Effects of twisted noncommutativity in multi-particle Hamiltonians

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.

Appendix: The unfolded Lie algebra \(\mathcal{G}\)

The structure constants of the unfolded Lie algebra $\mathcal{G}$ whose generators are introduced in (1), are explicitly given by

$$\begin{aligned} \begin{aligned} & [x_i,p_j]=i\hbar \delta_{ij}, \\ & [x_i,P_{jj}]= 2i\delta_{ij} p_j, \qquad [p_i,X_{jj}] = -2 \delta_{ij}x_j, \\ & [x_i,M_{jk}] = 2i\delta_{ik}x_j, \qquad [p_i, M_{jk}]= -2i \delta_{ij}p_k, \\ & [x_1,P_S]=2ip_2,\qquad [x_2,P_S]=2ip_1, \\ & [p_1,X_S]=-2ix_2, \qquad [p_2,X_S]=-2ix_1, \\ & [X_{ii},P_{jj}] = 2i\delta_{ij} M_{ij}, \\ & [X_{11},P_S]=[X_S,P_{22}]=2iM_{12}, \\ & [X_{22},P_S]=[X_S,P_{11}]=2iM_{21}, \\ & [X_S,P_S] = 2i(M_{11}+M_{22}), \\ & [X_{ii}, M_{ii}] = 4i X_{ii},\qquad [X_{11}, M_{21}]=[X_{22},M_{12}]=2i X_S, \\ & [P_{ii}, M_{ii}] = -4i P_{ii}, \\ & [P_{11}, M_{12}]=[P_{22},M_{21}]=-2i P_S, \\ & [X_S, M_{ii}]=2 i X_S,\qquad [X_S, M_{12}]=4iX_{11}, \\ & [X_S,M_{21}] = 4i X_{22}, \\ & [P_S, M_{ii}]=-2 i P_S,\qquad [P_S, M_{12}]=-4iP_{22}, \\ & [P_S,M_{21}] =- 4i P_{11}, \\ & [M_{11},M_{12}]=-[M_{22},M_{12}]=-2i M_{12}, \\ & [M_{11}, M_{21}]=-[M_{22}, M_{21}]=2iM_{21}, \\ & [M_{12}, M_{21}] = 2i(M_{22}-M_{11}). \end{aligned} \end{aligned}$$

(A.1)

The remaining commutators are vanishing.