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.
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