Polysymplectic formulation for topologically massive Yang–Mills field theory

Berra─Montiel, Jasel; Rio, E del; Molgado, Alberto

doi:10.1142/s0217751x17501019

Cited by 6 publications

(9 citation statements)

References 50 publications

(78 reference statements)

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“…Let us note here the recent discussion of the classical counterpart of this constraint in [33]. Now, by substituting (2.5) into (2.2) and using (2.7), we obtain…”

Section: )mentioning

confidence: 98%

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SchrÖdinger Wave Functional in Quantum Yang–Mills Theory from Precanonical Quantization

Kanatchikov

2018

Reports on Mathematical Physics

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A relation between the precanonical quantization of pure Yang-Mills fields and the standard functional Schrödinger representation in the temporal gauge is discussed. It is shown that the latter can be obtained from the former when the ultraviolet parameter κ introduced in precanonical quantization goes to infinity. In this limiting case, the Schrödinger wave functional can be expressed as the trace of the Volterra product integral of Clifford-algebra-valued precanonical wave functions restricted to a field configuration, and the canonical functional derivative Schrödinger equation together with the quantum Gauß constraint can be derived from the Dirac-like precanonical Schrödinger equation.

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“…Let us note here the recent discussion of the classical counterpart of this constraint in [33]. Now, by substituting (2.5) into (2.2) and using (2.7), we obtain…”

Section: )mentioning

confidence: 98%

“…Other recent discussions of gauge fields and gravity from the point of view of classical DW Hamiltonian theory and its geometrizations can be found in [30][31][32][33][34][35][36].…”

Section: Introductionmentioning

confidence: 99%

SchrÖdinger Wave Functional in Quantum Yang–Mills Theory from Precanonical Quantization

Kanatchikov

2018

Reports on Mathematical Physics

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“…In order to obtain the field equations from the decomposed Hamiltonian (38), we first need to determine which fields stand as canonical variables. To do so, we use the fundamental Poisson-Gerstenhaber bracket relations (11) for the SO(4, 1) Hamiltonian (30), namely, ee developed [16,17,18,19,36,37,38,39,40,41,42,43]. Some other recent references where other alternative geometric formalisms are addressed for models in General Relativity may be found in [44, 45, and references therein].…”

Section: De Donder-weyl Formulationmentioning

confidence: 99%

De Donder–Weyl Hamiltonian formalism of MacDowell–Mansouri gravity

Berra─Montiel

Molgado

Serrano-Blanco

2017

Class. Quantum Grav.

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We analyse the behaviour of the MacDowell-Mansouri action with internal symmetry group SO(4, 1) under the De Donder-Weyl Hamiltonian formulation. The field equations, known in this formalism as the De Donder-Weyl equations, are obtained by means of the graded Poisson-Gerstenhaber bracket structure present within the De Donder-Weyl formulation. The decomposition of the internal algebra so(4, 1) ≃ so(3, 1) ⊕ R 3,1 allows the symmetry breaking SO(4, 1) → SO(3, 1), which reduces the original action to the Palatini action without the topological term. We demonstrate that, in contrast to the Lagrangian approach, this symmetry breaking can be performed indistinctly in the polysymplectic formalism either before or after the variation of the De Donder-Weyl Hamiltonian has been done, recovering Einstein's equations via the Poisson-Gerstenhaber bracket.The equivalence between the action proposed by MacDowell and Mansouri and the Palatini action is made possible by the fact that the internal Lie algebra admits the orthogonal splitting so(4, 1) ≃ so(3, 1) ⊕R 3,1 , so that the symmetry breaking is achieved by projecting the associated SO(4, 1)-connection to its SO(3, 1) components, which turn out to be the standard Lorentz connection. As pointed out by Wise [2], a concise geometrical meaning can be given to the symmetry breaking process by identifying the SO(4, 1) gauge field as a connection of a Cartan geometry [11,12].Being a theory with a strong geometric background and also physically relevant due to its relation with General Relativity, the MacDowell-Mansouri model is a perfect candidate for analysis under the inherently geometric classical formulation known as the polysymplectic formalism. Based on the early work of De Donder, Weyl, Carathéodory, among others [13,14,15], the polysymplectic approach of field theory consists in a De Donder-Weyl Hamiltonian formalism endowed with a generalization of the symplectic structure which enables to define a Poisson-like bracket. This issue, in particular, sets an important distinction with most of the various versions of the multisymplectic formalism. One of the key points within the polysymplectic approach relies in changing the definition of the standard Hamiltonian momenta, here considered not as variations of the Lagrangian density with respect to time derivatives of the field variables but as variations with respect to the derivatives of the fields in every spacetime direction. These new fields, known within this context as polymomenta, define a De Donder-Weyl Legendre transformation which associates to the Lagrangian density a new function H DW called De Donder-Weyl Hamiltonian, and it dictates the physical behaviour of the system in the so-called polymomentum phase-space. Without a space-like foliation, the polysymplectic formalism provides a manifestly spacetime covariant formulation. The polymomentum phase-space is endowed with a canonical (n + 1)-form, called the polysymplectic form [16,17,18,19], which plays a similar role to the one of the standard symplectic 2-form. A ...

