Spin and statistics on the Groenewold-Moyal plane: Pauli-forbidden levels and transitions

The Groenewold-Moyal plane is the algebra A(0)(Rd+1) of functions on Rd+1 with the *-product as the multiplication law, and the commutator (x(mu), x(v)) = i theta(mu v) (mu,v = 0, 1,..., d) between the coordinate functions. Chaichian et al.(1) and Aschieri et al.(2) have proved that the Poincare group acts as automorphisms on A(theta)(Rd+1) if the coproduct is deformed. (See also the prior work of Majid,(3) Oeckl(4) and Grosse et al.(5)) In fact; the diffeomorphism group with a deformed coproduct also does so according to the results of Ref. 2. In this paper we show that for this new action, the Bose and Fermi commutation relations are deformed as well. Their potential applications to the quantum Hall effect are pointed out. Very striking consequences of these deformations are the occurrence of Pauli-forbidden energy levels and transitions. Such new effects are discussed in simple cases.