Higher-derivative non-Abelian gauge fields via the Faddeev–Jackiw formalism

R Bufalo,B M Pimentel

doi:10.1140/epjc/s10052-014-2993-1

Abstract

In this paper we analyze two higher-derivative theories, the generalized electrodynamics and the Alekseev–Arbuzov–Baikov effective Lagrangian from the point of view of the Faddeev–Jackiw symplectic approach. It is shown that the full set of constraints is obtained directly from the zero-mode eigenvectors, and that they are in accordance with well-known results from Dirac’s theory, a recurrent issue in the literature. The method shows to be rather economical in relation to the Dirac one, obviating thus unnecessary classification and calculations. Afterwards, to conclude we construct the transition amplitude of the non-Abelian theory following a constrained BRST method.

Highlights

A standard classical treatment of constrained theories was given originally by Dirac [1,2], it essentially analyzes the canonical structure of any theory, and it has been widely used in a great variety of quantum systems
In this paper we have presented a canonical study of higher-derivative theories, the Podolsky electrodynamics and its non-Abelian extension, the Alekseev–Arbuzov–Baikov effective Lagrangian, in the point of view of the symplectic Faddeev–Jackiw approach
The Dirac method remains as the standard method to deal with constrained systems, it has been recognized that some calculation is unnecessarily cumbersome there, and it is exactly in this point where the Faddeev– Jackiw (FJ) method shows to be an economical and rich framework for first-order Lagrangian functions, obviating mainly unnecessary calculations

Summary

Introduction

A standard classical treatment of constrained theories was given originally by Dirac [1,2], it essentially analyzes the canonical structure of any theory, and it has been widely used in a great variety of quantum systems. It should be realized that Dirac’s methodology is unnecessarily cumbersome and can be streamlined Within this context, Faddeev and Jackiw [3] suggested a symplectic approach for constrained systems based in a first-order Lagrangian. It is important to emphasize that sometimes it happens that the (iteratively deformed1) two-form matrix is singular and no new constraint is obtained from the corresponding zero-mode This is the case when one deals with gauge theories. Our main goal here would be exactly to study both higher-derivative theories, Podolsky’s electrodynamics and a non-Abelian [35,36] extension of the model, known as the Alekseev–Arbuzov–Baikov effective Lagrangian [37] in the framework of the Faddeev–Jackiw symplectic approach.

Faddeev–Jackiw symplectic method

Generalized electrodynamics

Transition amplitude via BFV

Concluding remarks