Osterwalder-Schrader Reconstruction and Diffeomorphism Invariance Abhay Ashtekar 1,5 , Donald Marolf 2,5 Jos´ e Mour˜ ao 3 and Thomas Thiemann 4,5 1. Center for Gravitational Physics and Geometry Department of Physics, The Pennsylvania State University University Park, PA 16802, USA 2. Department of Physics, Syracuse University, Syracuse, NY 13244, USA, 3. Departamento de F´ ısica, Instituto Superior T´ ecnico, Av. Rovisco Pais, 1049-001 Lisboa Codex, Portugal, 4. Albert-Einstein Institut, MPI f¨ ur Gravitationsphysik, Am M¨ uhlenberg, Haus 5, 14476 Golm, Germany, 5. Institute for Theoretical Physics, University of California, Santa Barbara, California, 93106, USA. Abstract The Osterwalder-Schrader reconstruction theorem enables one to obtain a Hilbert space quantum field theory from a measure on the space of (Eu- clidean) histories of a scalar quantum field. In this note we observe that, in an appropriate setting, this result can be extended to include more general theories. In particular, one can allow gauge fields and consider theories which are invariant under diffeomorphisms of the spacetime manifold. I. INTRODUCTION For scalar field theories in flat space-time, the Osterwalder-Schrader framework provides a valuable link between Euclidean and Minkowskian descriptions of the quantum field. In this note we will focus only on one aspect of that framework, namely the reconstruction theorem [1] which enables one to recover the Hilbert space of quantum states and the Hamiltonian operator, starting from an appropriate measure on the space of Euclidean paths. At least in simple cases, this procedure provides a precise correspondence between the path integral and canonical approaches to quantization. The purpose of this note is to extend the reconstruction theorem in two directions. The first direction is suggested by the fact that the standard formulation [2] is geared to “kinematically linear” systems —such as interacting scalar field theories— where the space of Euclidean paths has a natural vector space structure. More precisely, paths are assumed to belong to the space of Schwartz distributions and this assumption then permeates the entire framework. Although this assumption seems natural at first, in fact it imposes a rather 1