Local observables and particle statistics I

Abstract

We consider the family of those states which become asymptotically indistinguishable from the vacuum for observations in far away regions of space. The pure states of this family may be subdivided into superselection sectors labelled by generalized charge quantum numbers. The principle of locality implies that within this family one may define a natural product composition (leading for instance from single particle states ton-particle states). Intrinsically associated with then-fold product of states of one sector there is a unitary representation ofP ⁽ⁿ⁾, the permutation group ofn elements, analogous in its role to that arising in wave mechanics from the permutations of the arguments of ann-particle wave function. We show that each sector possesses a “statistics parameter” λ which determines the nature of the representation ofP ⁽ⁿ⁾ for alln and whose possible values are 0, ±d ⁻¹ (d a positive integer). A sector with λ ≠ 0 has a unique charge conjugate (“antiparticle” states); if λ=d ⁻¹ the states of the sector obey para-Bose statistics of orderd, if λ=−d ⁻¹ they obey para-Fermi statistics of orderd. Some conditions which restrict λ to ± 1 (ordinary Bose or Fermi statistics) are given.