  The two point correlation function of the stress-energy-momentum tensor
describes the propagation of a space-time "micro-earthquake" in the vacuum. In
the framework of the path integral formulation of field theory in curved
space-time, we derive the Ward-Takashi identity for two-point Green's function
of the stress-energy-momentum tensor for a general case of a non-conformal
theory. The identity constrains the longitudinal part of the correlator, with
the vacuum expectation value of the stress-energy-momentum, non-zero in a
non-conformal theory. The obtained formula is demonstrated on the free massive
Dirac fermion theory, treated at the one-loop level. This example befits a
class of phenomenological chiral quarks models which have been used
successfully in numerous applications in the soft non-perturbative regime of
strong interactions. We discuss the constraints following from the
Ward-Takahashi identity for the correlation functions in these models. We also
show how the temporal representation of the two-point correlators, which is an
object amenable to lattice QCD, displays an expected exponential fall-off.
