Equivalence of the (generalised) Hadamard and microlocal spectrum condition for (generalised) free fields in curved spacetime

We prove that the singularity structure of all n-point distributions of a state of a generalised real free scalar field in curved spacetime can be estimated if the two-point distribution is of Hadamard form. In particular this applies to the real free scalar field and the result has applications in perturbative quantum field theory, showing that the class of all Hadamard states is the state space of interest. In our proof we assume that the field is a generalised free field, i.e. that it satisies scalar (c-number) commutation relations, but it need not satisfy an equation of motion. The same argument also works for anti-commutation relations and it can be generalised to vector-valued fields. To indicate the strengths and limitations of our assumption we also prove the analogues of a theorem by Borchers and Zimmermann on the self-adjointness of field operators and of a very weak form of the Jost-Schroer theorem. The original proofs of these results in the Wightman framework make use of analytic continuation arguments. In our case no analyticity is assumed, but to some extent the scalar commutation relations can take its place.Comment: 18 page

An Algebraic Jost-Schroer Theorem for Massive Theories

We consider a purely massive local relativistic quantum theory specified by a family of von Neumann algebras indexed by the space-time regions. We assume that, affiliated with the algebras associated to wedge regions, there are operators which create only single particle states from the vacuum (so-called polarization-free generators) and are well-behaved under the space-time translations. Strengthening a result of Borchers, Buchholz and Schroer, we show that then the theory is unitarily equivalent to that of a free field for the corresponding particle type. We admit particles with any spin and localization of the charge in space-like cones, thereby covering the case of string-localized covariant quantum fields.Comment: 21 pages. The second (and crucial) hypothesis of the theorem has been relaxed and clarified, thanks to the stimulus of an anonymous referee. (The polarization-free generators associated with wedge regions, which always exist, are assumed to be temperate.

Cut Diagrams for High Energy Scatterings

A new approach is introduced to study QCD amplitudes at high energy and comparatively small momentum transfer. Novel cut diagrams, representing resummation of Feynman diagrams, are used to simplify calculation and to avoid delicate cancellations encountered in the usual approach. Explicit calculation to the 6th order is carried out to demonstrate the advantage of cut diagrams over Feynman diagrams.Comment: uu-encoded file containing a latex manuscript with 14 postscript figure