scholarly journals Null boundary phase space: slicings, news & memory

2021 ◽  
Vol 2021 (11) ◽  
Author(s):  
H. Adami ◽  
D. Grumiller ◽  
M. M. Sheikh-Jabbari ◽  
V. Taghiloo ◽  
H. Yavartanoo ◽  
...  
Keyword(s):  
Phase Space ◽  
Symmetry Algebra ◽  
Heisenberg Algebra ◽  
Einstein Gravity ◽  
Boundary Phase ◽  
Surface Charges ◽  
Direct Sum ◽  
Null Surface ◽  
Surface Expansion ◽  
Dimension One

Abstract We construct the boundary phase space in D-dimensional Einstein gravity with a generic given co-dimension one null surface $$ \mathcal{N} $$ N as the boundary. The associated boundary symmetry algebra is a semi-direct sum of diffeomorphisms of $$ \mathcal{N} $$ N and Weyl rescalings. It is generated by D towers of surface charges that are generic functions over $$ \mathcal{N} $$ N . These surface charges can be rendered integrable for appropriate slicings of the phase space, provided there is no graviton flux through $$ \mathcal{N} $$ N . In one particular slicing of this type, the charge algebra is the direct sum of the Heisenberg algebra and diffeomorphisms of the transverse space, $$ \mathcal{N} $$ N v for any fixed value of the advanced time v. Finally, we introduce null surface expansion- and spin-memories, and discuss associated memory effects that encode the passage of gravitational waves through $$ \mathcal{N} $$ N , imprinted in a change of the surface charges.

scholarly journals Chiral massive news: null boundary symmetries in topologically massive gravity

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
H. Adami ◽  
M.M. Sheikh-Jabbari ◽  
V. Taghiloo ◽  
H. Yavartanoo ◽  
C. Zwikel
Keyword(s):  
Central Charge ◽  
Three Dimensional ◽  
Virasoro Algebra ◽  
Symmetry Algebra ◽  
Massive Gravity ◽  
Surface Charges ◽  
Null Surface ◽  
Independent Functions ◽  
Chern Simons ◽  
Topologically Massive Gravity

Abstract We study surface charges on a generic null boundary in three dimensional topological massive gravity (TMG). We construct the solution phase space which involves four independent functions over the two dimensional null boundary. One of these functions corresponds to the massive chiral propagating graviton mode of TMG. The other three correspond to three surface charges of the theory, two of which can always be made integrable, while the last one can become integrable only in the absence of the chiral massive graviton flux through the null boundary. As the null boundary symmetry algebra we obtain Heisenberg ⊕ Virasoro algebra with a central charge proportional to the gravitational Chern-Simons term of TMG. We also discuss that the flux of the chiral massive gravitons appears as the (Bondi) news through the null surface.

scholarly journals Symmetries at null boundaries: two and three dimensional gravity cases

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
H. Adami ◽  
M.M. Sheikh-Jabbari ◽  
V. Taghiloo ◽  
H. Yavartanoo ◽  
C. Zwikel
Keyword(s):  
A Priori ◽  
Three Dimensional ◽  
Symmetry Algebra ◽  
State Dependence ◽  
Surface Charges ◽  
Full Generality ◽  
Codimension One ◽  
Null Surface ◽  
Dimensional Gravity ◽  
Symmetry Generators

Abstract We carry out in full generality and without fixing specific boundary conditions, the symmetry and charge analysis near a generic null surface for two and three dimensional (2d and 3d) gravity theories. In 2d and 3d there are respectively two and three charges which are generic functions over the codimension one null surface. The integrability of charges and their algebra depend on the state-dependence of symmetry generators which is a priori not specified. We establish the existence of infinitely many choices that render the surface charges integrable. We show that there is a choice, the “fundamental basis”, where the null boundary symmetry algebra is the Heisenberg⊕Diff(d − 2) algebra. We expect this result to be true for d > 3 when there is no Bondi news through the null surface.

scholarly journals Palatial Twistors from Quantum Inhomogeneous Conformal Symmetries and Twistorial DSR Algebras

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1309
Author(s):  
Jerzy Lukierski
Keyword(s):  
Phase Space ◽  
Hopf Algebras ◽  
Deformation Parameter ◽  
Heisenberg Algebra ◽  
Planck Length ◽  
Covariant Quantum ◽  
Drinfeld Twist ◽  
Quantum Deformations ◽  
Heisenberg Double ◽  
Conformal Symmetries

