Multivector fields and connections: Setting Lagrangian equations in field theories

Echeverría-Enríquez, A.; Muñoz-Lecanda, Miguel C.; Román‐Roy, Narciso

doi:10.1063/1.532525

Cited by 42 publications

(109 citation statements)

References 11 publications

(36 reference statements)

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“…The link between Hamiltonian n-vector fields and solutions of the field equations has already been indicated by Kanatchikov in [7]. Moreover, the sense in which multivector fields are related to distributions seems to be folklore and is written out explicitly in the work by Echeverría-Enríquez et al [3], see Appendix A of this paper. However, both use the smaller multisymplectic phase spaceP which requires the choice of a connection [14].…”

Section: Introductionmentioning

confidence: 90%

“…We are grateful to H.A. Kastrup for drawing our attention to the comprehensive review [9] and to E. Goldblatt for mentioning [3], where similar ideas have been developed. In particular, the link between multivector fields and distributions (in the sense of subspaces of the tangent space) can be found there in great detail.…”

Section: Acknowledgementsmentioning

confidence: 99%

“…This choice creates the last two terms in the above formula. Their precise expressions will not be important for what follows (for them, we refer to [3] …”

Section: Multisymplectic Formsmentioning

confidence: 99%

“…We add this material for the sake of completeness. It can be found for instance in [3,12]. Usually [10], when considering foliations of a given manifold M, one introduces the notion of distributions, i.e.…”

Section: Appendix a Distributions And Multivectorsmentioning

confidence: 99%

“…It is known that the DW Hamiltonian (6) constitutes a relation among coordinates of P that describes the image of FL. If one wants to extract a function H Γ , the global Hamiltonian function of [3], out of it one needs to employ a connection in E,…”

Section: Theorem 2 Let γ = (ϕ π ) Be a Solution Of The Dw Equation mentioning

confidence: 99%

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Geometry of Hamiltonian n-vector fields in multisymplectic field theory

Paufler

Römer

2002

Journal of Geometry and Physics

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Multisymplectic geometry-which originates from the well known De Donder-Weyl (DW) theory-is a natural framework for the study of classical field theories. Recently, two algebraic structures have been put forward to encode a given theory algebraically. Those structures are formulated on finite dimensional spaces, which seems to be surprising at first.In this paper, we investigate the correspondence of Hamiltonian functions and certain antisymmetric tensor products of vector fields. The latter turn out to be the proper generalisation of the Hamiltonian vector fields of classical mechanics. Thus we clarify the algebraic description of solutions of the field equations.

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Section: Introductionmentioning

confidence: 90%

Section: Acknowledgementsmentioning

confidence: 99%

“…This choice creates the last two terms in the above formula. Their precise expressions will not be important for what follows (for them, we refer to [3] …”

Section: Multisymplectic Formsmentioning

confidence: 99%

Section: Appendix a Distributions And Multivectorsmentioning

confidence: 99%

Section: Theorem 2 Let γ = (ϕ π ) Be a Solution Of The Dw Equation mentioning

confidence: 99%

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Geometry of Hamiltonian n-vector fields in multisymplectic field theory

Paufler

Römer

2002

Journal of Geometry and Physics

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Classical Field Theories from Hamiltonian Constraint: Local Symmetries and Static Gauge Fields

Zatloukal

2018

Adv. Appl. Clifford Algebras

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We consider the Hamiltonian constraint formulation of classical field theories, which treats spacetime and the space of fields symmetrically, and utilizes the concept of momentum multivector. The gauge field is introduced to compensate for non-invariance of the Hamiltonian under local transformations. It is a position-dependent linear mapping, which couples to the Hamiltonian by acting on the momentum multivector. We investigate symmetries of the ensuing gauged Hamiltonian, and propose a generic form of the gauge field strength. In examples we show how a generic gauge field can be specialized in order to realize gravitational and/or Yang-Mills interaction. Gauge field dynamics is not discussed in this article.Throughout, we employ the mathematical language of geometric algebra and calculus. * Electronic address: zatlovac@gmail.com; URL: http://www.zatlovac.eu arXiv:1611.02906v1 [math-ph] 9 Nov 2016 3 generic local transformations that possess additional position dependence. To compensate for this non-invariance, we introduce in Sec. II a gauge field with appropriate transformation properties, which acts as a q-dependent linear map on the momentum multivector P . The ensuing gauged Hamiltonian acquires thereupon an additional q-dependency, and the canonical equations (2) are modified to Eqs. (15). The structure of the latter canonical equations suggests to define, by Eq. (19) in Sec. III, the field strength corresponding to the gauge field, which is a linear q-dependent function that maps grade-r multivectors to grade-r + 1 multivectors. In Appendix B, it is interpreted as torsion corresponding to the Weitzenböck connection on the configuration space C .In the present article, the gauge field and the field strength are static background quantities in the sense that they do not obey their own dynamical equations of motion, but rather are prescribed by some external body. This is also why we do not attempt to include them in the set of partial observables, but treat them separately. We relegate the study of gauge field dynamics (presumably implemented by means of a suitable kinetic term) to the future.Conservation laws maintain the form of Eq. (7), where the left-hand side features the gauged Hamiltonian. Whether a vector field v is a symmetry generator thus depends on the concrete form of the gauge field. The issue is discussed in Sec. IV.The general theory of Sections II, III and IV is much illuminated through the examples of Sec. V. In Examples V A and V B, without any reference to a concrete form of the Hamiltonian, the partial observables are divided into D spacetime coordinates and N field components, and two subgroups of the group of all configuration-space diffeomorphisms are considered. In the first example, we "gauge" spacetime diffeomorphisms by a gauge field equivalent to the tetrad (or vierbein) in the tetrad formulation of gravity. It acts nontrivially only on the spacetime, and therefore has fewer degrees of freedom than a generic gauge field. In the second example, we consider rotations in the field ...

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On the Godbillon-Vey invariant of transversely parallelizable foliations

Rovenski

Walczak

2022

Differential Geometry and its Applications

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Multivector fields and connections: Setting Lagrangian equations in field theories

Cited by 42 publications

References 11 publications

Geometry of Hamiltonian n-vector fields in multisymplectic field theory

Geometry of Hamiltonian n-vector fields in multisymplectic field theory

Classical Field Theories from Hamiltonian Constraint: Local Symmetries and Static Gauge Fields

On the Godbillon-Vey invariant of transversely parallelizable foliations

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