  We introduce the Quantum Dissipative Villain (QDV) model as a prototype model
to study tunneling in dissipative quantum mechanics. Dissipation is provided by
a coupled linear environment. In the QDV model, the discrete character of a
tunneling degree of freedom coupled to an environment is explicit, leading to a
rich dual structure.
  We derive general exact mappings of the QDV model on several dual discrete
representations, including pairs of self-dual models, for general linear
environments and arbitrary temperatures. Self-duality allows to write exact
equations for each correlation function of each representation.
  Analogies with the theory of classical network transformations are also
presented. Finally we discuss the fundamental character of the QDV model.
  For instance, the standard Caldeira-Leggett model, which describes mesoscopic
Josephson junctions in a circuit and many other physical systems, is a special
QDV model. The self-dual structure of the QDV model allows then the exact
generalization of the Schmid approximate self-duality to general linear
environments and arbitrary temperatures.
