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

August 22, 2000

Some years ago, Radzikowski has found a characterization of Hadamard states
for scalar quantum fields on a four-dimensional globally hyperbolic spacetime
in terms of a specific form of the wavefront set of their two-point functions
(termed `wavefront set spectrum condition'), thereby initiating a major
progress in the understanding of Hadamard states and the further development of
quantum field theory in curved spacetime. In the present work, we extend this
important result on the equivalence of the wavefront set spectrum condition
with the Hadamard condition from scalar fields to vector fields (sections in a
vector bundle) which are subject to a wave-equation and are quantized so as to
fulfill the covariant canonical commutation relations, or which obey a Dirac
equation and are quantized according to the covariant anti-commutation
relations, in any globally hyperbolic spacetime having dimension three or
higher. In proving this result, a gap which is present in the published proof
for the scalar field case will be removed. Moreover we determine the
short-distance scaling limits of Hadamard states for vector-bundle valued
fields, finding them to coincide with the corresponding flat-space, massless
vacuum states.

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