Open group transformations within the Sp(2)-formalism

TL;DR

This paper extends the quantization of open groups within the Sp(2)-formalism, introducing new operators, equations, and solutions to generalize previous finite open group transformation results.

Contribution

It develops an Sp(2)-formalism for quantizing open groups, including a quantum master equation and explicit solutions, expanding the theoretical framework for open group transformations.

Findings

Generalized finite open group transformations to Sp(2)-formalism

Introduced an Sp(2)-version of the quantum master equation

Provided explicit solutions for the master equation

Abstract

Previously we have shown that open groups whose generators are in arbitrary involutions may be quantized within a ghost extended framework in terms of the nilpotent BFV-BRST charge operator. Here we show that they may also be quantized within an Sp(2)-frame in which there are two odd anticommuting operators called Sp(2)-charges. Previous results for finite open group transformations are generalized to the Sp(2)-formalism. We show that in order to define open group transformations on the whole ghost extended space we need Sp(2)-charges in the nonminimal sector which contains dynamical Lagrange multipliers. We give an Sp(2)-version of the quantum master equation with extended Sp(2)-charges and a master charge of a more involved form, which is proposed to represent the integrability conditions of defining operators of connection operators and which therefore should encode the generalized quantum Maurer-Cartan equations for arbitrary open groups. General solutions of this master equation are given in explicit form. A further extended Sp(2)-formalism is proposed in which the group parameters are quadrupled to a supersymmetric set and from which all results may be derived.