A recent method of constructing quantum mechanics in noncommutative coordinates, alternative to implying noncommutativity by means of star product is discussed. Within this approach we study Hall effect as well as quantum phases in noncommutative coordinates. The $θ$--deformed phases which we obtain are velocity independent.
Classical results of the axiomatic quantum field theory, namely the irreducibility of the set of field operators, Reeh and Schlieder's theorems and generalized Haag's theorem, are proven in $SO(1,1)$ invariant quantum field theory, of which an important example is noncommutative quantum field theory. New consequences of generalized Haag's theorem are obtained in $SO(1,3)$ invariant theories. It has been proven that the equality of four-point Wightman functions in two theories leads to the equality of elastic scattering amplitudes and thus the total cross-sections in these theories.
We analyze some features of the entanglement entropy for an integer quantum Hall state ($ν=1 $) in comparison with ideas from relativistic field theory and noncommutative geometry. The spectrum of the modular operator, for a restricted class of states, is shown to be similar to the case of field theory or a type ${\rm III}_1$ von Neumann algebra. We present arguments that the main part of the dependence of the entanglement entropy on background fields and geometric data such as the spin connection is given by a generalized Chern-Simons form. Implications of this result for bringing together id…
