  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.
