300 research outputs found
Conserved currents of massless fields of spin s>0
A complete and explicit classification of all locally constructed conserved
currents and underlying conserved tensors is obtained for massless linear
symmetric spinor fields of any spin s>0 in four dimensional flat spacetime.
These results generalize the recent classification in the spin s=1 case of all
conserved currents locally constructed from the electromagnetic spinor field.
The present classification yields spin s>0 analogs of the well-known
electromagnetic stress-energy tensor and Lipkin's zilch tensor, as well as a
spin s>0 analog of a novel chiral tensor found in the spin s=1 case. The chiral
tensor possesses odd parity under a duality symmetry (i.e., a phase rotation)
on the spin s field, in contrast to the even parity of the stress-energy and
zilch tensors. As a main result, it is shown that every locally constructed
conserved current for each s>0 is equivalent to a sum of elementary linear
conserved currents, quadratic conserved currents associated to the
stress-energy, zilch, and chiral tensors, and higher derivative extensions of
these currents in which the spin s field is replaced by its repeated
conformally-weighted Lie derivatives with respect to conformal Killing vectors
of flat spacetime. Moreover, all of the currents have a direct, unified
characterization in terms of Killing spinors. The cases s=2, s=1/2 and s=3/2
provide a complete set of conserved quantities for propagation of gravitons
(i.e., linearized gravity waves), neutrinos and gravitinos, respectively, on
flat spacetime. The physical meaning of the zilch and chiral quantities is
discussed.Comment: 26 pages; final version with minor changes, accepted in Proc. Roy.
Soc. A (London
Hamiltonian Flows of Curves in symmetric spaces G/SO(N) and Vector Soliton Equations of mKdV and Sine-Gordon Type
The bi-Hamiltonian structure of the two known vector generalizations of the
mKdV hierarchy of soliton equations is derived in a geometrical fashion from
flows of non-stretching curves in Riemannian symmetric spaces G/SO(N). These
spaces are exhausted by the Lie groups G=SO(N+1),SU(N). The derivation of the
bi-Hamiltonian structure uses a parallel frame and connection along the curves,
tied to a zero curvature Maurer-Cartan form on G, and this yields the vector
mKdV recursion operators in a geometric O(N-1)-invariant form. The kernel of
these recursion operators is shown to yield two hyperbolic vector
generalizations of the sine-Gordon equation. The corresponding geometric curve
flows in the hierarchies are described in an explicit form, given by wave map
equations and mKdV analogs of Schrodinger map equations.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA/ Minor changes made (typos
corrected and more discussion added about parallel frames and vector SG
equations
Parity violating spin-two gauge theories
Nonlinear covariant parity-violating deformations of free spin-two gauge
theory are studied in n>2 spacetime dimensions, using a linearized frame and
spin-connection formalism, for a set of massless spin-two fields. It is shown
that the only such deformations yielding field equations with a second order
quasilinear form are the novel algebra-valued types in n=3 and n=5 dimensions
already found in some recent related work concentrating on lowest order
deformations. The complete form of the deformation to all orders in n=5
dimensions is worked out here and some features of the resulting new
algebra-valued spin-two gauge theory are discussed. In particular, the internal
algebra underlying this theory on 5-dimensional Minkowski space is shown to
cause the energy for the spin-two fields to be of indefinite sign. Finally, a
Kaluza-Klein reduction to n=4 dimensions is derived, giving a parity-violating
nonlinear gauge theory of a coupled set of spin-two, spin-one, and spin-zero
massless fields.Comment: 17 page
Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation
The modified Camassa-Holm (mCH) equation is a bi-Hamiltonian system
possessing -peakon weak solutions, for all , in the setting of an
integral formulation which is used in analysis for studying local
well-posedness, global existence, and wave breaking for non-peakon solutions.
Unlike the original Camassa-Holm equation, the two Hamiltonians of the mCH
equation do not reduce to conserved integrals (constants of motion) for
-peakon weak solutions. This perplexing situation is addressed here by
finding an explicit conserved integral for -peakon weak solutions for all
. When is even, the conserved integral is shown to provide a
Hamiltonian structure with the use of a natural Poisson bracket that arises
from reduction of one of the Hamiltonian structures of the mCH equation. But
when is odd, the Hamiltonian equations of motion arising from the conserved
integral using this Poisson bracket are found to differ from the dynamical
equations for the mCH -peakon weak solutions. Moreover, the lack of
conservation of the two Hamiltonians of the mCH equation when they are reduced
to -peakon weak solutions is shown to extend to -peakon weak solutions
for all . The connection between this loss of integrability structure
and related work by Chang and Szmigielski on the Lax pair for the mCH equation
is discussed.Comment: Minor errata in Eqns. (32) to (34) and Lemma 1 have been fixe
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