κ-（A）dS非交换时空的量子构造与宇宙常数

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"（3 + 1）维κ-（A）dS非交换时空的构建与分析" 在物理学领域，κ-（A）dS非交换时空是一个重要的研究主题，它涉及量子群、宇宙常数、（反）德西特空间、κ-变形以及非交换空间时间的量子化。这篇由Angel Ballesteros、Ivan Gutierrez-Sagredo和Francisco J. Herranz发表在《Physics Letters B》的文章详细探讨了这一理论框架。 κ-（A）dS非交换时空是在κ-（A）dS泊松齐次空间的量子化基础上明确构建的。这一过程考虑了（3+1）维的空间，这是κ-闵可夫斯基时空在非零宇宙常数情况下的唯一合理扩展。κ-（A）dS时空的量子化要求对应量子（A）dS代数的时间平移生成元保持原生性，这是对κ-闵可夫斯基时空理论的重要扩展。文章指出，κ-（A）dS非交换时空具有一种特殊的结构特性，即存在一个局部空间坐标的二次子代数。这个子代数的定义依赖于宇宙学常数的参数。当宇宙常数参数变化时，一阶括号可以定义一个量子球。同时，时空坐标之间的交换子保持κ-闵可夫斯基时空的原有结构不变，这揭示了κ-（A）dS时空与κ-闵可夫斯基时空之间的内在联系。在环境坐标表示下，κ-（A）dS非交换时空呈现出非交换伪球的特性。这种表示方法有助于理解和分析时空的几何性质，特别是在非交换性导致的传统几何概念被重新诠释的情况下。该研究进一步深化了我们对非交换几何和量子引力的理解，因为非交换时空可以作为理解物理定律在极端尺度（如量子引力领域）如何表现的工具。κ-变形提供了处理非平凡宇宙常数背景下的时空结构的新视角，这在探索早期宇宙和强场物理等场景中具有重要意义。关键词：量子群、宇宙常数、（反）德西特空间、κ-变形、非交换空间时间、量子化 κ-（A）dS非交换时空的研究不仅扩展了我们对量子空间时间结构的认知，也为理解和统一量子力学与广义相对论提供了新的理论工具，尤其是在处理宇宙学和引力问题时。这一理论框架为未来的理论物理研究提供了丰富的理论基础和研究方向。

Physics Letters B 796 (2019) 93–101

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

The κ-(A)dS noncommutative spacetime

Angel Ballesteros

∗

, Ivan Gutierrez-Sagredo, Francisco J. Herranz

Departamento de Física, Universidad de Burgos, 09001 Burgos, Spain

a r t i c l e i n f o a b s t r a c t

Article history:

Received

29 May 2019

Received

in revised form 11 July 2019

Accepted

15 July 2019

Available

online 17 July 2019

Editor:

M. Cveti

Keywords:

Quantum

groups

Cosmological

constant

(Anti-)de

Sitter

Kappa-deformation

Noncommutative

spacetimes

Quantization

The (3+1)-dimensional κ-(A)dS noncommutative spacetime is explicitly constructed by quantizing its

semiclassical counterpart, which is the κ-(A)dS Poisson homogeneous space. This turns out to be the

only possible generalization of the well-known κ-Minkowski spacetime to the case of non-vanishing

cosmological constant, under the condition that the time translation generator of the corresponding

quantum (A)dS algebra is primitive. Moreover, the κ-(A)dS noncommutative spacetime is shown to

have a quadratic subalgebra of local spatial coordinates whose ﬁrst-order brackets in terms of the

cosmological constant parameter deﬁne a quantum sphere, while the commutators between time and

space coordinates preserve the same structure of the κ-Minkowski spacetime. When expressed in

ambient coordinates, the quantum κ-(A)dS spacetime is shown to be deﬁned as a noncommutative

pseudosphere.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

Noncommutative spacetimes and their associated uncertainty

relations are widely expected to provide suitable frameworks for

the description of minimum length or fuzziness features of the

spacetime arising in different approaches to Quantum Gravity (see,

for instance, [1–8] and references therein). As a consequence, sev-

eral

notions of noncommutative spacetimes have been proposed in

an attempt to describe this “quantum geometry”, and a remarkable

common feature of all these approaches is a shift from geome-

try

to algebra [9]. In particular, when considering noncommutative

spacetimes arising from quantum groups, the emphasis is put in

the introduction of some (deformed/quantum) symmetry, in such

a way that quantum spacetimes turn out to be covariant under the

action of a suitable quantum kinematical group of isometries.

Among

these noncommutative spacetimes with quantum group

symmetry, probably the most relevant example is provided by the

well-known κ-Minkowski noncommutative spacetime

[

]=−

, [

]=0, a, b =1, 2, 3,

(1)

where κ is a parameter proportional to the Planck mass (see

[10–13]). The algebra (1)deﬁnes a noncommutative spacetime

Corresponding author.

E-mail

addresses: angelb@ubu.es (A. Ballesteros), igsagredo@ubu.es

(I. Gutierrez-Sagredo),

fjherranz@ubu.es (F.J. Herranz).

which is covariant under the κ-Poincaré quantum group [12], a

“quantum deformation” of the group of isometries of Minkowski

spacetime which is the (Hopf algebra) dual of the κ-Poincaré

quantum algebra, that was obtained for the ﬁrst time in [13]

(see

also [14–16]) by making use of quantum group contraction

techniques [17–19] applied onto real forms of the Drinfel’d-Jimbo

quantum deformation for appropriate complex simple Lie alge-

bras [20,21].

Since then, the κ-Minkowski spacetime has provided

a privileged benchmark for the implementation of a number of

models aiming to describe different features of quantum geome-

try

at the Planck scale and their connections with ongoing phe-

nomenological

proposals. Without pretending to be exhaustive,

κ-Minkowski spacetime has been studied in relation with wave

propagation on noncommutative spacetimes [22], Deformed Spe-

cial

Relativity features [23], dispersion relations [24–26], relative

locality phenomena [27], curved momentum spaces and phase

spaces [28,29], noncommutative differential calculi [30,31], star

products [32], noncommutative ﬁeld theory [33–35], representa-

tion

theory [36,37], light cones [38]and noncommutative spaces

of worldlines [39].

However,

when cosmological distances are involved the inter-

play

between gravity and quantum spacetime should take into

consideration the spacetime curvature [40–43], and therefore a

natural (maximally symmetric) noncommutative spacetime to be

considered should be the quantum analogue of the (Anti-)de Sit-

ter

spacetime (hereafter (A)dS). Despite all the efforts devoted so

far in the literature to κ-deformations, the generalization of the

κ-Minkowski spacetime to the (A)dS case with non-vanishing cos-

https://doi.org/10.1016/j.physletb.2019.07.038

0370-2693/

SCOAP

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