Microlocal spectrum condition and hadamard form for vector-valued quantum fields in curved spacetime

Some years ago, Radzikowski has found a characterization of Hadamard states for scalar quantum fields on a four-dimensional globally hyperbolic spacet ime in terms of a specific form of the wavefront set of their two-point fun ctions (termed "wavefront set spectrum condition"), thereby initiating a ma jor progress in the understanding of Hadamard states and the further develo pment of quantum field theory in curved spacetime. In the present work, we extend this important result on the equivalence of the wavefront set spectr um condition with the Hadamard condition from scalar fields to vector field s (sections in a vector bundle) which are subject to a wave-equation and ar e quantized so as to fulfill the covariant canonical commutation relations, or which obey a Dirac equation and are quantized according to the covarian t anti-commutation relations, in any globally hyperbolic spacetime having d imension three or higher. In proving this result, a gap which is present in the published proof for t he scalar field case will be removed. Moreover we determine the short-dista nce scaling limits of Hadamard states for vector-bundle valued fields, find ing them to coincide with the corresponding flat-space, massless vacuum sta tes.