Symplectic geometry of radiative modes and conserved quantities at null infinity

1981 ◽  
Vol 376 (1767) ◽  
pp. 585-607 ◽  
Keyword(s):  
Angular Momentum ◽  
Gravitational Field ◽  
Phase Space ◽  
Degrees Of Freedom ◽  
Symplectic Geometry ◽  
Conserved Quantities ◽  
Null Infinity ◽  
Hamiltonian Description ◽  
Infinite Dimensional ◽  
Massless Spin

The Hamiltonian description of massless spin zero- and one-fields in Minkowski space is first recast in a way that refers only to null infinity and fields thereon representing radiative modes. With this framework as a guide, the phase space of the radiative degrees of freedom of the gravitational field (in exact general relativity) is introduced. It has the structure of an infinite-dimensional affine manifold (modelled on a Fréchet space) and is equipped with a continuous, weakly non-degenerate symplectic tensor field. The action of the Bondi-Metzner-Sachs group on null infinity is shown to induce canonical transformations on this phase space. The corresponding Hamiltonians – i. e. generating functions – are computed and interpreted as fluxes of supermomentum and angular momentum carried away by gravitational waves. The discussion serves three purposes: it brings out, via symplectic methods, the universality of the interplay between symmetries and conserved quantities; it sheds new light on the issue of angular momentum of gravitational radiation; and, it suggests a new approach to the quantization of the ‘true’ degrees of freedom of the gravitational field.

scholarly journals Poisson-Lie T-duality of WZW model via current algebra deformation

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Francesco Bascone ◽  
Franco Pezzella ◽  
Patrizia Vitale
Keyword(s):  
Phase Space ◽  
Configuration Space ◽  
Current Algebra ◽  
Degrees Of Freedom ◽  
Target Space ◽  
Moody Algebra ◽  
Hamiltonian Description ◽  
Direct Sum ◽  
Group Manifold ◽  
Two Parameter

Abstract Poisson-Lie T-duality of the Wess-Zumino-Witten (WZW) model having the group manifold of SU(2) as target space is investigated. The whole construction relies on the deformation of the affine current algebra of the model, the semi-direct sum $$ \mathfrak{su}(2)\left(\mathrm{\mathbb{R}}\right)\overset{\cdot }{\oplus}\mathfrak{a} $$ su 2 ℝ ⊕ ⋅ a , to the fully semisimple Kac-Moody algebra $$ \mathfrak{sl}\left(2,\mathrm{\mathbb{C}}\right)\left(\mathrm{\mathbb{R}}\right) $$ sl 2 ℂ ℝ . A two-parameter family of models with SL(2, ℂ) as target phase space is obtained so that Poisson-Lie T-duality is realised as an O(3, 3) rotation in the phase space. The dual family shares the same phase space but its configuration space is SB(2, ℂ), the Poisson-Lie dual of the group SU(2). A parent action with doubled degrees of freedom on SL(2, ℂ) is defined, together with its Hamiltonian description.

scholarly journals Gravitational memory effects and Bondi-Metzner-Sachs symmetries in scalar-tensor theories

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shaoqi Hou ◽  
Zong-Hong Zhu
Keyword(s):  
General Relativity ◽  
Angular Momentum ◽  
Evolution Equations ◽  
Equations Of Motion ◽  
Memory Effects ◽  
Scalar Tensor Theory ◽  
Null Infinity ◽  
Infinite Dimensional ◽  
Tensor Theory ◽  
Bondi Mass

Abstract The relation between gravitational memory effects and Bondi-Metzner-Sachs symmetries of the asymptotically flat spacetimes is studied in the scalar-tensor theory. For this purpose, the solutions to the equations of motion near the future null infinity are obtained in the generalized Bondi-Sachs coordinates with a suitable determinant condition. It turns out that the Bondi-Metzner-Sachs group is also a semi-direct product of an infinite dimensional supertranslation group and the Lorentz group as in general relativity. There are also degenerate vacua in both the tensor and the scalar sectors in the scalar-tensor theory. The supertranslation relates the vacua in the tensor sector, while in the scalar sector, it is the Lorentz transformation that transforms the vacua to each other. So there are the tensor memory effects similar to the ones in general relativity, and the scalar memory effect, which is new. The evolution equations for the Bondi mass and angular momentum aspects suggest that the null energy fluxes and the angular momentum fluxes across the null infinity induce the transition among the vacua in the tensor and the scalar sectors, respectively.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

Conservation laws

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Dynamical System ◽  
Angular Momentum ◽  
Conservation Laws ◽  
Total Energy ◽  
Superposition Principle ◽  
Momentum Conservation ◽  
Conserved Quantities ◽  
Angular Momentum Conservation ◽  
Third Law ◽  
The Law

This chapter defines the conserved quantities associated with an isolated dynamical system, that is, the quantities which remain constant during the motion of the system. The law of momentum conservation follows directly from Newton’s third law. The superposition principle for forces allows Newton’s law of motion for a body Pa acted on by other bodies Pa′ in an inertial Cartesian frame S. The law of angular momentum conservation holds if the forces acting on the elements of the system depend only on the separation of the elements. Finally, the conservation of total energy requires in addition that the forces be derivable from a potential.

