Weyl calculus and Noether currents: An application to cubic interactions

Expanded version of the lectures presented by X.B. at the 6th international spring school and workshop on quantum field theory and Hamiltonian systems (Calimanesti & Caciulata, Romania; May 2008).National audienceCubic couplings between a complex scalar field and an infinite tower of symmetric tensor gauge fields of each rank are investigated. A symmetric conserved current, bilinear in a free scalar field and containing r derivatives, is provided for any rank r>0 and is related to the corresponding rigid symmetry of Klein-Gordon's Lagrangian. Following Noether's method, the scalar field interacts with the tensor gauge fields via minimal coupling to the conserved currents. The corresponding cubic vertex is written in a very compact form by making use of Weyl's symbols. This enables the explicit computation of the non-Abelian gauge symmetry group, the four-scalar elastic scattering tree amplitude and the lower orders of the effective actions arising from integrating out either the scalar or the gauge fields. The tree scattering amplitude corresponding to the current exchanges between two scalar particles exhibits an exponential fall-off in the high-energy limit