Advanced Classical Field Theory

Giachetta, Giovanni; Mangiarotti, Luigi; Sardanashvily, G.

doi:10.1142/9789812838964

Cited by 36 publications

(172 citation statements)

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“…Now, let (Y, π XY , X) be a finite-dimensional fibre-bundle, where X stands for the base space, which is an n-dimensional smooth manifold without boundary and local coordinates denoted by {x µ } n−1 µ=0 , while π XY : Y → X is the standard projector map, and Y is the so-called total space whose fibres are m-dimensional smooth manifolds locally represented by {y a } m a=1 . In what follows, we will refer to the fibre-bundle (Y, π XY , X) just as π XY , details about the fibre-bundle formalism may be found in [7,27]. For an arbitrary point x ∈ X, we introduce Y x to be the fibre π −1 XY (x) of Y and denote by Y the space of all sections of π XY , such that, given the fibre coordinates (x µ , y a ) at the point y ∈ Y , a local section φ ∈ Y can be represented on Y by means of the composition function y • φ = (x µ , φ a ).…”

Section: Covariant Configuration Spacementioning

confidence: 99%

“…Here we have adopted the notation φ a µ := ∂ µ φ a . As we will see bellow, the fibre-bundle formalism provides a natural framework to study classical field theory, as discussed in [7,15,17,18,27].…”

Section: Covariant Configuration Spacementioning

confidence: 99%

“…which may be shown to be completely equivalent to the Euler-Lagrange field equations of the theory [7,16,18,22], namely…”

Section: Symmetries In the Lagrangian Formulationmentioning

confidence: 99%

“…As previously mentioned, we want to study symmetries of classical field theory within this covariant geometric framework, and in particular, the action of a Lie group on the covariant configuration bundle of the theory. To this end, let us consider a Lie group G whose Lie algebra is denoted by g, which acts on X and Y by diffeomorphisms and π XY -bundle automorphisms, respectively (see [7] for further details on fibre-bundle maps). Now, let η ∈ G, we denote by η X and η Y the corresponding transformations of X and Y associated to the group element η, respectively.…”

Section: Symmetries In the Lagrangian Formulationmentioning

confidence: 99%

“…This relation gives rise to the so-called BF gravity models [5,6]. Recently, BF theories have been exhaustively explored in the literature from different perspectives and approaches at both the classical and the quantum levels [1][2][3][7][8][9][10][11][12], mainly due to the features that distinguish these topological field theories which provide an extended framework to explore different aspects of gauge theories. In this context, of particular interest to us is the case of the 4-dimensional non-Abelian topological BF theory, which at the classical level has been studied from different perspectives including the Lagrangian [1,13], the pure Dirac-Hamiltonia [14], the Hamilton-Jacobi [10], and the covariant canonical [11] formalisms.…”

Section: Introductionmentioning

confidence: 99%

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Covariant momentum map for non-Abelian topological BF field theory

Molgado

Rodríguez–López

2019

Class. Quantum Grav.

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We analyze the inherent symmetries associated to the non-Abelian topological BF theory from the geometric and covariant perspectives of the Lagrangian and the multisymplectic formalisms. At the Lagrangian level, we classify the symmetries of the theory as natural and Noether symmetries and construct the associated Noether currents, while at the multisymplectic level the symmetries of the theory arise as covariant canonical transformations. These transformations allowed us to build within the multisymplectic approach, in a complete covariant way, the momentum maps which are analogous to the conserved Noether currents. The covariant momentum maps are fundamental to recover, after the space plus time decomposition of the background manifold, not only the extended Hamiltonian of the BF theory but also the generators of the gauge transformations which arise in the instantaneous Dirac-Hamiltonian analysis of the first-class constraint structure that characterizes the BF model under study. To the best of our knowledge, this is the first non-trivial physical model associated to General Relativity for which both natural and Noether symmetries have been analyzed at the multisymplectic level. Our study shed some light on the understanding of the manner in which the generators of gauge transformations may be recovered from the multisymplectic formalism for field theory.

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Section: Covariant Configuration Spacementioning

confidence: 99%

Section: Covariant Configuration Spacementioning

confidence: 99%

“…which may be shown to be completely equivalent to the Euler-Lagrange field equations of the theory [7,16,18,22], namely…”

Section: Symmetries In the Lagrangian Formulationmentioning

confidence: 99%

Section: Symmetries In the Lagrangian Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Covariant momentum map for non-Abelian topological BF field theory

Molgado

Rodríguez–López

2019

Class. Quantum Grav.

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Batalin-Vilkovisky Formalism in the Functional Approach to Classical Field Theory

Fredenhagen

Rejzner

2012

Commun. Math. Phys.

153

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We develop the Batalin-Vilkovisky formalism for classical field theory on generic globally hyperbolic spacetimes. A crucial aspect of our treatment is the incorporation of the principle of local covariance which amounts to formulate the theory without reference to a distinguished spacetime. In particular, this allows a homological construction of the Poisson algebra of observables in classical gravity. Our methods heavily rely on the differential geometry of configuration spaces of classical fields. 15 The field equations have to understood in the algebraic sense, see footnote 11.

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Classical higgs fields

Sardanashvily

2014

Theor Math Phys

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We consider classical gauge theory on a principal bundle P → X in a case of spontaneous symmetry breaking characterized by the reduction of a structure group G of P → X to its closed subgroup H. This reduction is ensured by the existence of global sections of the quotient bundle P/H → X treated as classical Higgs fields. Matter fields with an exact symmetry group H in such gauge theory are considered in the pairs with Higgs fields, and they are represented by sections of a composite bundle Y → P/H → X, where Y → P/H is a fiber bundle associated to a principal bundle P → P/H with a structure group H. A key point is that a composite bundle Y → X is proved to be associated to a principal G-bundle P → X. Therefore, though matter fields possess an exact symmetry group H ⊂ G, their gauge G-invariant theory in the presence of Higgs fields can be developed. Its gauge invariant Lagrangian factorizes through the vertical covariant differential determined by a connection on a principal H-bundle P → P/H. In a case of the Cartan decomposition of a Lie algebra of G, this connection can be expressed in terms of a connection on a principal bundle P → X, i.e., gauge potentials for a group of broken symmetries G.

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Advanced Classical Field Theory

Cited by 36 publications

References 0 publications

Covariant momentum map for non-Abelian topological BF field theory

Covariant momentum map for non-Abelian topological BF field theory

Batalin-Vilkovisky Formalism in the Functional Approach to Classical Field Theory

Classical higgs fields

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