Covariant phase space, constraints, gauge and the Peierls formula

Abstract

It is well known that both the symplectic structure and the Poisson brackets of classical field theory can be constructed directly from the Lagrangian in a covariant way, without passing through the non-covariant canonical Hamiltonian formalism. This is true even in the presence of constraints and gauge symmetries. These constructions go under the names of the covariant phase space formalism and the Peierls bracket. We review both of them, paying more careful attention, than usual, to the precise mathematical hypotheses that they require, illustrating them in examples. Also an extensive historical overview of the development of these constructions is provided. The novel aspect of our presentation is a significant expansion and generalization of an elegant and quite recent argument by Forger & Romero showing the equivalence between the resulting symplectic and Poisson structures without passing through the canonical Hamiltonian formalism as an intermediary. We generalize it to cover theories with constraints and gauge symmetries and formulate precise sufficient conditions under which the argument holds. These conditions include a local condition on the equations of motion that we call hyperbolizability, and some global conditions of cohomological nature. The details of our presentation may shed some light on subtle questions related to the Poisson structure of gauge theories and their quantization.

revised and updated material from arXiv:1211.1914; 73 pages; to appear as a review article in IJMPA. v2: fixed: arrows now appear in diagrams

References in corpus (4)

• Removal of violations of the Master Ward Identity in perturbative QFT
• Linear bosonic and fermionic quantum gauge theories on curved spacetimes
• Characteristics, Conal Geometry and Causality in Locally Covariant Field Theory
• Asymptotic generators of fermionic charges and boundary conditions preserving supersymmetry

Cited by in corpus (40)

• Covariant phase space with boundaries
• Symmetries and charges of general relativity at null boundaries
• Algebraic QFT in Curved Spacetime and quasifree Hadamard states: an introduction
• Geometric formulation of the Covariant Phase Space methods with boundaries
• Edge modes as reference frames and boundary actions from post-selection
• Algebra of diffeomorphism-invariant observables in Jackiw-Teitelboim Gravity
• Local and gauge invariant observables in gravity
• Celestial Holography: Lectures on Asymptotic Symmetries
• Homotopy colimits and global observables in Abelian gauge theory
• The initial boundary value problem and quasi-local Hamiltonians in General Relativity
• Radiative observables for linearized gravity on asymptotically flat spacetimes and their boundary induced states
• The Calabi complex and Killing sheaf cohomology
• Electromagnetic Potential in Pre-Metric Electrodynamics: Causal Structure, Propagators and Quantization
• Covariant canonical formulations of classical field theories
• Quantization of the linearised Einstein-Klein-Gordon system on arbitrary backgrounds and the special case of perturbations in Inflation
• Variational Tricomplex, Global Symmetries and Conservation Laws of Gauge Systems
• On the canonical formulation of gauge field theories and Poincare transformations
• Classical phase space and Hadamard states in the BRST formalism for gauge field theories on curved spacetime
• Soft edges: the many links between soft and edge modes
• Approaches to linear local gauge-invariant observables in inflationary cosmologies
• The dressing field method for diffeomorphisms: a relational framework
• Peierls brackets in non-Lagrangian field theory
• Flux Quantization on Phase Space
• Higher prequantum geometry
• Covariant Brackets for Particles and Fields
• Soft thermodynamics of gravitational shock wave
• Compatibility complexes of overdetermined PDEs of finite type, with applications to the Killing equation
• Poisson algebras for non-linear field theories in the Cahiers topos
• Green hyperbolic complexes on Lorentzian manifolds
• Covariant phase space and $L_{\infty }$ algebras
• Polysymplectic formulation for BF gravity with Immirzi parameter
• The Geometry of the solution space of first order Hamiltonian field theories I: from particle dynamics to free Electrodynamics
• Field Theory via Higher Geometry I: Smooth Sets of Fields
• On the global "two-sided" characteristic Cauchy problem for linear wave equations on manifolds
• On presymplectic structures for massless higher-spin fields
• Internal Lagrangians of PDEs as variational principles
• Historical Lagrangian Dynamics
• Gauge from holography and holographic gravitational observables
• Beyond Noether: A Covariant Study of Poisson-Lie Symmetries in Low Dimensional Field Theory
• Causal Poisson bracket via deformation quantization