  This dissertation presents a semiclassical analysis of conical topology
change in $1+1$ spacetime dimensions wherein, to lowest order, the ambient
spacetime is classical and fixed while the scalar field coupled to it is
quantized. The vacuum expectation value of the scalar field stress-energy
tensor is calculated via two different approaches. The first of these involves
the explicit determination of the so called Sorkin-Johnston state on the cone
and an original regularization scheme, while the latter employs the conformal
vacuum and the more conventional point-splitting renormalization. It is found
that conical topology change seems not to suffer from the same pathologies that
trousers-type topology change does. This provides tentative agreement with
conjectures due to Sorkin and Borde, which attempt to classify topology
changing spacetimes with respect to their Morse critical points and in
particular, that the cone and yarmulke in $1+1$ dimensions lack critical points
of unit Morse index.
