Faddeev–Jackiw quantization of topological invariants: Euler and Pontryagin classes

Abstract

The symplectic analysis for the four dimensional Pontryagin and Euler invariants is performed within the Faddeev–Jackiw context. The Faddeev–Jackiw constraints and the generalized Faddeev–Jackiw brackets are reported; we show that in spite of the Pontryagin and Euler classes give rise the same equations of motion, its respective symplectic structures are different to each other. In addition, a quantum state that solves the Faddeev–Jackiw constraints is found, and we show that the quantum states for these invariants are different to each other. Finally, we present some remarks and conclusions.