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

Exchange Symmetry Single Particle State Vacuum Representation Observable Algebra Weak Limit Point
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag Berlin Heidelberg 1992