  An analysis is presented of the phase transition of the quantum Ising model
with transverse field on the d-dimensional hypercubic lattice. It is shown that
there is a unique sharp transition. The value of the critical point is
calculated rigorously in one dimension. The first step is to express the
quantum Ising model in terms of a (continuous) classical Ising model in d+1
dimensions. A so-called `random-parity' representation is developed for the
latter model, similar to the random-current representation for the classical
Ising model on a discrete lattice. Certain differential inequalities are
proved. Integration of these inequalities yields the sharpness of the phase
transition, and also a number of other facts concerning the critical and
near-critical behaviour of the model under study.
