104 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
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
Symplectically-invariant soliton equations from non-stretching geometric curve flows
A moving frame formulation of geometric non-stretching flows of curves in the
Riemannian symmetric spaces and is
used to derive two bi-Hamiltonian hierarchies of symplectically-invariant
soliton equations. As main results, multi-component versions of the sine-Gordon
(SG) equation and the modified Korteweg-de Vries (mKdV) equation exhibiting
invariance are obtained along with their bi-Hamiltonian
integrability structure consisting of a shared hierarchy of symmetries and
conservation laws generated by a hereditary recursion operator. The
corresponding geometric curve flows in and
are shown to be described by a non-stretching wave map and a
mKdV analog of a non-stretching Schr\"odinger map.Comment: 39 pages; remarks added on algebraic aspects of the moving frame used
in the constructio
Hierarchy of Conservation Laws of Diffusion--Convection Equations
We introduce notions of equivalence of conservation laws with respect to Lie
symmetry groups for fixed systems of differential equations and with respect to
equivalence groups or sets of admissible transformations for classes of such
systems. We also revise the notion of linear dependence of conservation laws
and define the notion of local dependence of potentials. To construct
conservation laws, we develop and apply the most direct method which is
effective to use in the case of two independent variables. Admitting
possibility of dependence of conserved vectors on a number of potentials, we
generalize the iteration procedure proposed by Bluman and Doran-Wu for finding
nonlocal (potential) conservation laws. As an example, we completely classify
potential conservation laws (including arbitrary order local ones) of
diffusion--convection equations with respect to the equivalence group and
construct an exhaustive list of locally inequivalent potential systems
corresponding to these equations.Comment: 24 page
Conservation laws of scaling-invariant field equations
A simple conservation law formula for field equations with a scaling symmetry
is presented. The formula uses adjoint-symmetries of the given field equation
and directly generates all local conservation laws for any conserved quantities
having non-zero scaling weight. Applications to several soliton equations,
fluid flow and nonlinear wave equations, Yang-Mills equations and the Einstein
gravitational field equations are considered.Comment: 18 pages, published version in J. Phys. A:Math. and Gen. (2003).
Added discussion of vorticity conservation laws for fluid flow; corrected
recursion formula and operator for vector mKdV conservation law
On multi-graviton and multi-gravitino gauge theories
This paper studies nonlinear deformations of the linear gauge theory of any
number of spin-2 and spin-3/2 fields with general formal multiplication rules
in place of standard Grassmann rules for manipulating the fields, in four
spacetime dimensions. General possibilities for multiplication rules and
coupling constants are simultaneously accommodated by regarding the set of
fields equivalently as a single algebra-valued spin-2 field and single
algebra-valued spin-3/2 field, where the underlying algebra is factorized into
a field-coupling part and an internal multiplication part. The condition that
there exist a gauge invariant Lagrangian (to within a divergence) for these
algebra-valued fields is used to derive determining equations whose solutions
give all allowed deformation terms, yielding nonlinear field equations and
nonabelian gauge symmetries, together with all allowed formal multiplication
rules as needed in the Lagrangian for demonstration of invariance under the
gauge symmetries and for derivation of the field equations. In the case of
spin-2 fields alone, the main result of this analysis is that all deformations
(without any higher derivatives than appear in the linear theory) are
equivalent to an algebra-valued Einstein gravity theory. By a systematic
examination of factorizations of the algebra, a novel type of nonlinear gauge
theory of two or more spin-2 fields is found, where the coupling for the fields
is based on structure constants of an anticommutative, anti-associative
algebra, and with formal multiplication rules that make the fields
anticommuting (while products obey anti-associativity). Supersymmetric
extensions of these results are obtained in the more general case when spin-3/2
fields are included.Comment: 33 pages (latex
Quaternionic Soliton Equations from Hamiltonian Curve Flows in HP^n
A bi-Hamiltonian hierarchy of quaternion soliton equations is derived from
geometric non-stretching flows of curves in the quaternionic projective space
. The derivation adapts the method and results in recent work by one of
us on the Hamiltonian structure of non-stretching curve flows in Riemannian
symmetric spaces by viewing as a
symmetric space in terms of compact real symplectic groups and quaternion
unitary groups. As main results, scalar-vector (multi-component) versions of
the sine-Gordon (SG) equation and the modified Korteveg-de Vries (mKdV)
equation are obtained along with their bi-Hamiltonian integrability structure
consisting of a shared hierarchy of quaternionic symmetries and conservation
laws generated by a hereditary recursion operator. The corresponding geometric
curve flows in are shown to be described by a non-stretching wave map
and a mKdV analog of a non-stretching Schrodinger map.Comment: 25 pages; typos correcte
Potential Conservation Laws
We prove that potential conservation laws have characteristics depending only
on local variables if and only if they are induced by local conservation laws.
Therefore, characteristics of pure potential conservation laws have to
essentially depend on potential variables. This statement provides a
significant generalization of results of the recent paper by Bluman, Cheviakov
and Ivanova [J. Math. Phys., 2006, V.47, 113505]. Moreover, we present
extensions to gauged potential systems, Abelian and general coverings and
general foliated systems of differential equations. An example illustrating
possible applications of proved statements is considered. A special version of
the Hadamard lemma for fiber bundles and the notions of weighted jet spaces are
proposed as new tools for the investigation of potential conservation laws.Comment: 36 pages, extended versio
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