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首页统一描述不均匀正交群的κ-变形与广义相对论应用
本文主要探讨了正交群的不均匀Drinfeld型量子化的一般描述,这是一项针对物理学家和数学家的重要成果。研究者A. Borowiec和A. Pachoł在他们的论文《Eur.Phys.J.C(2014)74:2812》中,提出了一种统一的方法来处理不同维度时空中的任意签名度量张量所定义的正交群的量子化。这种量子化方法超越了常规的对角度量条件,允许度量张量包含κ-变形的效应,这是对经典代数结构的一种非平凡扩展。 κ-变形是一种量子化策略,它在量子力学和广义相对论等领域有着潜在的应用。在这个框架下,作者通过经典生成器的形式表达κ-变形的副产品,使得计算变得更为直观和通用。这种统一公式不仅适用于对角化后的度量张量,也适用于非对角线形式,这增加了理论的适用范围,对于处理非平凡几何背景下的物理问题具有重要意义。 论文的关键在于引入了一个额外的向量场,这个向量场用来参数化经典的r-矩阵,它是量子化过程中不可或缺的工具。通过这种方法,作者能够区分不同的非等价变形,这些变形对应于不同的稳定性子群类型。在文中,特别关注的是Lorentz群的κ-变形,这是在广义相对论中的一个关键组成部分,因为Lorentz群是描述四维时空旋转的结构群。 这篇论文提供了一种强大的数学工具,使研究人员能够在更广泛的量子几何背景下探讨引力理论的κ-化版本,这对理解量子引力、量子宇宙学以及量子场论在非平凡空间时间结构中的行为具有深远的影响。这项工作的开放获取特性也使得全球的研究者都能无障碍地获取和利用这一理论成果,促进了科学知识的共享和进一步发展。
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Eur. Phys. J. C (2014) 74:2812
DOI 10.1140/epjc/s10052-014-2812-8
Letter
Unified description for κ-deformations of orthogonal groups
A. Borowiec
1,a
, A. Pachoł
2,b
1
Institute for Theoretical Physics, pl. M. Borna 9, 50-204 Wrocław, Poland
2
Science Institute, University of Iceland, Dunhaga 3, 107 Reykjavik, Iceland
Received: 19 November 2013 / Accepted: 25 February 2014 / Published online: 19 March 2014
© The Author(s) 2014. This article is published with open access at Springerlink.com
Abstract In this paper we provide universal formulas
describing Drinfeld-type quantization of inhomogeneous
orthogonal groups determined by a metric tensor of an arbi-
trary signature living in a spacetime of arbitrary dimension.
The metric tensor does not need to be in diagonal form
and κ-deformed coproducts are presented in terms of clas-
sical generators. It opens the possibility for future applica-
tions in deformed general relativity. The formulas depend on
the choice of an additional vector field which parametrizes
classical r-matrices. Non-equivalent deformations are then
labeled by the corresponding type of stability subgroups.
For the Lorentzian signature it covers three (non-equivalent)
Hopf-algebraic deformations: time-like, space-like (a.k.a.
tachyonic) and light-like (a.k.a. light-cone) quantizations of
the Poincaré algebra. Finally the existence of the so-called
Majid–Ruegg (non-classical) basis is reconsidered.
1 Introduction
Deformations of relativistic symmetries have been fruitful for
the description of quantum symmetries governing physics
at the Planck scale. Such quantum deformations of space-
time symmetries are described within the Hopf-algebra lan-
guage and are controlled by classical r-matrices satisfy-
ing the classical Yang–Baxter (YB) equation: modified or
unmodified one. One of the most interesting deformations,
from the point of view of physical applications, the so-called
κ-deformation, has been found in [1–4]. The deformation
parameter corresponds to the Planck Mass; its inverse defin-
ing fundamental length can be considered as a quantum grav-
ity scale. The r-matrix for the κ-deformation of Poincaré
algebra is given then by r = M
0i
∧ P
i
and it satisfies the
modified (inhomogeneous) Yang–Baxter equation (MYBE):
[[r, r]] = M
μν
∧ P
μ
∧ P
ν
.Theκ-Poincaré Hopf algebra con-
a
e-mail: andrzej.borowiec@ift.uni.wroc.pl
b
e-mail: pachol@hi.is
stitutes the deformed symmetry of the κ-Minkowski algebra
[4,5] which is a quantum version of the standard Minkowski
spacetime. The κ-Minkowski spacetime has been mostly
studied in the so-called time-like version of κ-deformation,
distinguishing the ’time’ coordinate as the quantized one. The
r-matrix mentioned above corresponds to this case.
1
Another
option is the so-called light-like (null-plane) deformation cor-
responding to null-vectors, which was firstly considered in
[9] (then also in [10,11]) with quantum-deformed direction
on the light cone (x
+
= x
0
+ x
3
) and with the correspond-
ing symmetry the so-called ’null-plane quantum Poincaré Lie
algebra’. It was inspired by the central problem of quantum
relativistic systems in the Hamiltonian formulation, which
has been studied for the null-plane evolution. In this case the
information provided by the Poincaré invariance splits into a
dynamical and kinematical part which is also the case after
the deformation. One of the advantages of the deformation of
this type is that it is triangular i.e. it can be described by the
classical r-matrix satisfying classical Yang–Baxter equation
(CYBE) and the twisting element satisfying two-cocycle con-
dition do exist [12]. Moreover, the differential calculus for
the null-plane κ-Minkowski is shown to be bicovariant and
four-dimensional [13], which has been proved to be impossi-
ble to built for other kinds of κ-deformations (i.e. time- and
space-like) [14]. It was also shown [15] that after suitable
(nonlinear) change of basis the quantum algebra presented
in [9] can be identified with the κ-deformation, given i n [16]
for the choice of g
00
= 0.
Till now the most popular form of presentation of quan-
tum κ-Poincaré algebra is the one which uses formulas for
deformed coproducts found for the first time in [4] (with the
primitive energy generator P
0
). The corresponding system
of generators, known also as Majid–Ruegg or bicrossprod-
uct basis, satisfy classical commutation relations between
1
The corresponding classification of quantum deformations (complete
for Lorentz and almost-complete for Poincaré algebras) has been per-
formed in Refs. [6,7] (see also dual matrix quantum group version in
[8]).
123
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