Finite BRST-antiBRST Transformations in Generalized Hamiltonian Formalism

TL;DR

This paper develops finite BRST-antiBRST transformations within the Hamiltonian formalism, demonstrating their role in gauge fixing, gauge independence, and unitarity in constrained dynamical systems, especially Yang-Mills theory.

Contribution

It introduces explicit finite BRST-antiBRST transformations in the Hamiltonian formalism, including field-dependent cases, and connects these to gauge changes and unitarity proofs.

Findings

Finite transformations are quadratic in parameters.

Field-dependent transformations change gauge fixing without affecting vacuum functional.

Explicit connection between Hamiltonian and Lagrangian path integrals in Yang-Mills theory.

Abstract

We introduce the notion of finite BRST-antiBRST transformations for constrained dynamical systems in the generalized Hamiltonian formalism, both global and field-dependent, with a doublet $\lambda_{a}$, $a=1,2$, of anticommuting Grassmann parameters and find explicit Jacobians corresponding to these changes of variables in the path integral. It turns out that the finite transformations are quadratic in their parameters. Exactly as in the case of finite field-dependent BRST-antiBRST transformations for the Yang--Mills vacuum functional in the Lagrangian formalism examined in our previous paper [arXiv:1405.0790[hep-th]], special field-dependent BRST-antiBRST transformations with functionally-dependent parameters $\lambda_{a}=\int dt\(s_{a}\Lambda) $, generated by a finite even-valued function $\Lambda(t)$ and by the anticommuting generators $s_{a}$ of BRST-antiBRST transformations, amount to a precise change of the gauge-fixing function for arbitrary constrained dynamical systems. This proves the independence of the vacuum functional under such transformations. We derive a new form of the Ward identities, depending on the parameters $\lambda_{a}$, and study the problem of gauge-dependence. We present the form of transformation parameters which generates a change of the gauge in the Hamiltonian path integral, evaluate it explicitly for connecting two arbitrary $R_{\xi}$-like gauges in the Yang--Mills theory and establish, after integration over momenta, a coincidence with the Lagrangian path integral [arXiv:1405.0790[hep-th]], which justifies the unitarity of the $S$-matrix in the Lagrangian approach.