Lecture Notes in Loop Quantum Gravity. LN4: Hamiltonian framework

TL;DR

The work develops a covariant Hamiltonian framework for field theories using Legendre maps, phase and multisymplectic bundles, and Poincaré-Cartan forms, linking Lagrangian and Hamiltonian descriptions through $\Theta_L=(\mathbb{F})^*\Theta_H$ and boundary variational principles. It shows how degeneracies in the Legendre transform produce constraint equations that are intimately tied to boundary data, enabling a covariant characterization of physical states and dynamics without relying on a fixed Cauchy problem. By applying the formalism to Klein–Gordon, electromagnetism, and ABI gravity, the authors derive the corresponding Hamilton principal functionals and boundary equations, illustrating how constraints arise from symmetries and gauge structures. The perspective sets the stage for quantization of boundary pre-quantum configurations and informs the development of LQG by clarifying how connection spaces, holonomies, and spin networks may be treated within a covariant, boundary-driven framework.

Abstract

We discuss a covariant setting for Hamiltonian formalism in a relativistic field theory and we use this to obtain again the properties of Hamilton principal functional in Newtonian mechanics, relativistic mechanics, Klein-Gordon, electromagnetism, and Ashtekar-Barbero-Immirzi gravitational theory.