We construct some infinite-dimensional Lie algebras of local currents, compatible with the quantum kinematics of a theory based on a deformed Heisenberg-Poincare algebra with fundamental length scales, where the positional operators (in greater than 1+1 dimensions) no longer commute. Recent work by Chryssomalakos and Okon discusses the full set of possible stable deformations of Heisenberg-Poincare algebra, with explanation of the relevant cohomology theory; we base our present work on a specific proposal of Vilela Mendes.
We begin by clarifying the relation of the irreducible representations of a deformed subalgebra to those of the limiting Heisenberg algebra. Our construction of generalized kinetic energy and harmonic oscillator Hamiltonians in this framework leads to an answer different from that suggested by Vilela Mendes. Then we consider two approaches to local current algebra. First, we localize currents with respect to the discrete spectrum of the deformed position operator, and (as expected) see that the resulting Lie algebra necessarily includes elements having arbitrarily wide support. Second, we extend the usual nonrelativistic local current algebra of scalar functions and vector fields (and, correspondingly, the infinite-dimensional semidirect product groups of scalar functions and diffeomorphisms), whose irreducible representations describe a wide variety of quantum systems. The result is to localize with respect to an abstract single-particle configuration space, having one dimension more than the original physical space.
Thus, for example, the deformed (1+1)-dimensional theory entails self-adjoint representations of an infinite-dimensional Lie algebra of nonrelativistic, local currents on (2+1)-dimensional space-time. Interestingly, the local operators no longer act in a single irreducible representation of the (global, finite-dimensional) deformed Lie algebra, but connect the reducing subspaces in a direct integral of irreducible representations. Such an approach seems to open up some new possibilities. For example, representations previously interpreted as describing N indistinguishable anyonic particles in two-space, obeying braid statistics, might also provide local currents for a deformed algebra describing N-particle quantum mechanics in one spatial dimension having a fundamental length scale.