentum conservation law∇ α t α µ(r A ) ≈ 0.The Lagrangian partner of the Hamiltonian SEM–tensor T ΓA (r A ) is theSEM–tensor J ΓA (A) (5.407) on the solution A relative to the connectionΓ A on the bundle C. This is exactly the familiar symmetrized canonicalenergy–momentum tensor of gauge potentials.SEM Tensors of Matter FieldsIn gauge theory, matter fields possessing only internal symmetries aredescribed by sections of a vector bundle Y = (P × V )/G, associated witha principal bundle P [Sardanashvily (1998)]. It has a G−invariant fibremetric a Y . Because of the canonical vertical splitting V Y = Y × Y , themetric a Y is a fibre metric in the vertical tangent bundle V Y → X. Everyprincipal connection A on a principal bundle P yields the associatedconnectionΓ = dx α ⊗ [∂ α + A m µ (x)I mij y j ∂ i ], (5.435)where A m µ (x) are coefficients of the local connection 1–form and I m aregenerators of the structure group G on the standard fibre V of the bundleY .On the configuration space J 1 (X, Y ), the regular Lagrangian density ofmatter fields in the presence of a background connection Γ on Y readsL (m) = 1 2 aY ij[g µν (y i µ − Γ i µ)(y j ν − Γ j ν) − m 2 y i y j ] √ |g|ω. (5.436)The Legendre bundle of the vector bundle Y −→ X is Π = ∧ n T ∗ X ⊗T X ⊗ Y ∗ . The unique Hamiltonian form on Π associated with the La-