Covariant canonical formulations of classical field theories

Abstract

We review in simple terms the covariant approaches to the canonical formulation of classical relativistic field theories (in particular gauge field theories and general relativity) and we discuss the relationships between these approaches as well as the relation with the standard (non-covariant) Hamiltonian formulation. Particular attention is paid to conservation laws (notably related to geometric symmetries) within the different approaches. Moreover, for each of these approaches, the impact of space-time boundaries is also addressed. To make the text accessible to a wider audience, we have included an outline of Poisson and symplectic geometry for both classical mechanics and field theory.

178 pages, 9 figures, Dedicated to the memory of I. M. Singer, v4: Text cut modified above eq. (4.15)

References in corpus (76)

• Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism
• Setting the boundary free in AdS/CFT
• Surface charge algebra in gauge theories and thermodynamic integrability
• Edge modes of gravity -- I: Corner potentials and charges
• The $\Lambda$-BMS${}_4$ Charge Algebra
• Extended corner symmetry, charge bracket and Einstein's equations
• Edge modes of gravity -- II: Corner metric and Lorentz charges
• Quantum gravity in terms of topological observables
• Edge modes of gravity. Part III. Corner simplicity constraints
• Anomalies in gravitational charge algebras of null boundaries and black hole entropy
• Symmetries and conservation laws in Lagrangian gauge theories with applications to the mechanics of black holes and to gravity in three dimensions
• Hamiltonian dynamics on the symplectic extended phase space for autonomous and non-autonomous systems
• A general framework for gravitational charges and holographic renormalization
• Methods of Differential Geometry in Classical Field Theories: k-symplectic and k-cosymplectic approaches
• Geometric formulation of the Covariant Phase Space methods with boundaries
• Edge modes as reference frames and boundary actions from post-selection
• Momentum Maps and Classical Relativistic Fields. Part II: Canonical Analysis of Field Theories
• Tutorial on Scale and Conformal Symmetries in Diverse Dimensions
• Multisymplectic formulation of fluid dynamics using the inverse map
• First order parent formulation for generic gauge field theories
• Boundary and Corner Terms in the Action for General Relativity
• Topological censorship for Kaluza-Klein space-times
• Matter matters in Einstein-Cartan gravity
• Edge modes as dynamical frames: charges from post-selection in generally covariant theories
• Charges and Fluxes on (Perturbed) Non-expanding Horizons
• The Wald-Zoupas prescription for asymptotic charges at null infinity in general relativity
• Brown-York charges with mixed boundary conditions
• Null infinity as an open Hamiltonian system
• Hamiltonian formulation of an effective modified gravity with nondynamical background fields
• Wald-Zoupas prescription with (soft) anomalies
• Covariant canonical formalism for four-dimensional BF theory
• On covariant phase space methods
• Celestial Holography: Lectures on Asymptotic Symmetries
• Bundle geometry of the connection space, covariant Hamiltonian formalism, the problem of boundaries in gauge theories, and the dressing field method
• Matrix Quantization of Gravitational Edge Modes
• Tractor geometry of asymptotically flat spacetimes
• Covariant Symplectic Structure and Conserved Charges of Topologically Massive Gravity
• Dyons with potentials: duality and black hole thermodynamics
• Counting D1-D5-P Microstates in Supergravity
• Asymptotic symmetries in the BV-BFV formalism
• Ground state for a massive scalar field in BTZ spacetime with Robin boundary conditions
• Asymptotic symmetries of Yang-Mills fields in Hamiltonian formulation
• Covariant phase space for gravity with boundaries: metric vs tetrad formulations
• Presymplectic structures and intrinsic Lagrangians for massive fields
• Symplectic reduction of Yang-Mills theory with boundaries: from superselection sectors to edge modes, and back
• Presymplectic structures and intrinsic Lagrangians
• An introduction to infinite-dimensional differential geometry
• Ehrenfest Theorem in Precanonical Quantization
• Edge modes without edge modes
• Variational Tricomplex, Global Symmetries and Conservation Laws of Gauge Systems
• Ground State for the Klein-Gordon field in anti-de Sitter spacetime with dynamical Wentzell boundary conditions
• Weiss variation for general boundaries
• On the canonical formulation of gauge field theories and Poincare transformations
• Topological terms and the global symplectic geometry of the phase space in string theory
• Palatini gravity with nonmetricity, torsion, and boundaries in metric and connection variables
• Einstein-Yang-Mills Theory: Gauge Invariant Charges and Linearization Instability
• Peierls brackets in non-Lagrangian field theory
• BV equivalence with boundary
• More insights into symmetries in multisymplectic field theories
• Conserved currents in the Palatini formulation of general relativity
• Systematics of Boundary Actions in Gauge Theory and Gravity
• On the prequantisation map for 2-plectic manifolds
• Symmetries and Covariant Poisson brackets on pre-symplectic manifolds
• Improvement of a conserved current density versus adding a total derivative to a Lagrangian density
• Symmetries and gauge symmetries in multisymplectic first and second-order Lagrangian field theories: electromagnetic and gravitational fields
• Note About Covariant Hamiltonian Formalism for Strings, p-Branes and Unstable Dp-Branes
• Lie and Leibniz Algebras of Lower-Degree Conservation Laws
• Einstein and Jordan-Frame Covariant Hamiltonians for F(R) Gravity and Their Canonical Relationships
• Corners of gravity: the case of gravity as a constrained BF theory
• Leaky covariant phase spaces: Theory and application to $\Lambda$-BMS symmetry
• Peierls Brackets in Theoretical Physics
• Off-shell Noether currents and potentials for first-order general relativity
• The Covariant Phase Space of Gravity with Boundaries
• Covariant Guiding Laws for Fields
• Quantization of a New Canonical, Covariant, and Symplectic Hamiltonian Density
• Unimodular Gravity in Covariant Formalism

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• Symmetries and Covariant Poisson brackets on pre-symplectic manifolds
• Improvement of a conserved current density versus adding a total derivative to a Lagrangian density
• Canonical aspects of pregeometric vector-based first order gauge theory
• Field Theory via Higher Geometry I: Smooth Sets of Fields
• Spacetime boundaries do not break diffeomorphism and gauge symmetries
• Null infinity as a Killing horizon
• Using thermodynamics to learn gravitational wave physics
• On covariant and canonical Hamiltonian formalisms: Weakly Isolated Horizons
• Lectures on the Bondi--Metzner--Sachs group and related topics in infrared physics
• Mechanics as a general-relativistic gauge field theory, and Relational Quantization
• Toward Manifest Relationality in Transformers via Symmetry Reduction