  Here a recently observed weak first order transition in doped SrTiO3 is
argued to be a consequence of the coupling between strain and order parameter
fluctuations. Starting with a semi-microscopic action, and using
renormalization group equations for vertices, we write the free energy of such
a system. This fluctuation renormalized free energy is then used to discuss the
possibility of first order transition at zero temperature as well as at finite
temperature. An asymptotic analysis predicts small but a finite discontinuity
in the order parameter near a mean field quantum critical point at zero
temperature. In case of finite temperature transition, near quantum critical
point such a possibility is found to be extremely weak. Results are in accord
with some experimental findings on quantum paraelectrics such as SrTiO3 and
KTaO3.
