The Buchholz-Fredenhagen (BF)-Analysis

  • Rudolf Haag
Part of the Texts and Monographs in Physics book series (TMP)


The DHR-criterion (IV. 1.2) aimed at singling out the subset of states with vanishing matter density at infinity within a purely massive theory. With the same aim in mind Buchholz and Fredenhagen start from the consideration of a charge sector in which the space-time translations are implementable by unitary operators U(x) and the energy-momentum spectrum is as pictured in fig. (IV.3.1). It shall contain an isolated mass shell of mass m (single particle states) separated by a gap from the remainder of the spectrum which begins at mass values above M > m. In this situation they construct states with spectral support on a bounded part of the single particle mass shell which may be regarded as strongly localized in the following sense. Let π denote the representation of the observable algebra for this sector, ℋ the Hilbert space on which it acts and ψ the state vector of such a localized state. Then the effect of a finite translation on ψ can be reproduced by the action of an almost local operator on ψ, or, in terms of the infinitesimal generators (the energy-momentum operators in this representation) by
$$P_\mu\psi = B_\mu\psi;\quad B_\mu = B_\mu^\ast \in \pi({\frak A}_{\rm a.l.}).$$
Here 𝔄a.l. denotes the almost local part of 𝔄 i.e. the set of elements which can be approximated by local observables in a diamond of radius r with an error decreasing in norm faster than any inverse power of r.


Exchange Symmetry Single Particle State Vacuum Representation Observable Algebra Weak Limit Point 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Rudolf Haag
    • 1
  1. 1.IL Institut für Theoretische PhysikUniversität HamburgHamburg 50Fed. Rep. of Germany

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