  This paper is a continuation of earlier work where a classical history theory
of pure electrodynamics was developed in which the the history fields have
\emph{five} components. The extra component is associated with an extra
constraint, thus enlarging the gauge group of histories electrodynamics. In
this paper we quantise the classical theory developed previously by two
methods. Firstly we quantise the reduced classical history space, to obtain a
reduced quantum history theory. Secondly we quantise the classical
BRST-extended history space, and use the BRST charge to define a
`cohomological' quantum history theory. Finally we show that the reduced
history theory is isomorphic, as a history theory, to the cohomological history
theory.
