DEFORMATION QUANTIZATION: IS C1 NECESSARILY SKEW?

2002 ◽  
Vol 16 (14n15) ◽  
pp. 1925-1930
Author(s):  
CHRISTIAN FRONSDAL
Keyword(s):  
Poisson Bracket ◽  
Commutative Algebra ◽  
Deformation Quantization ◽  
Formal Series ◽  
Symplectic Space ◽  
Wide Spread ◽  
Algebra Of Functions ◽  
Associative Product

Deformation quantization (of a commutative algebra) is based on the introduction of a new associative product, expressed as a formal series, [Formula: see text]. In the case of the algebra of functions on a symplectic space the first term in the perturbation is often identified with the antisymmetric Poisson bracket. There is a wide-spread belief that every associative *-product is equivalent to one for which C1(f,g) is antisymmetric and that, in particular, every abelian deformation is trivial. This paper shows that this is far from being the case and illustrates the existence of abelian deformations by physical examples.

FOURIER ANALYSIS OVER HYPERBOLIC ALGEBRA, PSEUDO-DIFFERENTIAL OPERATORS, AND HYPERBOLIC DEFORMATION OF CLASSICAL MECHANICS

2007 ◽  
Vol 10 (03) ◽  
pp. 421-438 ◽  
Author(s):  
ANDREI KHRENNIKOV
Keyword(s):  
Quantum Mechanics ◽  
Fourier Analysis ◽  
Poisson Bracket ◽  
Commutative Algebra ◽  
Differential Operators ◽  
Classical Mechanics ◽  
Complex Numbers ◽  
Two Dimensional ◽  
Quantum Deformation ◽  
Pseudo Differential Operators

We develop Fourier analysis over hyperbolic algebra (the two-dimensional commutative algebra with the basis e1 = 1, e2 = j, where j2 = 1). We demonstrated that classical mechanics has, besides the well-known quantum deformation over complex numbers, another deformation — so-called hyperbolic quantum mechanics. The classical Poisson bracket can be obtained as the limit h → 0 not only of the ordinary Moyal bracket, but also a hyperbolic analogue of the Moyal bracket.

The algebra of differential forms on a full matric bialgebra

1993 ◽  
Vol 114 (1) ◽  
pp. 111-130 ◽  
Author(s):  
A. Sudbery
Keyword(s):  
Vector Space ◽  
Commutative Algebra ◽  
Differential Algebra ◽  
Differential Forms ◽  
Sufficient Conditions ◽  
Classical Case ◽  
Algebra Of Functions ◽  
The Matrix ◽  
Constant Coefficients ◽  
Necessary And Sufficient

AbstractWe construct a non-commutative analogue of the algebra of differential forms on the space of endomorphisms of a vector space, given a non-commutative algebra of functions and differential forms on the vector space. The construction yields a differential bialgebra which is a skew product of an algebra of functions and an algebra of differential forms with constant coefficients. We give necessary and sufficient conditions for such an algebra to exist, show that it is uniquely determined by the differential algebra on the vector space, and show that it is a non-commutative superpolynomial algebra in the matrix elements and their differentials (i.e. that it has the same dimensions of homogeneous components as in the classical case).

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.

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.

scholarly journals DEFORMATION QUANTIZATION OF CERTAIN NONLINEAR POISSON STRUCTURES

1998 ◽  
Vol 09 (05) ◽  
pp. 599-621 ◽  
Author(s):  
BYUNG-JAY KAHNG
Keyword(s):  
Poisson Bracket ◽  
Lie Algebra ◽  
Deformation Quantization ◽  
Dual Space ◽  
Poisson Brackets ◽  
Poisson Structures ◽  
Central Extensions ◽  
C Algebras ◽  
Lie Bracket ◽  
Compact Quantum Groups

As a generalization of the linear Poisson bracket on the dual space of a Lie algebra, we introduce certain nonlinear Poisson brackets which are "cocycle perturbations" of the linear Poisson bracket. We show that these special Poisson brackets are equivalent to Poisson brackets of central extension type, which resemble the central extensions of an ordinary Lie bracket via Lie algebra cocycles. We are able to formulate (strict) deformation quantizations of these Poisson brackets by means of twisted group C*-algebras. We also indicate that these deformation quantizations can be used to construct some specific non-compact quantum groups.

scholarly journals Peierls brackets in non-Lagrangian field theory

2014 ◽  
Vol 29 (27) ◽  
pp. 1450157 ◽  
Author(s):  
A. A. Sharapov
Keyword(s):  
Field Theory ◽  
Poisson Bracket ◽  
Path Integral ◽  
Deformation Quantization ◽  
Equations Of Motion ◽  
Integral Approach ◽  
Lagrangian Dynamics ◽  
Path Integral Approach ◽  
Lagrangian Field Theory

The concept of Lagrange structure allows one to systematically quantize the Lagrangian and non-Lagrangian dynamics within the path-integral approach. In this paper, I show that any Lagrange structure gives rise to a covariant Poisson bracket on the space of solutions to the classical equations of motion, be they Lagrangian or not. The bracket generalize the well-known Peierls' bracket construction and make a bridge between the path-integral and the deformation quantization of non-Lagrangian dynamics.

scholarly journals DEFORMATION QUANTIZATION OF POISSON MANIFOLDS IN THE DERIVATIVE EXPANSION

2009 ◽  
Vol 06 (02) ◽  
pp. 219-224 ◽  
Author(s):  
A. V. BRATCHIKOV
Keyword(s):  
Poisson Bracket ◽  
Lie Group ◽  
Deformation Quantization ◽  
Second Order ◽  
Poisson Manifolds ◽  
Poisson Structures ◽  
Derivative Expansion ◽  
Order Deformation ◽  
Bracket Algebra ◽  
Derivatives Of

Deformation quantization of Poisson manifolds is studied within the framework of an expansion in powers of derivatives of Poisson structures. We construct the Lie group associated with a Poisson bracket algebra which defines a second order deformation in the derivative expansion.

scholarly journals Non-commutative algebra of functions of 4-dimensional quantum Hall droplet

2002 ◽  
Vol 638 (1-2) ◽  
pp. 220-242 ◽  
Author(s):  
Yi-Xin Chen ◽  
Bo-Yu Hou ◽  
Bo-Yuan Hou
Keyword(s):  
Commutative Algebra ◽  
Quantum Hall ◽  
Algebra Of Functions
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