The Mathematical Structure of the Quantum BRST Constraint Method

This thesis describes 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. Rigorous models are constructed for the heuristic examples of BRST for quantum electromagnetism (BRST-QEM) and Hamiltonian BRST with a finite number of constraints. This facilitates comparison between the results produced by the BRST method, and the results of the T -procedure of Grundling and Hurst for the quantum Dirac constraint method. The different constraint methods are shown not to be equivalent for the examples of Hamiltonian BRST with a finite number of constraints that close, and a BRST-QEM model constructed using the Resolvent Algebra of Buchholz and Grundling with covariant test function space. Moreover, this leads to the following three consequences: The quantum BRST method, and quantum Dirac method of constraints, are not equivalent in general. Examples of quantum Hamiltonian BRST can be constructed to show that the BRST method does not remove the ghosts in the BRST physical algebra. This occurs since quantum Hamiltonian BRST selects multiple copies of the physical state space selected by the Dirac algorithm, and the ghosts are not removed from the BRST-physical state space. Extra selection criteria are required to select the correct physical space, which do not gaurantee correspondence between the Dirac and BRST physical algebras. Conversely, the BRST physical algebra and Dirac physical algebra coincide when QEM is encoded in the auxiliary Resolvent Algebra. This is a rigorous example of Lagrangian BRST, hence quantum Lagrangian and quantum Hamiltonian BRST are not equivalent constraint methods.