  We present the method that the pair production of particles is described as a
Fock space state vector, where the pair production stems from receiving energy
from, as an example, gravitational background field. At the same time we show a
sort of new mechanism that in the diffeomorphism-invariant Dirac sea the Weyl
symmetry is broken due to the pair production and this provides us a new
physical interpretation of the origin of Weyl anomaly. In the present paper at
the first place, we consider two kinds of background space-time which are
connected each other by the Weyl transformation. It is shown that the vacuum,
i.e. Dirac sea, in the "curved" space-time \underline{after} making Weyl
transformation is nothing but the pair produced state provided we consider the
state in terms of the Fock space basis in the "flat" space-time
\underline{before} making the Weyl transformation. When summing up with respect
to the time the contribution of pair production of each time-slice we obtain
the Ricci scalar $R$ and then it is related to the trace part $\left\langle
T^{\mu}_{\> \mu}\right\rangle$ of the energy-momentum tensor to produce Weyl
anomaly.
