Duality transformations play a very important role in theoretical physics. In this paper I propose new duality transformations for fermionic theories. They map the strong coupling regime of one theory to the weak coupling regime of another theory. These transformations are based on the functional integral representation of the fermionic theories in terms of Grassmann variables and rely heavily on the properties of Grassmann variables. Potential applications include the study of the strong coupling phase of the two dimensional Hubbard model.
A continuous sequence of infinitesimal unitary transformations, combined with an operator product expansion for vertex operators, is used to diagonalize the quantum sine-Gordon model for ${\ensuremath{\beta}}^{2}\ensuremath{\in}(2\ensuremath{\pi},\ensuremath{\infty})$. The leading order of this approximation already gives very accurate results for the single-particle gap in the strong-coupling phase. This approach can be understood as an extension of perturbative scaling theory since it links weak- to strong-coupling behavior in a systematic expansion. The method should also be useful for other strong-coupling problems that can be formulated in terms of vertex operators.