Star Products and Deformation Quantization

2017 ◽  
pp. 715-774
Author(s):  
A. Weinstein
Keyword(s):  
Deformation Quantization ◽  
Star Products

scholarly journals DEFORMATION QUANTIZATION OF CLASSICAL FIELDS

2001 ◽  
Vol 16 (14) ◽  
pp. 2533-2558 ◽  
Author(s):  
H. GARCÍA-COMPEÁN ◽  
J. F. PLEBAŃSKI ◽  
M. PRZANOWSKI ◽  
F. J. TURRUBIATES
Keyword(s):  
Gauge Theory ◽  
Deformation Quantization ◽  
The Other ◽  
Star Products ◽  
Abelian Gauge ◽  
Abelian Gauge Theory ◽  
Quantization Formalism ◽  
Classical Fields ◽  
Free Fields ◽  
Nontrivial Topology

We study the deformation quantization of scalar and Abelian gauge classical free fields. Stratonovich–Weyl quantizer, star products and Wigner functionals are obtained in field and oscillator variables. The Abelian gauge theory is particularly intriguing since the Wigner functional is factorized into a physical part and the other one containing the constraints only. Some effects of nontrivial topology within the deformation quantization formalism are also considered.

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.

scholarly journals Subalgebras with converging star products in deformation quantization: An algebraic construction forCPn

1996 ◽  
Vol 37 (12) ◽  
pp. 6311-6323 ◽  
Author(s):  
M. Bordemann ◽  
M. Brischle ◽  
C. Emmrich ◽  
S. Waldmann
Keyword(s):  
Deformation Quantization ◽  
Star Products ◽  
Algebraic Construction

scholarly journals GENERALIZED INVERSION OF THE HOCHSCHILD COBOUNDARY OPERATOR AND DEFORMATION QUANTIZATION

2011 ◽  
Vol 08 (01) ◽  
pp. 99-106 ◽  
Author(s):  
A. V. BRATCHIKOV
Keyword(s):  
Deformation Quantization ◽  
Generalized Inverse ◽  
Poisson Manifolds ◽  
Star Products ◽  
Generalized Inversion ◽  
Coboundary Operator ◽  
Differential Complex

Using a derivative decomposition of the Hochschild differential complex we define a generalized inverse of the Hochschild coboundary operator. It can be applied for systematic computations of star products on Poisson manifolds.

scholarly journals COVARIANT STAR PRODUCT ON SYMPLECTIC AND POISSON SPACE–TIME MANIFOLDS

2010 ◽  
Vol 25 (18n19) ◽  
pp. 3765-3796 ◽  
Author(s):  
M. CHAICHIAN ◽  
M. OKSANEN ◽  
A. TUREANU ◽  
G. ZET
Keyword(s):  
Deformation Quantization ◽  
Star Product ◽  
Differential Forms ◽  
Linear Connection ◽  
Poisson Manifold ◽  
Star Products ◽  
Tensor Fields ◽  
Smooth Functions ◽  
Poisson Space ◽  
Special Case

A covariant Poisson bracket and an associated covariant star product in the sense of deformation quantization are defined on the algebra of tensor-valued differential forms on a symplectic manifold, as a generalization of similar structures that were recently defined on the algebra of (scalar-valued) differential forms. A covariant star product of arbitrary smooth tensor fields is obtained as a special case. Finally, we study covariant star products on a more general Poisson manifold with a linear connection, first for smooth functions and then for smooth tensor fields of any type. Some observations on possible applications of the covariant star products to gravity and gauge theory are made.

scholarly journals SEIBERG–WITTEN EQUATIONS FROM FEDOSOV DEFORMATION QUANTIZATION OF ENDOMORPHISM BUNDLE

2011 ◽  
Vol 08 (02) ◽  
pp. 411-428 ◽  
Author(s):  
MICHAŁ DOBRSKI
Keyword(s):  
Gauge Group ◽  
Deformation Quantization ◽  
Star Products ◽  
Recursive Methods ◽  
Fedosov Construction ◽  
Arbitrary Gauge

It is shown how Seiberg–Witten equations can be obtained by means of Fedosov deformation quantization of endomorphism bundle and the corresponding theory of equivalences of star products. In such setting, Seiberg–Witten map can be iteratively computed for arbitrary gauge group up to any given degree with recursive methods of Fedosov construction. Presented approach can be also considered as a generalization of Seiberg–Witten equations to Fedosov type of noncommutativity.

scholarly journals From VOAs to Short Star Products in SCFT

2021 ◽  
Vol 384 (1) ◽  
pp. 245-277
Author(s):  
Mykola Dedushenko
Keyword(s):  
Star Products
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