scholarly journals BASIC SYMPLECTIC GEOMETRY FOR p-BRANES WITH THICKNESS IN A CURVED BACKGROUND

2004 ◽  
Vol 19 (27) ◽  
pp. 2069-2081 ◽  
Author(s):  
ALBERTO ESCALANTE
Keyword(s):  
Phase Space ◽  
Symplectic Geometry ◽  
Symplectic Structure ◽  
Canonical Formalism ◽  
Covariant Description ◽  
Relevant Role

We show that the Witten covariant phase space for p-branes with thickness in an arbitrary background is endowed of a symplectic potential, which although is not important to the dynamics of the system, plays a relevant role on the phase space, allowing us to generate a symplectic structure for the theory and therefore give a covariant description of canonical formalism for quantization.

scholarly journals COVARIANT DESCRIPTION OF PARAMETRIZED NONRELATIVISTIC HAMILTONIAN SYSTEMS

2004 ◽  
Vol 19 (15) ◽  
pp. 2473-2493 ◽  
Author(s):  
MAURICIO MONDRAGÓN ◽  
MERCED MONTESINOS
Keyword(s):  
Phase Space ◽  
Gauge Transformation ◽  
Hamiltonian Systems ◽  
Symplectic Structure ◽  
Canonical Formalism ◽  
Extended Phase Space ◽  
Extended Phase ◽  
Algebra Of Observables ◽  
Covariant Description ◽  
Constraint Surface

The various phase spaces involved in the dynamics of parametrized nonrelativistic Hamiltonian systems are displayed by using Crnkovic and Witten's covariant canonical formalism. It is also pointed out that in Dirac's canonical formalism there exists a freedom in the choice of the symplectic structure on the extended phase space and in the choice of the equations that define the constraint surface with the only restriction that these two choices combine in such a way that any pair (of these two choices) generates the same gauge transformation. The consequence of this freedom on the algebra of observables is also discussed.

scholarly journals Poly-symplectic geometry and the AKSZ formalism

2021 ◽  
pp. 2150030
Author(s):  
Ivan Contreras ◽  
Nicolás Martínez Alba
Keyword(s):  
Phase Space ◽  
General Theory ◽  
Symplectic Geometry ◽  
Symplectic Structure ◽  
Sigma Model ◽  
Type Theorem ◽  
Poisson Structures ◽  
Action Functional ◽  
Symplectic Structures ◽  
Poisson Sigma Model

In this paper, we extend the AKSZ formulation of the Poisson sigma model to more general target spaces, and we develop the general theory of graded geometry for poly-symplectic and poly-Poisson structures. In particular, we prove a Schwarz-type theorem and transgression for graded poly-symplectic structures, recovering the action functional and the poly-symplectic structure of the reduced phase space of the poly-Poisson sigma model, from the AKSZ construction.

Covariant statistical mechanics of non-Hamiltonian systems

2018 ◽  
Vol 15 (02) ◽  
pp. 1850017
Author(s):  
Vahid Hosseinzadeh ◽  
Kourosh Nozari
Keyword(s):  
Phase Space ◽  
Statistical Mechanics ◽  
Hamiltonian Systems ◽  
Symplectic Geometry ◽  
Symplectic Structure ◽  
Differential Forms ◽  
Volume Element ◽  
Covariant Formulation ◽  
Locally Conformal

In this paper, using the elegant language of differential forms, we provide a covariant formulation of the equilibrium statistical mechanics of non-Hamiltonian systems. The key idea of the paper is to focus on the structure of phase space and its kinematical and dynamical roles. While in the case of Hamiltonian systems, the structure of the phase space is a symplectic structure (a nondegenerate closed two-form), we consider an almost symplectic structure for the more general case of non-Hamiltonian systems. An almost symplectic structure is a nondegenerate but not necessarily closed two-form structure. Consequently, the dynamics becomes non-Hamiltonian and based on the fact that the structure is nondegenerate, we can also define a volume element. With a well-defined volume in hand, we derive the Liouville equation and find an invariant statistical state. Recasting non-Hamiltonian systems in terms of the almost symplectic geometry has at least two advantages: the formalism is covariant and therefore does not depend on coordinates and there is no confusion in the determination of the natural volume element of the system. For clarification, we investigate the application of the formalism in two examples in which the underlying geometry of the phase space is locally conformal symplectic geometry.

