Representations of a local current algebra in nonsimply connected space and the Aharonov–Bohm effect

Abstract

A recent paper established technical conditions for the construction of a class of induced representations of the nonrelativistic current group 𝒮Λ𝒦, where 𝒮 is Schwartz’s space of rapidly decreasing C∞ functions, and 𝒦 is a group of C∞ diffeomorphisms of Rs. Bose and Fermi N-particle systems were recovered as unitarily inequivalent induced representations of the group by lifting the action of 𝒦 on an orbit Δ⊆𝒮′ to its universal covering space δ̃. For s⩾3, δ̃ is the coordinate space for N particles, which is simply connected. In two-dimensional space, however, the coordinate space is multiply connected, implying induced representations other than those describing the usual Bose or Fermi statistics; these are explored in the present paper. Likewise the Aharonov–Bohm effect is described by means of induced representations of the local observables, defined in a nonsimply connected region of Rs. The vector potential plays no role in this description of the Aharonov–Bohm effect.