Local Quantum Physics pp 174-184 | Cite as

# The Buchholz-Fredenhagen (BF)-Analysis

Chapter

## Abstract

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
Here 𝔄

*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$$P_\mu\psi = B_\mu\psi;\quad B_\mu = B_\mu^\ast \in \pi({\frak A}_{\rm a.l.}).$$

(IV.3.1)

_{a.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*.### Keywords

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1992