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

Integrable generalizations of Schrodinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces

A moving frame formulation of non-stretching geometric curve flows in Euclidean space is used to derive a 1+1 dimensional hierarchy of integrable SO(3)-invariant vector models containing the Heisenberg ferromagnetic spin model as well as a model given by a spin-vector version of the mKdV equation. These models describe a geometric realization of the NLS hierarchy of soliton equations whose bi-Hamiltonian structure is shown to be encoded in the Frenet equations of the moving frame. This derivation yields an explicit bi-Hamiltonian structure, recursion operator, and constants of motion for each model in the hierarchy. A generalization of these results to geometric surface flows is presented, where the surfaces are non-stretching in one direction while stretching in all transverse directions. Through the Frenet equations of a moving frame, such surface flows are shown to encode a hierarchy of 2+1 dimensional integrable SO(3)-invariant vector models, along with their bi-Hamiltonian structure, recursion operator, and constants of motion, describing a geometric realization of 2+1 dimensional bi-Hamiltonian NLS and mKdV soliton equations. Based on the well-known equivalence between the Heisenberg model and the Schrodinger map equation in 1+1 dimensions, a geometrical formulation of these hierarchies of 1+1 and 2+1 vector models is given in terms of dynamical maps into the 2-sphere. In particular, this formulation yields a new integrable generalization of the Schrodinger map equation in 2+1 dimensions as well as a mKdV analog of this map equation corresponding to the mKdV spin model in 1+1 and 2+1 dimensions.Comment: Published version with typos corrected. Significantly expanded version of a talk given by the first author at the 2008 BIRS workshop on "Geometric Flows in Mathematics and Physics

Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces

Universal bi-Hamiltonian hierarchies of group-invariant (multicomponent) soliton equations are derived from non-stretching geometric curve flows \map(t,x) in Riemannian symmetric spaces $M=G/H$, including compact semisimple Lie groups $M=K$ for $G=K\times K$, $H={\rm diag} G$. The derivation of these soliton hierarchies utilizes a moving parallel frame and connection 1-form along the curve flows, related to the Klein geometry of the Lie group $G\supset H$ where $H$ is the local frame structure group. The soliton equations arise in explicit form from the induced flow on the frame components of the principal normal vector N=\covder{x}\mapder{x} along each curve, and display invariance under the equivalence subgroup in $H$ that preserves the unit tangent vector T=\mapder{x} in the framing at any point $x$ on a curve. Their bi-Hamiltonian integrability structure is shown to be geometrically encoded in the Cartan structure equations for torsion and curvature of the parallel frame and its connection 1-form in the tangent space T_\map M of the curve flow. The hierarchies include group-invariant versions of sine-Gordon (SG) and modified Korteweg-de Vries (mKdV) soliton equations that are found to be universally given by curve flows describing non-stretching wave maps and mKdV analogs of non-stretching Schrodinger maps on $G/H$. These results provide a geometric interpretation and explicit bi-Hamiltonian formulation for many known multicomponent soliton equations. Moreover, all examples of group-invariant (multicomponent) soliton equations given by the present geometric framework can be constructed in an explicit fashion based on Cartan's classification of symmetric spaces.Comment: Published version, with a clarification to Theorem 4.5 and a correction to the Hamiltonian flow in Proposition 5.1

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

Designing Virtuous Sex Robots

We propose that virtue ethics can be used to address ethical issues central to discussions about sex robots. In particular, we argue virtue ethics is well equipped to focus on the implications of sex robots for human moral character. Our evaluation develops in four steps. First, we present virtue ethics as a suitable framework for the evaluation of human–robot relationships. Second, we show the advantages of our virtue ethical account of sex robots by comparing it to current instrumentalist approaches, showing how the former better captures the reciprocal interaction between robots and their users. Third, we examine how a virtue ethical analysis of intimate human–robot relationships could inspire the design of robots that support the cultivation of virtues. We suggest that a sex robot which is equipped with a consent-module could support the cultivation of compassion when used in supervised, therapeutic scenarios. Fourth, we discuss the ethical implications of our analysis for user autonomy and responsibility

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