量子κ-BMS对称性：变形与物理应用

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"κ-变形的BMS对称性" κ-变形的BMS对称性是一种在量子重力理论中出现的创新概念，它扩展了传统的边界或无限远处的对称性。BMS（Bondi-Metzner-Sachs）对称性最初是在经典广义相对论中被引入的，用来描述在引力背景下的无限远点的局部群结构。这个对称性组在引力理论中扮演着重要角色，因为它与引力波、能量动量张量的定义以及黑洞物理紧密相关。 κ-变形，又称为κ-庞加莱代数，是对庞加莱对称性的非平凡量子化，它引入了非局部性和非平凡的代数结构。κ-庞加莱代数的构建是通过一种称为扭曲的操作来实现的，这种扭曲使得原来的经典代数运算规则发生改变。在这个研究中，作者们通过将κ-庞加莱代数的轻度变形方法应用于BMS对称性，得到了κ-BMS Hopf代数。 Hopf代数是具有乘法和共轭操作的代数结构，它在量子群理论中扮演核心角色。κ-BMS Hopf代数的形成表明，κ-变形不仅限于基本的庞加莱对称性，还可以扩展到其更复杂的子结构，如BMS对称性。这一发现对于理解量子引力中的对称性和无限远点的行为具有重要意义。文章中提到，κ-BMS对称性的物理相关性在于其可能作为量子重力的渐近对称性。这暗示着在极端条件下，如黑洞事件视界附近或者宇宙的大尺度结构中，κ-BMS对称可能成为描述物理现象的关键工具。这种对称性的存在可能会对引力波的检测、黑洞信息悖论的解决方案，甚至可能的量子引力理论的构造产生深远影响。此外，κ-变形的引入还可能导致新的数学结构和物理概念的发展。例如，κ-BMS代数可能需要新的边界条件，以适应非平凡的代数结构，这可能会对量子场论在时空中如何定义和演化产生新的洞见。同时，κ-变形也可能影响到黑洞熵的计算和黑洞热力学的理解。 κ-变形的BMS对称性是一个前沿的研究领域，它结合了量子群理论、广义相对论和量子引力，有望为我们提供更深入的理解，尤其是在引力理论的边界和无限远行为方面。随着理论的进一步发展，κ-BMS对称性的应用可能不仅限于物理学，也可能在数学和其他相关科学领域产生重要的连锁反应。

Physics Letters B 790 (2019) 415–420

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

κ-deformed BMS symmetry

Andrzej Borowiec

, Lennart Brocki

, Jerzy Kowalski-Glikman

a,b,∗

, Josua Unger

Institute for Theoretical Physics, University of Wrocław, pl. M. Borna 9, 50-204 Wrocław, Poland

National Centre for Nuclear Research, Ho ˙za 69, 00-681 Warsaw, Poland

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

Article history:

Received

27 November 2018

Received

in revised form 23 January 2019

Accepted

30 January 2019

Available

online 4 February 2019

Editor:

M. Cveti

Keywords:

BMS

symmetry

Symmetry

deformations

Hopf

algebras

We present the quantum κ-deformation of BMS symmetry, by generalizing the lightlike κ-Poincaré

Hopf algebra. On the technical level our analysis relies on the fact that the lightlike κ-deformation of

Poincaré algebra is given by a twist and the lightlike deformation of any algebra containing Poincaré as a

subalgebra can be done with the help of the same twisting element. We brieﬂy comment on the physical

relevance of the obtained κ-BMS Hopf algebra as a possible asymptotic symmetry of quantum gravity.

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

In the recent years there is a surge of interest in BMS symmetry

at null inﬁnity of asymptotically ﬂat spacetimes [1], [2], [3]. This

renewed interest in the seemingly exotic aspects of classical gen-

eral

relativity was fueled by the discovery of a surprising and close

relationship between asymptotic symmetries and soft gravitons

theorem [4] that, as it turned out, has its roots in Ward identities

for supertranslations [5], [6]. Moreover, the gravitational memory

effect [7] appears to be related to the two [8], so that there is

a triangle of interrelationships, which vertices are BMS symmetry,

Weinberg’s soft graviton theorem, and the memory effect. The ex-

tensive

discussion of these effects can be found in recent reviews

[9] and [10].

has also been argued recently [11] that the similarity of null

inﬁnity and black hole horizon suggests that the charges associ-

ated

with the BMS symmetry might be present at the black hole

horizon, so that the black hole might have an inﬁnite number of

hairs, so-called soft hairs, in addition to the three classical one:

mass, charge, and angular momentum. It was argued in [11] and

[12] that the presence of these charges may help solving the black

hole information paradox.

this letter we will investigate the properties of a κ-deformed

generalization of the BMS symmetry. There are several reasons to

be interested in such a generalization. The main one is that it

is believed that investigations of κ-deformation might shed some

light on the properties of elusive quantum gravity. The deforma-

tion

parameter of κ-deformation of Poincaré algebra [13–17], see

Corresponding author.

E-mail

address: jerzy.kowalski-glikman@ift.uni.wroc.pl (J. Kowalski-Glikman).

[18]for a recent review and more references, has dimension of

mass, and therefore it is natural to identify it with Planck mass, or

inverse Planck length, which in turn suggests a possible close rela-

tionship

between this deformation and quantum gravity. In fact, it

was shown in [19] that in the case of gravity in 2 + 1dimensions

the κ-Poincaré algebra is the algebra of symmetries of quantum

ﬂat space in non-perturbatively quantized quantum gravity, and

therefore it plays in quantum gravity the role analogous to the

fundamental role played by the Poincaré algebra in quantum ﬁeld

theory.

There

are three incarnations in which κ-deformation can ap-

pear.

First, there is the mostly studied deformation in time direc-

tion,

in which case the rotations subalgebra of the Poincaré algebra

is kept undeformed. This deformation satisﬁes a natural require-

ment

that rotations are not deformed, and in particular that the

angular momentum and spin composition laws are not modiﬁed. It

is for this same reason that space-like κ-deformation with a pre-

ferred

spacial direction of deformation was never considered to be

a phenomenologically viable possibility, and is treated as a mathe-

matical

curiosity with no real relevance for physics.

The

third possibility is the lightlike κ-deformation [20–27]. In

contrast to the previous two cases it is implemented by a trian-

gular

r-matrix, which satisﬁes the Classical Yang–Baxter Equation.

A remarkable property of this so-called non-standard deformation

is that it is given by a twist which makes its structure much sim-

pler

to construct [28]. Particularly, any embedding of the algebra

into a larger Lie algebra preserves the triangular structure and al-

lows

to perform the deformation by the same twist element. Since

the BMS algebra is ultimately tied in with null inﬁnity and con-

tains

the Poincaré subalgebra, in this paper we deform it using

https://doi.org/10.1016/j.physletb.2019.01.063

0370-2693/

SCOAP

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