Connections in Classical and Quantum Field Theory

The relation between the Jacobi morphism and the Hessian in gauge-natural field theories

“…For the proof of the theorem, we note that the Hamilton equations for the Hamiltonian connection γ H (λ, K) (equivalent to the kernel of the Euler-Lagrange morphism of the Lagrangian ω(λ, K); see, e.g., [35]) coincide with K because of Theorem 3 and are therefore satisfied identically. Generalized Bergmann-Bianchi identities then appear as constraints for such an equivalence to hold.…”

“…Every connection on J 2s Y ζ × X V A (r,k) defines a Hamiltonian form; vice versa, every Hamiltonian form H admits a Hamiltonian connection γ H (λ, K) such that γ H (λ, K) Ω = dH(λ, K). Then let γ H (λ, K) be the corresponding Hamiltonian connection form (see [35]). It is a well-known fact that when dim X > 1, there is in general a set of Hamiltonian connections γ H depending on a linear connection on X.…”

Normal frames for general connections on differentiable fibre bundles

“…16 Let (E, π, M) be a C 3 bundle endowed with C 1 connection ∆ h . Let k ∈ N, k ≤ dim M, and J k be an open set in R k .…”

“…11 Such an identification is justified by the definition of ∇ via the parallel transport assigned to ∆ h or via a projection, generated by ∆ h , of a suitable Lie derivative on X(E)-see [19]. 12 Usually the affine connections are defined on affine bundles [21,16]. In vector bundles they can be introduced as follows.…”

Noether versus Killing symmetry of conformally flat Friedmann metric

“…In the hope that symmetries other than conventional ones may provide additional information about physics of spacetimes, Noether symmetries (symmetries arising from studying the invariance of the functional involving Lagrangians [13,14]) of some static spacetimes geometries were investigated [15]. It was shown that Noether symmetries provide nontrivial new conservation laws not given by kvs.…”