Micro-local approach to the Hadamard condition in quantum field theory on curved space-time
Due to the lack of translational invariance there is in general no preferred vacuum state for a quantum field theory on a general spacetime. For free scalar fields this problem is in general solved by the Hadamard condition, which distinguishes a whole class of quasi-free states by certain singularity properties of their two-point function. The subject of the present paper is the study of this condition from the micro-local point of view by relating the global Hadamard condition to the theory of distinguished parametrices and to a condition on the wave front set of the two-point function. To this end, the author reviews Kay's and Wald's rigorous definition of the global Hadamard condition [\textit{B. S. Kay} and \textit{R. M. Wald}, Phys. Rep. 207, 49-136 (1991)] and the theory of distinguished parametrices of Duistermaat and Hörmander, specializing to the case of a Klein-Gordon operator on a globally hyperbolic Lorentzian manifold \((M,g)\). In the main theorem it is then proven that the following is equivalent: (1) the global Hadamard condition, (2) the property that the Feynman propagator of the two-point function is a distinguished parametrix and (3) the property that the wave front set of the two-point function \(\omega_2\) is given by \[ WF'(\omega_2) =\{((x_1,k_1), (x_2,k_2)) \in T^*M \setminus 0 \times T^* M\setminus 0\mid(x_1,k_1) \sim(x_2,k_2),\quad k_1 \triangleright 0\}, \] where \((x_1,k_1)\sim(x_2, k_2)\) means that \((x_1,k_1)\) and \((x_2,k_2)\) are on the same null geodesic strip and \(k_1\triangleright 0\) is a shorthand for: \(k_1\) is future directed. The last condition is called the ``wave front set spectrum condition'' because it is reminiscent of the spectral condition on Minkowski space. It represents a remarkable progress in quantum field theory on Minkowski space, since it allows the mathematically rigorous treatment of topics which were available up to know only on Minkowski space, e.g. perturbative renormalizability. A first step in this direction is the definition of Wick ordered products of free fields in \textit{R. Brunetti, K. Fredenhagen} and \textit{M. Köhler} [Commun. Math. Phys. 180, No. 3, 633-652 (1996)] which is based on a variant of the wave front set spectrum condition.
scientific article; zbMATH DE number 912969
scientific article