Multisymplectic and Polysymplectic Structures on Fiber Bundles

Forger, Michael; Gomes, Leandro

doi:10.1142/s0129055x13500189

Cited by 25 publications

(46 citation statements)

References 14 publications

(60 reference statements)

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“…By substituting the (n − 1)-form variables from the fundamental brackets (15) into (16) we reproduce the DW Hamiltonian equations (4). Note that equation (16) generalises the Poisson bracket form of the equations of motion of a function on the phase space F (q, p, t) in mechanics: …”

Section: Poisson Brackets In Dw Hamiltonian Formulationmentioning

confidence: 99%

Ehrenfest Theorem in Precanonical Quantization

Kanatchikov

2015

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Abstract.We discuss the precanonical quantization of fields which is based on the De Donder-Weyl (DW) Hamiltonian formulation and treats the space and time variables on an equal footing. Classical field equations in DW Hamiltonian form are derived as the equations for the expectation values of precanonical quantum operators. This field-theoretic generalization of the Ehrenfest theorem demonstrates the consistency of three aspects of precanonical field quantization: (i) the precanonical representation of operators in terms of the Clifford (Dirac) algebra valued partial differential operators, (ii) the Dirac-like precanonical generalization of the Schrödinger equation without the distinguished time dimension, and (iii) the definition of the scalar product for calculation of expectation values of operators using the precanonical wave functions.

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Section: Poisson Brackets In Dw Hamiltonian Formulationmentioning

confidence: 99%

Ehrenfest Theorem in Precanonical Quantization

Kanatchikov

2015

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“…In the literature (see [3,17]) first statement is known as poly-Lagrangian subspace and the existence of such subbundle is equivalent to the existence of a Darboux coordinate system for poly-symplectic manifolds. In contrast to the usual symplectic manifolds, there exists polysymplectic manifolds that do not allow Darboux coordinates: Example 2.11.…”

Section: Lagrangian and Coisotropic Submanifoldsmentioning

confidence: 99%

Poly-Poisson sigma models and their relational poly-symplectic groupoids

Contreras

Alba

2018

Journal of Mathematical Physics

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The main idea of this note is to describe the integration procedure for poly-Poisson structures, that is, to find a poly-symplectic groupoid integrating a poly-Poisson structure, in terms of topological field theories, namely via the path-space construction. This will be given in terms of the poly-Poisson sigma model (P P SM ) and we prove that every poly-Poisson structure has a natural integration via relational poly-symplectic groupoids, extending the results in [8] and [26]. We provide familiar examples (trivial, linear, constant and symplectic) within this formulation and we give some applications of this construction regarding the classification of poly-symplectic integrations, as well as Morita equivalence of poly-Poisson manifolds. arXiv:1706.06014v2 [math-ph] 14 Nov 2017 Poly-Poisson manifoldsWe begin with the definition of the main structures to consider in this paper:Definition 2.1. A Poly-Poisson structure of order r, or simply an r-Poisson structure, on a manifold M is a pair (S, P ), where S → M is a vector subbundle of T * M ⊗R r and P : S → T M is a vector-bundle morphism (covering the identity) such that the following conditions hold:(i) i P (η) η = 0, for all η ∈ S,the space of section Γ(S) is closed under the bracket (2.1) η, γ := L P (η) γ − i P (γ) dη for γ, η ∈ Γ(S),

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“…We aim to define the geometrical objects which, contracted with the polysymplectic form (6), maintain vertical information. With this in mind, let [30,31,36,37]. We will call p X a Hamiltonian multivector field, if there exists a unique (n − p; 0)-horizontal form…”

Section: Polysymplectic Formalismmentioning

confidence: 99%

“…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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Multisymplectic and Polysymplectic Structures on Fiber Bundles

Cited by 25 publications

References 14 publications

Ehrenfest Theorem in Precanonical Quantization

Ehrenfest Theorem in Precanonical Quantization

Poly-Poisson sigma models and their relational poly-symplectic groupoids

De Donder–Weyl Hamiltonian formalism of MacDowell–Mansouri gravity

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