  In this paper we address the relationship between zero temperature Glauber
dynamics and the diffusion-annihilation problem in the free fermion case. We
show that the well-known duality transformation between the two problems can be
formulated as a similarity transformation if one uses appropriate (toroidal)
boundary conditions. This allow us to establish and clarify the precise nature
of the relationship between the two models. In this way we obtain a one-to-one
correspondence between observables and initial states in the two problems. A
random initial state in Glauber dynamics is related to a short range correlated
state in the annihilation problem. In particular the long-time behaviour of the
density in this state is seen to depend on the initial conditions. Hence, we
show that the presence of correlations in the initial state determine the
dependence of the long time behaviour of the density on the initial conditions,
even if such correlations are short-ranged. We also apply a field-theoretical
method to the calculation of multi-time correlation functions in this initial
state.