scholarly journals Quantum gravity, dynamical phase-space and string theory

2014 ◽  
Vol 23 (12) ◽  
pp. 1442006 ◽  
Author(s):  
Laurent Freidel ◽  
Robert G. Leigh ◽  
Djordje Minic
Keyword(s):  
Quantum Theory ◽  
Phase Space ◽  
String Theory ◽  
Quantum Gravity ◽  
Momentum Space ◽  
Symplectic Geometry ◽  
Complex Geometry ◽  
Hamiltonian Dynamics ◽  
Natural Extension ◽  
Dynamical Phase

In a natural extension of the relativity principle, we speculate that a quantum theory of gravity involves two fundamental scales associated with both dynamical spacetime as well as dynamical momentum space. This view of quantum gravity is explicitly realized in a new formulation of string theory which involves dynamical phase-space and in which spacetime is a derived concept. This formulation naturally unifies symplectic geometry of Hamiltonian dynamics, complex geometry of quantum theory and real geometry of general relativity. The spacetime and momentum space dynamics, and thus dynamical phase-space, is governed by a new version of the renormalization group (RG).

scholarly journals QUANTIZATION OF THE TOPOLOGICAL σ-MODEL AND THE MASTER EQUATION OF THE BV FORMALISM

1994 ◽  
Vol 09 (06) ◽  
pp. 491-500 ◽  
Author(s):  
S. AOYAMA
Keyword(s):  
Phase Space ◽  
Master Equation ◽  
Symplectic Structure ◽  
Kähler Manifold ◽  
Quantum Master Equation ◽  
Kahler Manifold ◽  
Invariant States

We quantize the topological σ-model. The quantum master equation of the Batalin-Vilkovisky formalism ΔρΨ=0 appears as a condition which eliminates the exact states from the BRST invariant states Ψ defined by QΨ=0. The phase space of the BV formalism is a supermanifold with a specific symplectic structure, called the fermionic Kähler manifold.

scholarly journals Black hole thermodynamics in Snyder phase space

2017 ◽  
Vol 14 (11) ◽  
pp. 1750164
Author(s):  
Sara Saghafi ◽  
Kourosh Nozari
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Phase Space ◽  
Symplectic Structure ◽  
Symplectic Manifolds ◽  
Black Hole Thermodynamics ◽  
Schwarzschild Black Hole ◽  
Noncommutative Phase Space ◽  
Physics Of Black Holes ◽  
First Time

By defining a noncommutative symplectic structure, we study thermodynamics of Schwarzschild black hole in a Snyder noncommutative phase space for the first time. Since natural cutoffs are the results of compactness of symplectic manifolds in phase space, the physics of black holes in such a space would be affected mainly by these cutoffs. In this respect, this study provides a basis for more deeper understanding of the black hole thermodynamics in a pure mathematical viewpoint.

scholarly journals COMPLEX MODULI OF PHYSICAL QUANTA

2005 ◽  
Vol 20 (12) ◽  
pp. 869-874
Author(s):  
JOSÉ M. ISIDRO
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Symplectic Structure ◽  
Classical Mechanics ◽  
Differentiable Structure ◽  
Complex Moduli ◽  
Classical Phase Space

Classical mechanics can be formulated using a symplectic structure on classical phase space, while quantum mechanics requires a complex-differentiable structure on that same space. Complex-differentiable structures on a given real manifold are often not unique. This paper is devoted to analysing the dependence of the notion of a quantum on the complex-differentiable structure chosen on classical 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.

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