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  • American Institute of Physics (AIP)  (3)
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  • 1
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 31 (1990), S. 725-743 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The general relationship between local symmetries occurring in a Lagrangian formulation of a field theory and the corresponding constraints present in a phase space formulation are studied. First, a prescription—applicable to an arbitrary Lagrangian field theory—for the construction of phase space from the manifold of field configurations on space-time is given. Next, a general definition of the notion of local symmetries on the manifold of field configurations is given that encompasses, as special cases, the usual gauge transformations of Yang–Mills theory and general relativity. Local symmetries on phase space are then defined via projection from field configuration space. It is proved that associated to each local symmetry which suitably projects to phase space is a corresponding equivalence class of constraint functions on phase space. Moreover, the constraints thereby obtained are always first class, and the Poisson bracket algebra of the constraint functions is isomorphic to the Lie bracket algebra of the local symmetries on the constraint submanifold of phase space. The differences that occur in the structure of constraints in Yang–Mills theory and general relativity are fully accounted for by the manner in which the local symmetries project to phase space: In Yang–Mills theory all the "field-independent'' local symmetries project to all of phase space, whereas in general relativity the nonspatial diffeomorphisms do not project to all of phase space and the ones that suitably project to the constraint submanifold are "field dependent.'' As by-products of the present work, definitions are given of the symplectic potential current density and the symplectic current density in the context of an arbitrary Lagrangian field theory, and the Noether current density associated with an arbitrary local symmetry. A number of properties of these currents are established and some relationships between them are obtained.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 33 (1992), S. 248-255 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The usual approach to analyze the linear stability of a static solution of some system of equations consists of searching for linearized solutions which satisfy suitable boundary conditions spatially and which grow exponentially in time. In the case of the n=1 Einstein–Yang–Mills (EYM) black hole, an interesting situation occurs. There exists a perturbation which grows exponentially in time−and spatially decreases to zero at the horizon−but nevertheless is physically singular on the horizon. Thus, this unstable mode is unacceptable as initial data, and the question arises as to whether the n=1 EYM black hole is stable. We analyze this issue here in the more general case of a scalar field φ satisfying the wave equation ∂2φ/∂t2 = (DaDa − V)φ on a manifold R×M, where Da is the derivative operator associated with a complete Riemannian metric on M and V is a bounded function on M whose derivatives also are bounded. We prove that if the operator A = −DaDa + V fails to be a strictly positive operator on the Hilbert space L2(M), then there exists smooth initial data of compact support in M which give rise to a solution which grows unboundedly with time. This implies that the n=1 EYM black hole and other mathematically similar systems are unstable despite the nonexistence of physically acceptable exponentially growing modes. Rigorous criteria for linear stability are also obtained.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 31 (1990), S. 2378-2384 
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Let M be an n-dimensional manifold with derivative operator ∇a and let B(M) be an arbitrary vector bundle over M, equipped with a connection. A cross section of B defines a field φ on M. Let α be a p-form on M (with p〈n) which is locally constructed from φ and finitely many of its derivatives (as well as, possibly, some "background fields'' ψ and their derivatives) such that dα=0 for all cross sections φ. Suppose further that α=0 for the zero cross section, φ=0. It is proven here that there exists a (p−1)-form β that also is a local function of φ,ψ and finitely many of their derivatives, such that α=dβ. A number of applications of this result are described. In particular, gauge invariance is established for the charges and the total fluxes derived from gauge-dependent conserved currents, and severe limitations are established on the the possibilities for gravitational analogs of magnetic charges.
    Type of Medium: Electronic Resource
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