Equivalence of domains for hyperbolic Hubbard-Stratonovich transformations

We settle a long standing issue concerning the traditional derivation of non-compact non-linear sigma models in the theory of disordered electron systems: the hyperbolic Hubbard-Stratonovich (HS) transformation of Pruisken-Schafer type. Only recently the validity of such transformations was proved in the case of U(p, q) (non-compact unitary) and O(p, q) (non-compact orthogonal) symmetry. In this article we give a proof for general non-compact symmetry groups. Moreover, we show that the Pruisken-Schafer type transformations are related to other variants of the HS transformation by deformation of the domain of integration. In particular we clarify the origin of surprising sign factors which were recently discovered in the case of orthogonal symmetry. (C) 2011 American Institute of Physics. [doi:10.1063/1.3585672]