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

TL;DR

This paper applies Faddeev-Jackiw symplectic analysis to four-dimensional topological invariants, revealing differences in their symplectic structures and quantum states despite similar equations of motion.

Contribution

It provides a novel symplectic analysis of Pontryagin and Euler classes, highlighting their distinct quantum states within the Faddeev-Jackiw framework.

Findings

Pontryagin and Euler classes have different symplectic structures.

Quantum states for these invariants are distinct.

Faddeev-Jackiw brackets are explicitly derived.

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.