Local Currents for a Deformed Heisenberg–Poincaré Lie Algebra of Quantum Mechanics, and Anyon Statistics

Abstract

We set out to construct a Lie algebra of local currents whose space integrals, or “charges”, form a subalgebra of the deformed Heisenberg–Poincare algebra of quantum mechanics discussed by Vilela Mendes, parameterized by a fundamental length scale l. One possible technique is to localize with respect to an abstract single-particle configuration space having one dimension more than the original physical space. Then in the limit l→0, the extra dimension becomes an unobservable, internal degree of freedom. The deformed (1+1)-dimensional theory entails self-adjoint representations of an infinite-dimensional Lie algebra of nonrelativistic, local currents modeled on (2+1)-dimensional space-time. This suggests a new possible interpretation of such representations of the local current algebra, not as describing conventional particles satisfying bosonic, fermionic, or anyonic statistics in two-space, but as describing systems obeying these statistics in a deformed one-dimensional quantum mechanics. In this context, we have an interesting comparison with earlier results of Hansson, Leinaas, and Myrheim on the dimensional reduction of anyon systems. Thus motivated, we introduce irreducible, anyonic representations of the deformed global symmetry algebra. We also compare with the technique of localizing currents with respect to the discrete positional spectrum.