Eur. Phys. J. C (2013) 73: 2483
https://doi.org/10.1140/epjc/s10052-013-2483-x

Regular Article - Theoretical Physics

Effects of twisted noncommutativity in multi-particle Hamiltonians

¹ UFABC, Rua Santa Adélia 166, Bangu, cep 09210-170, Santo André (SP), Brazil
² CBPF, Rua Dr. Xavier Sigaud 150, Urca, cep 22290-180, Rio de Janeiro (RJ), Brazil

^* e-mail: toppan@cbpf.br

Received: 3 February 2013
Revised: 3 June 2013
Published online: 12 July 2013

Abstract

The non-commutativity induced by a Drinfel’d twist produces Bopp-shift-like transformations for deformed operators. In a single-particle setting the Drinfel’d twist allows to recover the non-commutativity obtained from various methods which are not based on Hopf algebras. In multi-particle sector, on the other hand, the Drinfel’d twist implies novel features. In conventional approaches to non-commutativity, deformed primitive operators are postulated to act additively. A Drinfel’d twist implies non-additive effects which are controlled by the coproduct. We stress that in our framework, the central element denoted as ħ is associated to an additive operator whose physical interpretation is that of the Particle Number operator.

We illustrate all these features for a class of (abelian twist-deformed) 2D Hamiltonians. Suitable choices of the parameters lead to the Hamiltonian of the non-commutative Quantum Hall Effect, the harmonic oscillator, the quantization of the configuration space. The non-additive effects in the multi-particle sector, leading to results departing from the existing literature, are pointed out.