Canonical Noether symmetries and commutativity properties for gauge systems

Gràcia, Xavier; Pons, Josep M.

doi:10.1063/1.1289825

Cited by 6 publications

(6 citation statements)

References 21 publications

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“…These pullbacks yield Noether Lagrangian symmetries. For details see [2,26] This restriction to solution trajectories is intimately related to our demonstration that all G[ξ A ] generators (with ξ ∼ 0 > 0) can be interpreted as Hamiltonians (for time evolution, in the sense discussed in Section VII).…”

Section: Discussionmentioning

confidence: 97%

Gauge group and reality conditions in Ashtekar’s complex formulation of canonical gravity

2000

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We discuss reality conditions and the relation between spacetime diffeomorphisms and gauge transformations in Ashtekar's complex formulation of general relativity. We produce a general theoretical framework for the stabilization algorithm for the reality conditions, which is different from Dirac's method of stabilization of constraints. We solve the problem of the projectability of the diffeomorphism transformations from configuration-velocity space to phase space, linking them to the reality conditions. We construct the complete set of canonical generators of the gauge group in the phase space which includes all the gauge variables. This result proves that the canonical formalism has all the gauge structure of the Lagrangian theory, including the time diffeomorphisms.

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Section: Discussionmentioning

confidence: 97%

Gauge group and reality conditions in Ashtekar’s complex formulation of canonical gravity

2000

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“…Essentially, it was done in [17], but here we use, in the Lagrangian case, the geometric content of vector fields along π k+1,k and, in the Hamiltonian case, the new tools of Hamiltonian dynamics in the cotangent bundle T * E.…”

Section: Discussionmentioning

confidence: 99%

“…We also studied Noether symmetries that come from a function defined on the dual affine bundle, imposing conditions on them to guarantee, in some sense, in which cases the symmetry commutes with the dynamic vector field. Essentially, it was done in [16], but here we use, in the Lagrangian case, geometric content of vector fields along π k+1,k and, in the Hamiltonian case, the new tools of Hamiltonian dynamics in the cotangent bundle T * E.…”

Section: Discussionmentioning

confidence: 99%

“…In this section we are going to investigate the Lie brackets of the dynamical vector fields (in both the Lagrangian and the Hamiltonian formalisms) and the vector field that generates the symmetry, generalizing to the time-dependent formalism the results of which are stated in [17]. We will assume that we already performed the constraint algorithm and we have arrived to a consistent solution.…”

Section: Commutation Relationsmentioning

confidence: 99%

See 1 more Smart Citation

On Second Noether's Theorem and Gauge Symmetries in Mechanics

Cariñena

Lázaro-Camí

Martínez

2006

Int. J. Geom. Methods Mod. Phys.

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We review the geometric formulation of the second Noether's theorem in time-dependent mechanics. The commutation relations between the dynamics on the final constraint manifold and the infinitesimal generator of a symmetry are studied. We show an algorithm for determining a gauge symmetry which is closely related to the process of stabilization of constraints, both in Lagrangian and Hamiltonian formalisms. The connections between both formalisms are established by means of the time-evolution operator.It is an easy exercise to prove the result of this theorem writing down the variation of the action under such an infinitesimal variation, performing an appropriate integration by parts, and using that the function ε is arbitrary. However, we want to understand it from a more intrinsic point of view.The geometric setting of the Lagrangian formalism for field theories [10] uses the theory of jet bundles [24]. One starts with a fiber bundle π : E → R and for any pair of integer numbers k, l with k > l ≥ 0 defines the natural projections π k,l : J k π → J l π, where J 0 π = E and denote π k : J k π → R the composition π k = π • π k,0 . The coordinate in R will be denoted (t), and we will refer to it as the time, and in E by (t, q α ). Similarly, coordinates in J k π will be denoted t, q α , q α(1) , . . . , q α (k) . In the particular case of k = 1, J 1 π, usually called evolution space, its coordinates will be denoted in the more usual way (t, q α , v α ) and for k = 2, J 2 π, (t, q α , v α , a α ). When we choose a trivialization E R × Q then the jet bundles also trivialize J k π R × T k Q, but, in general, there is no preferred trivialization.The dynamics is defined as follows: given a π 1 -semibasic 1-form L = L dt, we define the Poincaré-Cartan forms, Θ L = dL • S + L dt ∈ 1 (J 1 π), with S being Int. J. Geom. Methods Mod. Phys. 2006.03:471-487. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 07/13/15. For personal use only.

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“…Thus, each Lagrangian Noether infinitessimal symmetry can be obtained from a Hamiltonian generator of symmetries, which is a conserved quantity (see [21], [22], [26], [28], [29], [32]). …”

Section: Introductionmentioning

confidence: 99%

Untitled

Echeverría-Enríquez¹,

Marín-Solano²,

Muñoz-Lecanda³

et al. 2003

Acta Applicandae Mathematicae

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The "time-evolution K-operator" (or "relative Hamiltonian vector field") in mechanics is a powerful tool which can be geometrically defined as a vector field along the Legendre map. It has been extensively used by several authors for studying the structure and properties of the dynamical systems (mainly the non-regular ones), such as the relation between the Lagrangian and Hamiltonian formalisms, constraints, and higher-order mechanics.This paper is devoted to defining a generalization of this operator for field theories, in a covariant formulation. In order to do this, we use sections along maps, in particular multivector fields (skew-symmetric contravariant tensor fields of order greater than 1), jet fields and connection forms along the Legendre map. As a relevant result, we use these geometrical objects to obtain the solutions of the Lagrangian and Hamiltonian field equations, and the equivalence among them (specially for non-regular field theories).

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Canonical Noether symmetries and commutativity properties for gauge systems

Cited by 6 publications

References 21 publications

Gauge group and reality conditions in Ashtekar’s complex formulation of canonical gravity

Gauge group and reality conditions in Ashtekar’s complex formulation of canonical gravity

On Second Noether's Theorem and Gauge Symmetries in Mechanics

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