scholarly journals HERMITIAN VECTOR FIELDS AND COVARIANT QUANTUM MECHANICS OF A SPIN PARTICLE

2010 ◽  
Vol 07 (04) ◽  
pp. 599-623 ◽  
Author(s):  
DANIEL CANARUTTO
Keyword(s):  
Quantum Mechanics ◽  
Lie Algebra ◽  
Vector Fields ◽  
Spin Particle ◽  
Covariant Quantum ◽  
Complex Dimension ◽  
Algebra Of Functions ◽  
Quantum Vector ◽  
Classical Phase Space ◽  
Phase Functions

In the context of Covariant Quantum Mechanics for a spin particle, we classify the "quantum vector fields", i.e. the projectable Hermitian vector fields of a complex bundle of complex dimension 2 over spacetime. Indeed, we prove that the Lie algebra of quantum vector fields is naturally isomorphic to a certain Lie algebra of functions of the classical phase space, called "special phase functions". This result provides a covariant procedure to achieve the quantum operators generated by the quantum vector fields and the corresponding observables described by the special phase functions.

scholarly journals ON THE ⋆-PRODUCT QUANTIZATION AND THE DUFLO MAP IN THREE DIMENSIONS

2012 ◽  
Vol 27 (35) ◽  
pp. 1250207 ◽  
Author(s):  
LUIGI ROSA ◽  
PATRIZIA VITALE
Keyword(s):  
Quantum Mechanics ◽  
Hydrogen Atom ◽  
Lie Algebra ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Moyal Product ◽  
Product Quantization ◽  
Algebra Of Functions ◽  
Duflo Map

We analyze the ⋆-product induced on ℱ(ℝ3) by a suitable reduction of the Moyal product defined on ℱ(ℝ4). This is obtained through the identification ℝ3≃𝔤*, with 𝔤 a three-dimensional Lie algebra. We consider the 𝔰𝔲(2) case, exhibit a matrix basis and realize the algebra of functions on 𝔰𝔲(2)* in such a basis. The relation to the Duflo map is discussed. As an application to quantum mechanics we compute the spectrum of the hydrogen atom.

scholarly journals Deforming 𝔥-trivial the Lie algebra Vect(S1) inside the Lie algebra of pseudodifferential operators Ψ𝒟𝒪

2017 ◽  
Vol 14 (06) ◽  
pp. 1750082 ◽  
Author(s):  
Imed Basdouri ◽  
Issam Bartouli ◽  
Jean Lerbet
Keyword(s):  
Lie Algebra ◽  
Cotangent Bundle ◽  
Vector Fields ◽  
Pseudodifferential Operators ◽  
Sufficient Conditions ◽  
Lie Derivative ◽  
Smooth Vector ◽  
Algebra Of Functions ◽  
Infinitesimal Deformations ◽  
Necessary And Sufficient

In this paper, we consider the action of Vect(S1) by Lie derivative on the spaces of pseudodifferential operators [Formula: see text]. We study the [Formula: see text]-trivial deformations of the standard embedding of the Lie algebra Vect(S1) of smooth vector fields on the circle, into the Lie algebra of functions on the cotangent bundle [Formula: see text]. We classify the deformations of this action that become trivial once restricted to [Formula: see text], where [Formula: see text] or [Formula: see text]. Necessary and sufficient conditions for integrability of infinitesimal deformations are given.

