  We prove a priori estimates and, as sequel, existence of Euclidean Gibbs
states for quantum lattice systems. For this purpose we develop a new
analytical approach, the main tools of which are: first, a characterization of
the Gibbs states in terms of their Radon-Nikodym derivatives under shift
transformations as well as in terms of their logarithmic derivatives through
integration by parts formulae, and second, the choice of appropriate Lyapunov
functionals describing stabilization effects in the system. The latter
technique becomes applicable since on the basis of the integration by parts
formulae the Gibbs states are characterized as solutions of an infinite system
of partial differential equations. Our existence result generalize essentially
all previous ones. In particular, superquadratic growth of the interaction
potentials is allowed and $N$-particle interactions for $N\in \mathbb{N}\cup
\{\infty \}$ are included. We also develop abstract frames both for the
necessary single spin space analysis and for the lattice analysis apart from
their applications to our concrete models. Both types of general results
obtained in these two frames should be also of their own interest in infinite
dimensional analysis.
