scholarly journals A COHOMOLOGICAL CONSTRUCTION OF MODULES OVER FEDOSOV DEFORMATION QUANTIZATION ALGEBRA

2008 ◽  
Vol 05 (04) ◽  
pp. 547-556
Author(s):  
S. A. POL'SHIN
Keyword(s):  
Left Ideal ◽  
Symplectic Manifold ◽  
Commutative Algebra ◽  
Deformation Quantization ◽  
Separation Of Variables ◽  
Star Product ◽  
Arbitrary Point ◽  
Type Star ◽  
Vanishing Ideal ◽  
Deformed Algebra

In certain neighborhood U of an arbitrary point of a symplectic manifold M we construct a Fedosov-type star-product *L such that for an arbitrary leaf ℘ of a given polarization [Formula: see text] the vanishing ideal of ℘ ∩ U in the commutative algebra C∞(U)[[h]] is a left ideal in the deformed algebra (C∞(U)[[h]],*L). With certain additional assumptions on M, *L becomes a so-called star-product with separation of variables.

Deformation quantization and boundary value problems

2016 ◽  
Vol 13 (02) ◽  
pp. 1650007
Author(s):  
Nikolai Tarkhanov
Keyword(s):  
Harmonic Oscillator ◽  
Boundary Value Problems ◽  
Symplectic Manifold ◽  
Deformation Quantization ◽  
Boundary Value ◽  
Index Theorem ◽  
Manifold With Boundary ◽  
Quantum Observables ◽  
Compact Symplectic Manifold

We describe a natural construction of deformation quantization on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator.

DISSIPATIVE SCALAR FIELD THEORY VIA DEFORMATION QUANTIZATION

2013 ◽  
Vol 28 (16) ◽  
pp. 1350068
Author(s):  
ILIANA CARRILLO-IBARRA ◽  
HUGO GARCÍA-COMPEÁN ◽  
FRANCISCO J. TURRUBIATES
Keyword(s):  
Field Theory ◽  
Scalar Field ◽  
Deformation Quantization ◽  
Star Product ◽  
Quantum Mechanical ◽  
Expectation Values ◽  
Scalar Field Theory ◽  
Oscillation Modes ◽  
Quantization Formalism ◽  
Damped Oscillation

The dissipative scalar field theory by means of the deformation quantization formalism is studied. Following the ideas presented by G. Dito and F. J. Turrubiates [Phys. Lett. A352, 309 (2006)] for quantum mechanics, a star product which contains the dissipative effect for the damped oscillation modes of the field is constructed. Employing this approach the expectation values of some observables in the quantum mechanical case as well as certain correlation functions for the field case are obtained under a particular dissipative process.

scholarly journals FUZZY COMPLEX GRASSMANNIAN SPACES AND THEIR STAR PRODUCTS

2003 ◽  
Vol 18 (11) ◽  
pp. 1935-1958 ◽  
Author(s):  
BRIAN P. DOLAN ◽  
OLIVER JAHN
Keyword(s):  
Commutative Algebra ◽  
Star Product ◽  
Matrix Multiplication ◽  
Noncommutative Algebra ◽  
Infinite Dimensional ◽  
Complex Projective Spaces ◽  
Finite Dimensional ◽  
Algebra Of Functions ◽  
Finite Sum ◽  
Complex Projective

We derive an explicit expression for an associative star product on noncommutative versions of complex Grassmannian spaces, in particular for the case of complex two-planes. Our expression is in terms of a finite sum of derivatives. This generalizes previous results for complex projective spaces and gives a discrete approximation for the Grassmannians in terms of a noncommutative algebra, represented by matrix multiplication in a finite-dimensional matrix algebra. The matrices are restricted to have a dimension which is precisely determined by the harmonic expansion of functions on the commutative Grassmannian, truncated at a finite level. In the limit of infinite-dimensional matrices we recover the commutative algebra of functions on the complex Grassmannians.

GENERALIZED COHERENT STATES APPROACH TO DEFORMATION QUANTIZATION

2012 ◽  
Vol 27 (18) ◽  
pp. 1250095 ◽  
Author(s):  
S. A. A. GHORASHI ◽  
R. ROKNIZADEH ◽  
M. BAGHERI HAROUNI
Keyword(s):  
Coherent State ◽  
Deformation Quantization ◽  
Coherent States ◽  
Star Product ◽  
Eigenvalue Equation ◽  
Generalized Coherent States

Generalized (f)-coherent state approach in deformation quantization framework is investigated by using a *-eigenvalue equation. For this purpose we introduce a new Moyal star product called f-star product, so that by using this *f-eigenvalue equation one can obtain exactly the spectrum of a general Hamiltonian of a deformed system. Eventually the method is supported with some examples.

scholarly journals Comparison and continuity of Wick-type star products on certain coadjoint orbits

2019 ◽  
Vol 31 (5) ◽  
pp. 1203-1223
Author(s):  
Chiara Esposito ◽  
Philipp Schmitt ◽  
Stefan Waldmann
Keyword(s):  
Star Product ◽  
Coadjoint Orbit ◽  
Type Star ◽  
Coadjoint Orbits ◽  
Star Products ◽  
Continuity Properties ◽  
Fréchet Algebra

AbstractIn this paper, we discuss continuity properties of the Wick-type star product on the 2-sphere, interpreted as a coadjoint orbit. Star products on coadjoint orbits in general have been constructed by different techniques. We compare the constructions of Alekseev–Lachowska and Karabegov, and we prove that they agree in general. In the case of the 2-sphere, we establish the continuity of the star product, thereby allowing for a completion to a Fréchet algebra.

scholarly journals ON RELATION BETWEEN WEYL AND KONTSEVICH QUANTUM PRODUCTS: DIRECT EVALUATION UP TO THE ℏ3-ORDER

2001 ◽  
Vol 16 (10) ◽  
pp. 615-625 ◽  
Author(s):  
A. ZOTOV
Keyword(s):  
Deformation Quantization ◽  
Jacobi Identity ◽  
Star Product ◽  
Star Products ◽  
Moyal Product ◽  
Third Order ◽  
Direct Evaluation ◽  
The Third ◽  
Arbitrary Change ◽  
Poisson Bivector

In his celebrated paper Kontsevich has proved a theorem which manifestly gives a quantum product (deformation quantization formula) and states that changing coordinates leads to gauge equivalent star products. To illuminate his procedure, we make an arbitrary change of coordinates in the Weyl (Moyal) product and obtain the deformation quantization formula up to the third order. In this way, the Poisson bivector is shown to depend on ℏ and not to satisfy the Jacobi identity. It is also shown that the values of coefficients in the formula obtained follow from associativity of the star product.

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