Symplectic geometry on an infinite-dimensional phase space and an asymptotic representation of quantum averages by Gaussian functional integrals

2008 ◽  
Vol 72 (1) ◽  
pp. 127-148
Author(s):  
A Yu Khrennikov
Keyword(s):  
Phase Space ◽  
Asymptotic Representation ◽  
Symplectic Geometry ◽  
Functional Integrals ◽  
Infinite Dimensional ◽  
Dimensional Phase Space

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.

DISSIPATIVE SOLITON PULSATIONS WITH PERIODS BEYOND THE LASER CAVITY ROUND TRIP TIME

2005 ◽  
Vol 14 (02) ◽  
pp. 177-194 ◽  
Author(s):  
N. AKHMEDIEV ◽  
J. M. SOTO-CRESPO ◽  
M. GRAPINET ◽  
Ph. GRELU
Keyword(s):  
Phase Space ◽  
Nonlinear System ◽  
Fiber Laser ◽  
Laser Cavity ◽  
Continuous Model ◽  
Round Trip Time ◽  
Round Trip ◽  
Infinite Dimensional ◽  
Trip Time ◽  
Dimensional Phase Space

We review recent results on periodic pulsations of the soliton parameters in a passively mode-locked fiber laser. Solitons change their shape, amplitude, width and velocity periodically in time. These pulsations are limit cycles of a dissipative nonlinear system in an infinite-dimensional phase space. Pulsation periods can vary from a few to hundreds of round trips. We present a continuous model of a laser as well as a model with parameter management. The results of the modeling are supported with experimental results obtained using a fiber laser.

NONLOCAL LAGRANGIANS AND HAMILTONIAN FORMALISM

1994 ◽  
Vol 03 (01) ◽  
pp. 211-214
Author(s):  
LLOSA J. ◽  
VIVES J.
Keyword(s):  
Phase Space ◽  
Classical Mechanics ◽  
Hamiltonian Formalism ◽  
Lagrangian Systems ◽  
Infinite Dimensional ◽  
First Order ◽  
Dimensional Phase Space ◽  
Set Up

A presymplectic formalism is set up for nonlocal Lagrangian systems. The method is based on an ‘equivalent’ first order Lagrangian that is processed by standard ways of classical mechanics. The Hamiltonian formalism for the latter is then pulled back onto the infinite dimensional phase space of the nonlocal system.

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