Secondary Calculus and the Covariant Phase Space

Abstract

The covariant phase space of a Lagrangian field theory is the solution space of the associated Euler-Lagrange equations. It is, in principle, a nice environment for covariant quantization of a Lagrangian field theory. Indeed, it is manifestly covariant and possesses a canonical (functional) "presymplectic structure" w (as first noticed by Zuckerman in 1986) whose degeneracy (functional) distribution is naturally interpreted as the Lie algebra of gauge transformations. We propose a fully rigorous approach to the covariant phase space in the framework of secondary calculus. In particular we describe the degeneracy distribution of w. As a byproduct we rederive the existence of a Lie bracket among gauge invariant functions on the covariant phase space.

40 pages, typos corrected

References in corpus (4)

• Surface charge algebra in gauge theories and thermodynamic integrability
• On covariant phase space methods
• Involutive distributions of operator-valued evolutionary vector fields
• Involutive distributions of operator-valued evolutionary vector fields and their affine geometry. II

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• Covariant canonical formulations of classical field theories
• Hamilton-Jacobi Diffieties
• A connection between the classical r-matrix formalism and covariant Hamiltonian field theory
• Partial Differential Hamiltonian Systems
• The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs
• Field Theory via Higher Geometry I: Smooth Sets of Fields
• The notion of Observables in Multisymplectic Geometry
• Gauge from holography and holographic gravitational observables
• The geometry of the space of Cauchy data of nonlinear PDEs
• Internal symmetry of the $L_{⩽3}$ algebra arising from a Lie pair