We construct recently introduced palatial NC twistors by considering the pair of conjugated (Born-dual) twist-deformed D=4 quantum inhomogeneous conformal Hopf algebras Uθ(su(2,2)⋉T4) and Uθ¯(su(2,2)⋉T¯4), where T4 describes complex twistor coordinates and T¯4 the conjugated dual twistor momenta. The palatial twistors are suitably chosen as the quantum-covariant modules (NC representations) of the introduced Born-dual Hopf algebras. Subsequently, we introduce the quantum deformations of D=4 Heisenberg-conformal algebra (HCA) su(2,2)⋉Hℏ4,4 (Hℏ4,4=T¯4⋉ℏT4 is the Heisenberg algebra of twistorial oscillators) providing in twistorial framework the basic covariant quantum elementary system. The class of algebras describing deformation of HCA with dimensionfull deformation parameter, linked with Planck length λp, is called the twistorial DSR (TDSR) algebra, following the terminology of DSR algebra in space-time framework. We describe the examples of TDSR algebra linked with Palatial twistors which are introduced by the Drinfeld twist and the quantization map in Hℏ4,4. We also introduce generalized quantum twistorial phase space by considering the Heisenberg double of Hopf algebra Uθ(su(2,2)⋉T4).

scholarly journals Edge modes of gravity. Part II. Corner metric and Lorentz charges

2020 ◽  
Vol 2020 (11) ◽  
Author(s):  
Laurent Freidel ◽  
Marc Geiller ◽  
Daniele Pranzetti
Keyword(s):  
Phase Space ◽  
Lorentz Invariance ◽  
Discrete Series ◽  
Continuum Theory ◽  
Symmetry Algebra ◽  
Area Spectrum ◽  
Area Element ◽  
Bf Theory ◽  
Tetrad Gravity ◽  
The Continuum

Abstract In this second paper of the series we continue to spell out a new program for quantum gravity, grounded in the notion of corner symmetry algebra and its representations. Here we focus on tetrad gravity and its corner symplectic potential. We start by performing a detailed decomposition of the various geometrical quantities appearing in BF theory and tetrad gravity. This provides a new decomposition of the symplectic potential of BF theory and the simplicity constraints. We then show that the dynamical variables of the tetrad gravity corner phase space are the internal normal to the spacetime foliation, which is conjugated to the boost generator, and the corner coframe field. This allows us to derive several key results. First, we construct the corner Lorentz charges. In addition to sphere diffeomorphisms, common to all formulations of gravity, these charges add a local $$ \mathfrak{sl} $$ sl (2, ℂ) component to the corner symmetry algebra of tetrad gravity. Second, we also reveal that the corner metric satisfies a local $$ \mathfrak{sl} $$ sl (2, ℝ) algebra, whose Casimir corresponds to the corner area element. Due to the space-like nature of the corner metric, this Casimir belongs to the unitary discrete series, and its spectrum is therefore quantized. This result, which reconciles discreteness of the area spectrum with Lorentz invariance, is proven in the continuum and without resorting to a bulk connection. Third, we show that the corner phase space explains why the simplicity constraints become non-commutative on the corner. This fact requires a reconciliation between the bulk and corner symplectic structures, already in the classical continuum theory. Understanding this leads inevitably to the introduction of edge modes.

scholarly journals A comparative review of four formulations of noncommutative quantum mechanics

2016 ◽  
Vol 31 (19) ◽  
pp. 1630025 ◽  
Author(s):  
Laure Gouba
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Path Integral ◽  
Heisenberg Algebra ◽  
Noncommutative Quantum Mechanics ◽  
Comparative Review ◽  
Noncommutative Quantum

Four formulations of quantum mechanics on noncommutative Moyal phase spaces are reviewed. These are the canonical, path-integral, Weyl–Wigner and systematic formulations. Although all these formulations represent quantum mechanics on a phase space with the same deformed Heisenberg algebra, there are mathematical and conceptual differences which we discuss.

scholarly journals PLANAR SYSTEM AND w∞ ALGEBRA

2005 ◽  
Vol 20 (37) ◽  
pp. 2861-2871 ◽  
Author(s):  
JAMILA DOUARI
Keyword(s):  
Phase Space ◽  
Heisenberg Algebra ◽  
Two Dimensional ◽  
Planar System ◽  
Exotic Particles ◽  
Dimensional Phase Space