Shape Dynamics and the Linking Theory

2018 ◽  
Author(s):  
Flavio Mercati
Keyword(s):  
Phase Space ◽  
Degrees Of Freedom ◽  
Line Element ◽  
Hamiltonian Formulation ◽  
Operational Definition ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Problem Of Time ◽  
Definition Of ◽  
Matter Fields

This chapter explains in detail the current Hamiltonian formulation of SD, and the concept of Linking Theory of which (GR) and SD are two complementary gauge-fixings. The physical degrees of freedom of SD are identified, the simple way in which it solves the problem of time and the problem of observables in quantum gravity are explained, and the solution to the problem of constructing a spacetime slab from a solution of SD (and the related definition of physical rods and clocks) is described. Furthermore, the canonical way of coupling matter to SD is introduced, together with the operational definition of four-dimensional line element as an effective background for matter fields. The chapter concludes with two ‘structural’ results obtained in the attempt of finding a construction principle for SD: the concept of ‘symmetry doubling’, related to the BRST formulation of the theory, and the idea of ‘conformogeometrodynamics regained’, that is, to derive the theory as the unique one in the extended phase space of GR that realizes the symmetry doubling idea.

scholarly journals Infinite-dimensional fermionic symmetry in supersymmetric gauge theories

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Thomas T. Dumitrescu ◽  
Temple He ◽  
Prahar Mitra ◽  
Andrew Strominger
Keyword(s):  
Ward Identity ◽  
Gauge Theories ◽  
Soft Photon ◽  
Supersymmetric Gauge Theories ◽  
Null Infinity ◽  
Infinite Dimensional ◽  
Charged Matter ◽  
Gauge Symmetries ◽  
Supersymmetric Gauge

Abstract We establish the existence of an infinite-dimensional fermionic symmetry in four-dimensional supersymmetric gauge theories by analyzing semiclassical photino dynamics in abelian $$ \mathcal{N} $$ N = 1 theories with charged matter. The symmetry is parametrized by a spinor-valued function on an asymptotic S2 at null infinity. It is not manifest at the level of the Lagrangian, but acts non-trivially on physical states, and its Ward identity is the soft photino theorem. The infinite-dimensional fermionic symmetry resides in the same $$ \mathcal{N} $$ N = 1 supermultiplet as the physically non-trivial large gauge symmetries associated with the soft photon theorem.

A REPRESENTATION OF THE BELAVKIN EQUATION VIA PHASE SPACE FEYNMAN PATH INTEGRALS

2004 ◽  
Vol 07 (04) ◽  
pp. 507-526 ◽  
Author(s):  
S. ALBEVERIO ◽  
G. GUATTERI ◽  
S. MAZZUCCHI
Keyword(s):  
Phase Space ◽  
Continuous Measurement ◽  
Quantum Particle ◽  
Oscillatory Integral ◽  
Feynman Path Integral ◽  
Feynman Path ◽  
Infinite Dimensional ◽  
Feynman Path Integrals ◽  
Characteristics Method ◽  
Stochastic Characteristics

The Belavkin equation, describing the continuous measurement of the momentum of a quantum particle, is studied. The existence and uniqueness of its solution is proved via analytic tools. A stochastic characteristics method is applied. A rigorous representation of the solution by means of an infinite dimensional oscillatory integral (Feynman path integral) defined on the phase space is also given.

Infinitely entangled states

2003 ◽  
Vol 3 (4) ◽  
pp. 281-306
Author(s):  
M. Keyl ◽  
D. Schlingemann ◽  
R.F. Werner
Keyword(s):  
Hilbert Spaces ◽  
Degrees Of Freedom ◽  
Entangled States ◽  
Entangled State ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Density Operators ◽  
Fundamental Paper ◽  
Extended Notion

For states in infinite dimensional Hilbert spaces entanglement quantities like the entanglement of distillation can become infinite. This leads naturally to the question, whether one system in such an infinitely entangled state can serve as a resource for tasks like the teleportation of arbitrarily many qubits. We show that appropriate states cannot be obtained by density operators in an infinite dimensional Hilbert space. However, using techniques for the description of infinitely many degrees of freedom from field theory and statistical mechanics, such states can nevertheless be constructed rigorously. We explore two related possibilities, namely an extended notion of algebras of observables, and the use of singular states on the algebra of bounded operators. As applications we construct the essentially unique infinite analogue of maximally entangled states, and the singular state used heuristically in the fundamental paper of Einstein, Rosen and Podolsky.

scholarly journals About the non-standard viewpoint on the dynamics of closed vortex filament

2018 ◽  
Vol 32 (33) ◽  
pp. 1850410 ◽  
Author(s):  
S. V. Talalov
Keyword(s):  
Phase Space ◽  
Effective Mass ◽  
Explicit Formula ◽  
Vortex Filament ◽  
Hamiltonian Description ◽  
Suggested Approach ◽  
Galilei Group ◽  
Filament Dynamics

In this paper, we construct the Hamiltonian description of the closed vortex filament dynamics in terms of non-standard variables, phase space and constraints. The suggested approach makes obvious interpretation of the considered system as a structured particle that possesses certain external and internal degrees of the freedom. The constructed theory is invariant under the transformation of Galilei group. The appearance of this group allows for a new viewpoint on the energy of a closed vortex filament with zero thickness. The explicit formula for the effective mass of the structured particle “closed vortex filament” is suggested.

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