Universal Groups of Effect Spaces

Gerald A. Goldin, Ralph Menikoff, D. H. Sharp

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.

Stanley Gudder

G. F. Tuthill

The author has investigated percolation properties of unfrustrated plaquettes in the two-dimensional Ising model with nearest-neighbour bonds +or-J distributed at random. The effects of plaquette correlations are considered using a two-parameter position-space renormalisation group (PSRG) approach, and it is found (within the framework of this approximate PSRG) that this correlated site percolation problem is in the same universality class as random site percolation. The percolation threshold is then found from large-cell PSRG methods to be indistinguishable from that for randomly placed sites.

E. C. G. Sudarshan

Tomasz Maciążek, Adam Sawicki

Abstract We show that non-abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces. In particular, we formulate a general framework for describing quantum statistics of particles constrained to move in a topological spaceX. The framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space ofXwhich can be achieved by determining its homology groups. We apply this methodology for configuration spaces of graphs. As a conclusion, we provide families of graphs which are good candidates for studying simple effective models of anyon dynamics as well as models of non-abelian anyons on networks that are used in quantum computing. These conclusions are based on our solution of the so-called universal presentation problem for homology groups of graph configuration spaces for certain families of graphs.

Ivan Chajda, Helmut Länger