n -symplectic algebra of observables in covariant Lagrangian field theory

Abstract

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 → J 1 π we lift a Lagrangian L :and then use L to define a "modified n-symplectic potential"θL on LπE, the Cartan-Hamilton-Poincaré (CHP) R n -valued 1-form. If the lifted Lagrangian is non-zero then (LπE, dθL)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(θ|Q L )) is shown to be an n-symplectic manifold. We prove the theorem thatθL = φ * L (θ|Q L ), 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, dθL). 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.

SYMMETRIES AND CONSERVATION LAWS IN THE GÜNTHER k-SYMPLECTIC FORMALISM OF FIELD THEORY

“…Let φ : R k → Q be a solution to the Euler-Lagrange equations, then from (22), (43), (44) and (45) we obtain…”

“…Let us remark here that the polysymplectic formalism developed by Sardanashvily [13], based on a vector-valued form defined on some associated fiber bundle, is a different description of classical field theories of first order than the polysymplectic (or k-symplectic) formalism proposed by Günther (see also [22] for more details). We must also remark that the soldering form on the linear frames bundle is a polysymplectic form, and its study and applications to field theory, constitute the n-symplectic geometry developed by L. K. Norris in [39,40,41,42,43].…”

On a kind of Noether symmetries and conservation laws in k-cosymplectic field theory

“…A refinement of this concept led to define k-symplectic manifolds [2,3,4], which are polysymplectic manifolds admiting Darboux-type coordinates [19]. (Other different polysymplectic formalisms for describing field theories have been also proposed [8,12,22,25,26,29]).…”

Quantization of polysymplectic manifolds

“…, ω k ), which are collectively nondegenerate in the sense that ∩ i ker ω i vanishes as a distribution on T M . Similar ideas have appeared in the theories of k-almost cotangent structures [19], generalized symplectic geometry [56] and n-symplectic geometry [57], and have found applications in, for example, Lie group thermodynamics [5].…”