Covariant canonical formulations of classical field theories

Gieres, François

doi:10.48550/arxiv.2109.07330

Cited by 9 publications

(13 citation statements)

References 195 publications

(555 reference statements)

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“…From a technical standpoint, we will employ covariant phase space methods [55][56][57][58][59][60][61][62][63] (see [64,65] for reviews) to describe the dynamics of fields in a (bounded) spacetime region M . For definiteness, we will take M to have the topology of a cylinder, with time-like boundary Γ (but generalizing the construction to the case of a null boundary should be reasonably straightforward).…”

Section: Jhep02(2022)172mentioning

confidence: 99%

“…A has two components once pulled-back to Γ; imposing boundary conditions on a Lagrangian submanifold amounts to fixing just one of them. 65 In local inertial coordinates (t, x), 65 While we will ultimately be imposing boundary conditions on the components of the edge-frame-dressed connection A rad , we will here briefly explore the frame-(in)dependence of boundary conditions for the gaugedependent A, as the discussion applies in the same manner to A rad , but is notation wise simpler.…”

Section: Abelian Chern-simons Theorymentioning

confidence: 99%

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Edge modes as reference frames and boundary actions from post-selection

Carrozza

Hoehn

2022

J. High Energ. Phys.

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We introduce a general framework realizing edge modes in (classical) gauge field theory as dynamical reference frames, an often suggested interpretation that we make entirely explicit. We focus on a bounded region M with a co-dimension one time-like boundary Γ, which we embed in a global spacetime. Taking as input a variational principle at the global level, we develop a systematic formalism inducing consistent variational principles (and in particular, boundary actions) for the subregion M. This relies on a post-selection procedure on Γ, which isolates the subsector of the global theory compatible with a general choice of gauge-invariant boundary conditions for the dynamics in M. Crucially, the latter relate the configuration fields on Γ to a dynamical frame field carrying information about the spacetime complement of M; as such, they may be equivalently interpreted as frame-dressed or relational observables. Generically, the external frame field keeps an imprint on the ensuing dynamics for subregion M, where it materializes itself as a local field on the time-like boundary Γ; in other words, an edge mode. We identify boundary symmetries as frame reorientations and show that they divide into three types, depending on the boundary conditions, that affect the physical status of the edge modes. Our construction relies on the covariant phase space formalism, and is in principle applicable to any gauge (field) theory. We illustrate it on three standard examples: Maxwell, Abelian Chern-Simons and non-Abelian Yang-Mills theories. In complement, we also analyze a mechanical toy-model to connect our work with recent efforts on (quantum) reference frames.

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Section: Jhep02(2022)172mentioning

confidence: 99%

Section: Abelian Chern-simons Theorymentioning

confidence: 99%

Edge modes as reference frames and boundary actions from post-selection

Carrozza

Hoehn

2022

J. High Energ. Phys.

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“…Here, Stokes' theorem was applied for the last equality, see reference [15] for the notation of the hypersurface integration measure. Thus, if the variation δϕ and its derivatives vanish at the boundary ∂Ω of Ω, the variation δS vanishes identically for all of these field configurations and so does its functional derivative with respect to ϕ, i.e.…”

Section: Some Reminders 21 Lagrangian Density Given By a Total Deriva...mentioning

confidence: 99%

“…As a matter of fact, the general formulation [5][6][7][8][9] of Noether's first theorem states that there is a one-to-one correspondence between equivalence classes of (global) variational symmetries and equivalence classes of (on-shell) conserved currents. (For a review, see for instance references [9,15].) More precisely, two infinitesimal global symmetry transformations are considered to be equivalent if they differ by a gauge symmetry transformation and/or an "equation of motion symmetry transformation", i.e.…”

Section: Noether's First Theorem and Improvement Of Currentsmentioning

confidence: 99%

Improvement of a conserved current density versus adding a total derivative to a Lagrangian density

Gieres

2022

Preprint

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who made a large variety of original contributions to diverse fields of theoretical and mathematical physics, and in particular to classical and quantum field theory. The discussions with him have always been quite pleasant and enlightening. By his humbleness, attentiveness, kindness and generosity he has been a great example and steady encouragement. His precious advice, fine mind, humor and reassuring presence are deeply missed.

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“…These are equivalent. Although the form-canonical equations of the CCF are equivalent to modified De Donder-Weyl equations [8], the Poisson bracket of the CCF is not equivalent to it of the De Donder-Weyl theory [12,13]. Reference [10] introduced the generator of the CCF and studied it of the local Lorentz transformation of gravity in the first order formalism.…”

Section: Introductionmentioning

confidence: 99%

Noether currents and generators of local gauge transformations in the covariant canonical formalism

Nakajima¹

2022

Preprint

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We investigate generators of local transformations in the covariant canonical formalism (CCF) for a gauge field and the second order formalism of gravity. The CCF treats space and time on an equal footing regarding the differential forms as the basic variables. The conjugate forms π A are defined as derivatives of the Lagrangian d-form L(ψ A , dψ A ) with respect to dψ A , namely π A := ∂L/∂dψ A , where ψ A are p-form dynamical fields. The formcanonical equations are derived from the form-Legendre transformation of the Lagrangian form H := dψ A ∧ π A − L. We show that the Noether current form is the generator of an infinitesimal transformation ψ A → ψ A + δψ A if the transformation of the Lagrangian form is given by δL = dl and δψ A and l are described by only ψ A and the parameters. As an instance, we study the local gauge transformation for the gauge field and the local Lorentz transformation for the second order formalism of gravity.

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Covariant canonical formulations of classical field theories

Cited by 9 publications

References 195 publications

Edge modes as reference frames and boundary actions from post-selection

Edge modes as reference frames and boundary actions from post-selection

Improvement of a conserved current density versus adding a total derivative to a Lagrangian density

Noether currents and generators of local gauge transformations in the covariant canonical formalism

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