Symplectic structure and conserved quantities for some new conformally covariant systems

Abstract

Let M be the n-dimensional Minkowski space, n ⩾ 3. One consequence of [1] is that the null space of the equation {(n − 2k + 2)d ∗d + (n − 2k − 2)dd ∗} Φ = 0 on differential k-forms Φ in M is conformally covariant. The same is true of a nonlinear equation obtained by adding to the above a term homogeneous of degree (n + 2) (n − 2) . This generalizes the well-known conformal covariance properties of the wave equation and the equations φ ± φ (n + 2) (n − 2) = 0 when k = 0, and of Maxwell's equations on a vector potential when k = (n ± 2) 2 (and n is even). We define a natural (conformally invariant) symplectic structure for the new equations, and use it to calculate the (n + 1)(n + 2) 2 conserved quantities corresponding to the standard conformal group generators.