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“…Such canonical form, known as the polysymplectic form, encodes the relevant physical data of a given classical field theory in order to construct a well-defined Poisson-Gerstenhaber bracket for a set of appropriately prescribed differential Hamiltonian forms, thus allowing us to analyze an arbitrary classical field theory in a covariant Poisson-Hamiltonian framework [21][22][23]. Some physically motivated examples for which the multisymplectic and polysymplectic formalisms have been applied may be encountered in references [5,8,10,21,[24][25][26][27][28][29][30][31][32][33][34][35]. Despite their mathematical elegance, from our point of view, the analysis of the gauge content for a given classical field theory from the perspective of such geometric formulations has been rarely exploited, especially when considering certain highly non-trivial gauge models associated with general relativity.…”

Section: Introductionmentioning

confidence: 99%

A review on geometric formulations for classical field theory: the Bonzom–Livine model for gravity

Berra─Montiel

Molgado

Rodríguez–López

2021

Class. Quantum Grav.

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Motivated by the study of physical models associated with general relativity, we review some finite-dimensional, geometric and covariant formulations that allow us to characterize in a simple manner the symmetries for classical field theory by implementing an appropriate fibre-bundle structure, either at the Lagrangian, the Hamiltonian multisymplectic or the polysymplectic levels. In particular, we are able to formulate Noether’s theorems by means of the covariant momentum maps and to systematically introduce a covariant Poisson–Hamiltonian framework. Also, by focusing on the space plus time decomposition for a generic classical field theory and its relation to these geometric formulations, we are able to successfully recover the gauge content and the true local degrees of freedom for the theory. In order to illustrate the relevance of these geometric frameworks, we center our attention to the analysis of a model for three-dimensional theory of general relativity that involves an arbitrary Immirzi-like parameter. At the Lagrangian level, we reproduce the field equations of the system which for this model turn out to be equivalent to the vanishing torsion condition and the Einstein equations. We also concentrate on the analysis of the gauge symmetries of the system in order to obtain the Lagrangian covariant momentum map associated with the theory and, consequently, its corresponding Noether currents. Next, within the Hamiltonian multisymplectic approach, we aim our attention to describing how the gauge symmetries of the model yield covariant canonical transformations on the covariant multimomenta phase-space, thus giving rise to the existence of a covariant momentum map. Besides, we analyze the physical system under consideration within the De Donder–Weyl canonical theory implemented at the polysymplectic level, thus establishing a relation from the covariant momentum map to the conserved currents of the theory within this covariant Hamiltonian approach. Finally, after performing the space plus time decomposition of the space-time manifold, and taking as a starting point the Hamiltonian multisymplectic formulation, we are able to recover both the extended Hamiltonian and the gauge structure content that characterize the gravity model of our interest within the instantaneous Dirac–Hamiltonian formulation.

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Polysymplectic formulation for topologically massive Yang–Mills field theory

Cited by 6 publications

References 50 publications

SchrÖdinger Wave Functional in Quantum Yang–Mills Theory from Precanonical Quantization

SchrÖdinger Wave Functional in Quantum Yang–Mills Theory from Precanonical Quantization

De Donder–Weyl Hamiltonian formalism of MacDowell–Mansouri gravity

A review on geometric formulations for classical field theory: the Bonzom–Livine model for gravity

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