Jackiw and Pi reply

Abstract

Jackiw and Pi Reply: That our soliton profiles satisfy the requisite equations is of course verified without encountering distributions: The original field equations are nonsingular, the profiles are nonsingular, and straightforward differentiation establishes that the profiles indeed solve the equations. To derive our solutions one may use singular coordinates, as we do in Ref. 1. Or one may ignore singular points in the coordinate system by invoking continuity of the nonsingular solutions; this is the strategy of Ref, 2. Hagen 3 adopts a method, which has already appeared in the literature, 4 wherein singular coordinates are avoided. Still other approaches are available. 5 It matters not whether one uses singular or nonsingular coordinates, provided singularities are treated consistently and are absent from the final solutions. That we achieve this desired goal is self-evident.In one approach to the consistent handling of singularities, we use an unfamiliar and unconventional representation for the planar 8 function, !

g=2as a gauge condition

“…The strength parameter m carries dimensions of mass and the term has been posited previously in a gauge-and parity-invariant mass generation mechanism for a gauge theory, where also the gauge transformation (11) was introduced [5]. The spin-2 case does not exhibit exactly the same behavior as above, yet something similar, but less direct, does hold.…”

Clebsch (string) decomposition in d=3 field theory

“…A nonabelian generalization of the Clebsch decomposition has recently been suggested [9]. It shares the abelian property that the CS term is a pure divergence and a similar degeneracy in the field equations occurs.…”

Particle creation via relaxing hypermagnetic knots

“…the identity map S 3 → S 3 ). Perform an Abelian projection via G k ( x) = tr (U † ∂ k Uσ 3 ), then the Abelian gauge field G k is proportional to the gauge field (7), and the magnetic field ǫ jkl ∂ k G l is proportional to the knotted hypermagnetic field (8), [24]. Now we need the following important result on zero modes of the above spatial Dirac equation (5) for the specific gauge field (7).…”

“…Geometrically, B is tangent to the closed curves χ = const (see e.g. [23,25,26,24]). The Hopf index N H of χ may be computed from B via…”