The n-symplectic Algebra of Observables in Covariant Lagrangian Field Theory‡

n-symplectic geometry on the adapted frame bundle λ : LπE → E of an n = (m + k)-dimensional fiber bundle π : E → M is used to set up an algebra of observables for covariant Lagrangian field theories. Using the principle bundle ρ : LπE → Jπ we lift a Lagrangian L : Jπ → R to a Lagrangian L := ρ∗(L) : LπE → R, and then use L to define a ”modified n-symplectic potential” θ̂L on LπE, the Cartan-Hamilton-Poincaré (CHP) R-valued 1-form. If the lifted Lagrangian is non-zero then (LπE, dθ̂L) is an n-symplectic manifold. To characterize the observables we define a lifted Legendre transformation φL from LπE into LE. The image QL := φL(LπE) is a submanifold of LE, and (QL, d(θ̂|QL)) is shown to be an n-symplectic manifold. We prove the theorem that θ̂L = φ ∗ L(θ|QL), and pull back the reduced canonical n-symplectic geometry on QL to LπE to define the algebras of observables on the n-symplectic manifold (LπE, dθ̂L). To find the reduced n-symplectic algebra on QL we set up the equations of nsymplectic reduction, and apply the general theory to the model of a k-tuple of massless scalar fields on Minkowski spacetime. The formalism set forth in this paper lays the ground work for a geometric quantization theory of fields.