We study the exotic particles symmetry in the background of noncommutative two-dimensional phase-space leading to realize in physics the deformed version of Cλ-extended Heisenberg algebra and ω∞ symmetry.

scholarly journals Noncommutative Heisenberg algebra in the neighbourhood of a generic null surface

2018 ◽  
Vol 934 ◽  
pp. 557-577 ◽  
Author(s):  
Krishnakanta Bhattacharya ◽  
Bibhas Ranjan Majhi
Keyword(s):  
Heisenberg Algebra ◽  
Null Surface

scholarly journals Некоммутативная задача Ландау о фазовом пространстве при наличии минимальной длины

2020 ◽  
pp. 188-198
Author(s):  
F.A. Dossa ◽  
J.T. Koumagnon ◽  
J.V. Hounguevou ◽  
G.Y.H. Avossevou
Keyword(s):  
Electromagnetic Field ◽  
Phase Space ◽  
Momentum Space ◽  
Hypergeometric Functions ◽  
Heisenberg Algebra ◽  
Minimal Length ◽  
Wave Functions ◽  
Energy Eigenvalues ◽  
Uvarov Method

The deformed Landau problem under a electromagnetic field is studied, where the Heisenberg algebra is constructed in detail in non-commutative phase space in the presence of a minimal length. We show that, in the presence of a minimal length, the momentum space is more practical to solve any problem of eigenvalues. From the Nikiforov-Uvarov method, the energy eigenvalues are obtained and the corresponding wave functions are expressed in terms of hypergeometric functions. The fortuitous degeneration observed in the spectrum shows that the formulation of the minimal length complements that of the non-commutative phase space. Изучается деформированная задача Ландау в электромагнитном поле, в которой алгебра Гейзенберга подробно строится в некоммутативном фазовом пространстве при наличии минимальной длины. Мы показываем, что при наличии минимальной длины импульсное пространство более практично для решения любой проблемы собственных значений. С помощью метода Никифорова-Уварова получаются собственные значения энергии, а соответствующие волновые функции выражаются через гипергеометрические функции. Случайное вырождение, наблюдаемое в спектре, показывает, что формулировка минимальной длины дополняет формулировку некоммутативного фазового пространства.

The Transformation of Commutative Phase Space to Noncommutative One, and its Lorentz Transformation-Like Forms

2021 ◽  
pp. 188-198
Author(s):  
Makoto Nakamura ◽  
Hiroshi Kakuhata ◽  
Kouichi Toda
Keyword(s):  
Phase Space ◽  
Lorentz Transformation ◽  
Linear Transformation ◽  
Arbitrary Dimension ◽  
Two Dimensional ◽  
Exact Form ◽  
Noncommutative Phase Space ◽  
Two Dimension ◽  
Direct Sum

Noncommutative phase space of arbitrary dimension is discussed. We introduce momentum-momentum noncommutativity in addition to co-ordinate-coordinate noncommutativity. We find an exact form for the linear transformation which relates a noncommutative phase space to the corresponding ordinary one. By using this form, we show that a noncommutative phase space of arbitrary dimension can be represented by the direct sum of two-dimensional noncommutative ones. In two-dimension, we obtain the transformation which relates a noncommutative phase space to commutative one. The transformation has the Lorentz transformation-like forms and can also describe the Bopp's shift.

scholarly journals On a spectral sequence for the cohomology of a nilpotent Lie algebra

2014 ◽  
Vol 14 (01) ◽  
pp. 1450078
Author(s):  
Viviana del Barco
Keyword(s):  
Spectral Sequence ◽  
Lie Algebra ◽  
Cohomology Groups ◽  
Nilpotent Lie Algebra ◽  
Lie Algebra Cohomology ◽  
Direct Sum ◽  
Differential Complex ◽  
Dimension One

Given a nilpotent Lie algebra 𝔫 we construct a spectral sequence which is derived from a filtration of its Chevalley–Eilenberg differential complex and converges to the Lie algebra cohomology of 𝔫. The limit of this spectral sequence gives a grading for the Lie algebra cohomology, except for the cohomology groups of degree 0, 1, dim 𝔫 - 1 and dim 𝔫 as we shall prove. We describe the spectral sequence associated to a nilpotent Lie algebra which is a direct sum of two ideals, one of them of dimension one, in terms of the spectral sequence of the co-dimension one ideal. Also, we compute the spectral sequence corresponding to each real nilpotent Lie algebra of dimension less than or equal to six.

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