Locality and the structure of particle states

Abstract

Starting from the principle of locality of observables we derive localization properties of massive particle states which hold in all models of relativistic quantum theory, including gauge theories. It turns out that particles may always be regarded as well localized distributions of matter, although their mathematical description might require the introduction of non-local (unobservable) fields, which are assigned to infinite string-like regions. In spite of the non-locality of these fields one can show that such particles obey Bose-or Fermi (para) statistics, that to each particle there exists an antiparticle and that collision states of particles exist. A selfcontained exposition of the underlying physical ideas is given in the Introduction, and some perspectives for the structure of field-theoretic models arising from our analysis are discussed in the Conclusions.

The Spin-Statistics Theorem for Anyons and Plektons in d = 2+1

“…Fields are localizable to the same extent to which the charges are localizable which they carry. In the case of braid group statistics, the charges cannot be localized in bounded regions of spacetime [4], but they can be localized, in the massive case, in regions which extend to infinity in some space-like direction, namely, in space-like cones [3]. Now the manifold of space-like directions,…”

“…But our framework, using a restricted notion of charged fields, can be set up starting from the standard assumptions [12] of local relativistic quantum theory on the observabels plus weak Haag Duality, together with our assumptions on the mass spectrum. For the convenience of the reader, we sketch in Appendix A how this may be done and indicate the relation with the notions used in the literature [3,5,8]. 4 Two other possibilities are: To introduce a reference space-like cone from which all allowed localization cones have to keep space-like separated (this cone playing the role of a "cut" in the context of multivalued functions) [3]; or a cohomology theory of nets of operator algebras as introduced by Roberts [19][20][21].…”

“…3 States and Fields. Denoting the quantum numbers of our sector collectively by χ, the space of states of the sector corresponds to a Hilbert space H χ .…”

“…Two pathsẽ 1 andẽ 2 starting at e 0 and ending in C will be called equivalent w.r.t. C iff the pathẽ 2 * ẽ −1 1 (the inverse ofẽ 1 followed byẽ 2 ) is fixed-endpoint 3 Recall that in the case of braid group statistics there is no canonical way to construct a field algebra from the observables [18]. But our framework, using a restricted notion of charged fields, can be set up starting from the standard assumptions [12] of local relativistic quantum theory on the observabels plus weak Haag Duality, together with our assumptions on the mass spectrum.…”

“…In [2], Buchholz and Epstein extended the theorem to massive particles carrying a non-localizable charge. In the purely massive case, such charges are still localizable in space-like cones [3], i.e., cones in spacetime which extend to space-like infinity. 1 The analysis of Buchholz and Epstein was carried out in four-dimensional spacetime, in which case λ is a real number associated with a unitary representation of the permutation group (λ > 0 corresponding to Bosons and λ < 0 corresponding to Fermions).…”