Jet Modules

2007 ◽  
Vol 59 (4) ◽  
pp. 712-729 ◽  
Author(s):  
Yuly Billig
Keyword(s):  
Lie Algebra ◽  
Vector Fields ◽  
Tensor Fields ◽  
Indecomposable Modules ◽  
Important Family ◽  
Algebra Of Functions

AbstractIn this paper we classify indecomposable modules for the Lie algebra of vector fields on a torus that admit a compatible action of the algebra of functions. An important family of suchmodules is given by spaces of jets of tensor fields.

scholarly journals Nonassociative Algebras: A Framework for Differential Geometry

2003 ◽  
Vol 2003 (60) ◽  
pp. 3777-3795 ◽  
Author(s):  
Lucian M. Ionescu
Keyword(s):  
Differential Geometry ◽  
Lie Algebra ◽  
Vector Fields ◽  
Bianchi Identity ◽  
Affine Connection ◽  
Natural Generalization ◽  
Nonassociative Algebra ◽  
Lie Bracket ◽  
Algebra Of Functions ◽  
Starting Point

A nonassociative algebra endowed with a Lie bracket, called atorsion algebra, is viewed as an algebraic analog of a manifold with an affine connection. Its elements are interpreted as vector fields and its multiplication is interpreted as a connection. This provides a framework for differential geometry on a formal manifold with a formal connection. A torsion algebra is a natural generalization of pre-Lie algebras which appear as the “torsionless” case. The starting point is the observation that the associator of a nonassociative algebra is essentially the curvature of the corresponding Hochschild quasicomplex. It is a cocycle, and the corresponding equation is interpreted as Bianchi identity. The curvature-associator-monoidal structure relationships are discussed. Conditions on torsion algebras allowing to construct an algebra of functions, whose algebra of derivations is the initial Lie algebra, are considered. The main example of a torsion algebra is provided by the pre-Lie algebra of Hochschild cochains of aK-module, with Lie bracket induced by Gerstenhaber composition.

scholarly journals HERMITIAN VECTOR FIELDS AND SPECIAL PHASE FUNCTIONS

2006 ◽  
Vol 03 (04) ◽  
pp. 719-754 ◽  
Author(s):  
JOSEF JANYŠKA ◽  
MARCO MODUGNO
Keyword(s):  
Electromagnetic Field ◽  
Line Bundle ◽  
Lie Algebra ◽  
Vector Fields ◽  
Riemannian Metric ◽  
Absolute Time ◽  
Lorentzian Metric ◽  
First Case ◽  
Fibred Manifold ◽  
Phase Functions

We start by analyzing the Lie algebra of Hermitian vector fields of a Hermitian line bundle. Then, we specify the base space of the above bundle by considering a Galilei, or an Einstein spacetime. Namely, in the first case, we consider, a fibred manifold over absolute time equipped with a spacelike Riemannian metric, a spacetime connection (preserving the time fibring and the spacelike metric) and an electromagnetic field. In the second case, we consider a spacetime equipped with a Lorentzian metric and an electromagnetic field. In both cases, we exhibit a natural Lie algebra of special phase functions and show that the Lie algebra of Hermitian vector fields turns out to be naturally isomorphic to the Lie algebra of special phase functions. Eventually, we compare the Galilei and Einstein cases.

A note on the algebra of Poisson brackets

1984 ◽  
Vol 96 (1) ◽  
pp. 45-60 ◽  
Author(s):  
C. J. Atkin
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Symplectic Manifold ◽  
Vector Fields ◽  
Poisson Brackets ◽  
Infinite Dimensional

In a long sequence of notes in the Comptes Rendus and elsewhere, and in the papers [1], [2], [3], [6], [7], Lichnerowicz and his collaborators have studied the ‘classical infinite-dimensional Lie algebras’, their derivations, automorphisms, co-homology, and other properties. The most familiar of these algebras is the Lie algebra of C∞ vector fields on a C∞ manifold. Another is the Lie algebra of ‘Poisson brackets’, that is, of C∞ functions on a C∞ symplectic manifold, with the Poisson bracket as composition; some questions concerning this algebra are of considerable interest in the theory of quantization – see, for instance, [2] and [3].

scholarly journals Representations of the Lie algebra of vector fields on a sphere

2019 ◽  
Vol 223 (8) ◽  
pp. 3581-3593 ◽  
Author(s):  
Yuly Billig ◽  
Jonathan Nilsson
Keyword(s):  
Lie Algebra ◽  
Vector Fields
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