Spin and Statistics on the Groenewold–moyal Plane: Pauli-Forbidden Levels and Transitions

Abstract

The Groenewold-Moyal plane is the algebra A θ (R d+1 ) of functions on R d+1 with the * -product as the multiplication law, and the commutator [x µ ,x ν ] = iθ µν (µ, ν = 0, 1, ..., d) between the coordinate functions. Chaichian et al. [1] and Aschieri et al. [2] have proved that the Poincaré group acts as automorphisms on A θ (R d+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 [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.PACS numbers: 11.10. Nx, 11.30.Cp Dedicated to Rafael Sorkin, our friend and teacher, and a true and creative seeker of knowledge. *

Twisted statistics inκ-Minkowski spacetime

“…It may be noted that although the twist element F ϕ depends explicitly on the choice of the φ realization, it is a remarkable fact that the twisted flip operator τ ϕ as obtained in (43) is independent of the class of realizations (8) and the corresponding orderings. This conclusion follows naturally from the properties of the twist element (22) and the twisted flip operator (43) in the κ space. By construction, any symmetrization or antisymmetrization of two particle states carried out using the flip operator τ ϕ would be preserved under the action of the Poincare group with a twisted coproduct.…”

“…We now restrict our attention to the special cases of the ϕ realizations given in (8). For this particular special class of realizations, the twist element is given in (22). Using (22) and (42), we obtain an explicit expression for the twisted flip operator τ ϕ for the class (8) of realizations as…”

“…The lack of locality leads to a failure of causality [30], and the spin-statistics theorem also fails to be valid. In particular, the Pauli exclusion principle is not expected to be valid for theories based on the algebra (31), and bounds on the deformation parameter a can be obtained from the bounds on the validity of the Pauli exclusion principle [22,23].…”

“…This issue has naturally arisen in the context of noncommutative quantum mechanics and field theory [20,21,22,23,24]. In the noncommutative case, the issue of statistics is closely related to the symmetry of the noncommutative spacetime on which the dynamics is being studied.…”

“…In the presence of noncommutativity, it is thus natural to demand that the statistics is invariant under the action of the twisted symmetry. This condition leads to a new twisted flip operator, which is compatible with the twisted coproduct of the symmetry group [22,23]. The operators projecting to the symmetric and antisymmetric sectors of the Hilbert space are then constructed from the twisted flip operator.…”