Non-Commutative Time, the Quantum Hall Effect and Twistor Theory

Abstract

It is proposed that Maxwell theory, with a topological term, in four non-commutative dimensions, where the co-ordinates obey the Heisenberg algebra, is an umbrella theory for the description of the two-dimensional Quantum Hall Effect (following the fluidic approach of Susskind and Polychronakos), the twistor analyses of self-dual gravity, solitons and instantons and the twistor theory of isolated horizons in space-time. Applied to the Quantum Hall case, the underlying metric is Lorentzian: two canonically conjugate co-ordinates are spatial, the other two co-ordinates naturally are null and represent time and a conjugate energy variable.