Hamilton's formalism for systems with constraints

Abstract

The main goal of these lectures is to introduce and review the Hamiltonian formalism for classical constrained systems and in particular gauge theories. Emphasis is put on the relation between local symmetries and constraints and on the relation between Lagrangean and Hamiltonian symmetries.

Quasi-Local Energy-Momentum and Angular Momentum in General Relativity

“…In the standard Hamiltonian formulation of the dynamics of the classical matter fields on a given (not necessarily flat) spacetime (see, e.g., [283, 558] and references therein) the configuration and momentum variables, ϕ A and π A , respectively, are fields on a connected three-manifold Σ, which is interpreted as the typical leaf of a foliation Σ t of the spacetime. The foliation can be characterized on Σ by a function N , called the lapse.…”

“…However, since the velocity Ṅ a cannot be expressed by the canonical variables (see e.g. [558, 63]), K H [ K ] can be written as a function on the ADM phase space only if the boundary conditions at ∂ Σ ensure the vanishing of the integral of v a Ṅ a / N .…”

Quasi-Local Energy-Momentum and Angular Momentum in GR: A Review Article

“…In the standard Hamiltonian formulation of the dynamics of the classical matter fields on a given (not necessarily flat) spacetime (see for example [212, 396] and references therein) the configuration and momentum variables, φ A and π A , respectively, are fields on a connected 3-manifold Σ, which is interpreted as the typical leaf of a foliation Σ t of the spacetime. The foliation can be characterized on Σ by a function N , called the lapse.…”

“…Thus K H [ K ] is a well-defined function of the configuration and velocity variables ( N , N a , h ab ) and ( Ṅ , Ṅ a , ḣ ab ), respectively. However, since the velocity Ṅ a cannot be expressed by the canonical variables [396, 46], K H [ K ] can be written as a function on the ADM phase space only if the boundary conditions at ∂ Σ ensure the vanishing of the integral of v a Ṅ a / N .…”

Entangled States in Quantum Cosmology and the Interpretation of Λ

“…The role of Friedmann cosmology in this approach is to link the emerging value of Ω Λ to the various epochs since cosmological parameters depend on the redshift z. In this picture, the ESA hypothesis suggests a precise role of a cosmological states, instead of standard techniques, where geometric and matter degrees of freedom have to be decomposed to achieve quantization in generic curved space-times [42][43][44][51][52][53][54][55][56][57][58][59][60][61][62].…”