Abstract
We consider two quantum phase spaces which can be described by two Hopf algebroids linked with the well-known $\theta_{\mu \nu }$-deformed $D=4$ Poincare-Hopf algebra $\mathbb{H}$. The first algebroid describes $\theta_{\mu \nu }$-deformed relativistic phase space with canonical NC space-time (constant $\theta_{\mu \nu }$ parameters) and the second one incorporates dual to $\mathbb{H}$ quantum $\theta_{\mu \nu }$-deformed Poincare-Hopf group algebra $\mathbb{G}$, which contains noncommutative space-time translations given by $\Lambda $-dependent $\Theta_{\mu \nu }$ parameters ($% \Lambda $ $\equiv \Lambda_{\mu \nu }$ parametrize classical Lorentz group). The canonical $\theta_{\mu \nu }$-deformed space-time algebra and its quantum phase space extension is covariant under the quantum Poincare transformations described by $\mathbb{G}$. We will also comment on the use of Hopf algebroids for the description of multiparticle structures in quantum phase spaces.
Highlights
There have been proposed in recent years various models of noncommutative (NC) space-times which characterizes space-time geometry if the quantum gravity (QG) effects are included
In this paper we shall study the canonical case of quantum space-times, with NC counterparts xμ (μ 1⁄4 0, 1, 2, 3) of space-time coordinates satisfying the well-known formula
If λ is proportional to λP, from formula (2) due to the presence of Planck constant ħ, one can deduce the quantum-mechanical and QG origin of relation (1)
Summary
There have been proposed in recent years various models of noncommutative (NC) space-times which characterizes space-time geometry if the quantum gravity (QG) effects are included (see e.g., [1,2,3,4,5,6,7,8]). The well-known θμν-deformed quantum space-times [see (1)] and associated quantum phase spaces are generated by the following Abelian twist [9,10]: It defines θμν-deformed quantum Poincare-Hopf algebra H, with nondeformed Poincarealgebra sectors, with modified coproducts and antipodes [11]. Given by the particular choice of semidirect product G⋊H called smash product, one obtains (10 þ 10)dimensional generalized θμν-deformed quantum phase space Hð10þ10Þ 1⁄4 ðξμ; Λμν; pμ; MμνÞ Such phase space can be employed in physical applications for the description of NC dynamics on algebraic θμν-deformed quantum Poincaregroup manifold.. A. Twist deformed quantum Poincarealgebra H The classical D 1⁄4 4 Poincare-Hopf algebra looks as follows: 4See [18] for the use of star product to represent the quantum group transformations. From (3) and (5), it follows that in the considered case of θμν-deformations U 1⁄4 1 and the antipodes remain unchanged, i.e., SF ðhÞ 1⁄4 S0ðhÞ 1⁄4 −h
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