The Mathematical Structure of the Quantum BRST Constraint Method

TL;DR

This thesis explores the mathematical foundations of the quantum BRST constraint method within a C*-algebraic framework, comparing it to the Dirac method and revealing key differences and conditions for equivalence.

Contribution

It formulates the quantum BRST structures in a C*-algebraic context and compares them rigorously to the Dirac method, highlighting their differences and specific cases of equivalence.

Findings

Quantum BRST and Dirac methods are not generally equivalent.

BRST does not always remove ghosts in the physical algebra.

BRST and Dirac physical algebras coincide in a C*-algebraic model of QEM.

Abstract

This thesis describes the mathematical structures of the quantum BRST constraint method. Ultimately, the quantum BRST structures are formulated in a C*-algebraic context, leading to comparison of the quantum BRST and the Dirac constraint method in a mathematically consistent framework. Comparisons of rigorous models leads to the following three consequences: The quantum BRST method and quantum Dirac method of constraints are not equivalent in general; The BRST method does not remove the ghosts in the BRST physical algebra in general. Extra selection criteria are required to select the correct physical space, which do not guarantee correspondence between the Dirac and BRST physical algebras; Conversely, the BRST physical algebra and Dirac physical algebra coincide for a C*-algebraic model of QEM. This is a rigorous example of Lagrangian BRST, implying quantum Lagrangian and quantum Hamiltonian BRST are not equivalent constraint methods.