Faddeev-Jackiw Approach to Gauge Theories and Ineffective Constraints

Garcı́a, J. Antonio; Pons, Josep M.

doi:10.1142/s0217751x98001736

Cited by 11 publications

(14 citation statements)

References 16 publications

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“…We denote this transformation by 10) where D µ is the Yang-Mills covariant derivative (in its action on spacetime scalars and Yang-Mills vectors). Under this transformation, the field transforms as…”

Section: A Yang-mills Gauge Transformationsmentioning

confidence: 99%

Gauge transformation in Einstein–Yang–Mills theories

Pons

Salisbury

Shepley

2000

Journal of Mathematical Physics

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We discuss the relation between spacetime diffeomorphisms and gauge transformations in theories of the Yang-Mills type coupled with Einstein's General Relativity. We show that local symmetries of the Hamiltonian and Lagrangian formalisms of these generally covariant gauge systems are equivalent when gauge transformations are required to induce transformations which are projectable under the Legendre map. Although pure Yang-Mills gauge transformations are projectable by themselves, diffeomorphisms are not. Instead the projectable symmetry group arises from infinitesimal diffeomorphism-inducing transformations which must depend on the lapse function and shift vector of the spacetime metric plus associated gauge transformations. Our results are generalizations of earlier results by ourselves and by Salisbury and Sundermeyer. 04.20.Fy, 04.40.-b, 11.10.Ef, 11.15.-q gr-qc/9912086

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Section: A Yang-mills Gauge Transformationsmentioning

confidence: 99%

Gauge transformation in Einstein–Yang–Mills theories

Pons

Salisbury

Shepley

2000

Journal of Mathematical Physics

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“…This method, when rephrased, as we do here, in terms of the canonical Lagrangian, is just a reduction procedure from L c to L that can be independently justified. 18 We prove in the appendix that it is legitimate to substitute within the Lagrangian (L c in our case) the auxiliary variables, that is, the variables (p and λ in our case) that can be isolated by using their own equations of motion. Now, given a generalized Noether transformation δ c q i associated, according to (61), with a constant of motion G c , we can readily prove that δq i := (δ c q i )| (p=p,λ=v) (λ = v includes, obviously,λ =v = (∂v/∂q)q +q(∂v/∂q), and so on forλ, etc.)…”

Section: Back To the Original Lagrangianmentioning

confidence: 92%

Rigid and Gauge Noether Symmetries for Constrained Systems

Garcı́a

Pons

2000

Int. J. Mod. Phys. A

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We develop the general theory of Noether symmetries for constrained systems, that is, systems that are described by singular Lagrangians. In our derivation, the Dirac bracket structure with respect to the primary constraints appears naturally and plays an important role in the characterization of the conserved quantities associated to these Noether symmetries. The issue of projectability of these symmetries from tangent space to phase space is fully analyzed, and we give a geometrical interpretation of the projectability conditions in terms of a relation between the Noether conserved quantity in tangent space and the presymplectic form defined on it. We also examine the enlarged formalism that results from taking the Lagrange multipliers as new dynamical variables; we find the equation that characterizes the Noether symmetries in this formalism, and we also prove that the standard formulation is a particular case of the enlarged one. The algebra of generators for Noether symmetries is discussed in both the Hamiltonian and Lagrangian formalisms. We find that a frequent source for the appearance of open algebras is the fact that the transformations of momenta in phase space and tangent space only coincide on shell. Our results apply with no distinction to rigid and gauge symmetries; for the latter case we give a general proof of the existence of Noether gauge symmetries for theories with first and second class constraints that do not exhibit tertiary constraints in the stabilization algorithm. Among some examples that illustrate our results, we study the Noether gauge symmetries of the Abelian Chern-Simons theory in 2n + 1 dimensions. An interesting feature of this example is that its primary first class constraints can only be identified after the determination of the secondary constraint. The example is worked out retaining all the original set of variables. 4681 Int. J. Mod. Phys. A 2000.15:4681-4721. Downloaded from www.worldscientific.com by UNIVERSITY OF TORONTO on 10/08/14. For personal use only.

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“…In the Refs. [10,11] irregular systems of the type II were discussed. It was pointed out that there was a possible loss of dynamical information in some cases.…”

Section: Nonlinear Constraintsmentioning

confidence: 99%

“…In Refs. [10,11] it was observed that Dirac's conjecture may not hold in the presence of irregular constraints of type II.…”

Section: Commentsmentioning

confidence: 99%

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Dynamical structure of irregular constrained systems

Mišković

Zanelli

2003

Journal of Mathematical Physics

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Hamiltonian systems with functionally dependent constraints (irregular systems), for which the standard Dirac procedure is not directly applicable, are discussed. They are classified according to their behavior in the vicinity of the constraint surface into two fundamental types. If the irregular constraints are multilinear (type I), then it is possible to regularize the system so that the Hamiltonian and Lagrangian descriptions are equivalent. When the constraints are power of a linear function (type II), regularization is not always possible and the Hamiltonian and Lagrangian descriptions may be dynamically inequivalent. It is shown that the inequivalence between the two formalisms can occur if the kinetic energy is an indefinite quadratic form in the velocities. It is also shown that a system of type I can evolve in time from a regular configuration into an irregular one, without any catastrophic changes. Irregularities have important consequences in the linearized approximation to nonlinear theories, as well as for the quantization of such systems. The relevance of these problems to Chern-Simons theories in higher dimensions is discussed.

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Faddeev-Jackiw Approach to Gauge Theories and Ineffective Constraints

Cited by 11 publications

References 16 publications

Gauge transformation in Einstein–Yang–Mills theories

Gauge transformation in Einstein–Yang–Mills theories

Rigid and Gauge Noether Symmetries for Constrained Systems

Dynamical structure of irregular constrained systems

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