Star product and star function

2019 ◽  
pp. 281-292
Author(s):  
Akira YOSHIOKA
Keyword(s):  
Complex Space ◽  
Star Product ◽  
Star Products ◽  
Exponential Functions ◽  
Product Algebra

We give a brief review on star products and star functions [8, 9]. We introduce a star product on polynomials. Extending the product to functions on complex space, we introduce exponential element in the star product algebra. By means of the star exponential functions we can define several functions called star functions in the algebra.We show certain examples.

scholarly journals Hermitian realizations of κ-Minkowski space–time

2015 ◽  
Vol 30 (03) ◽  
pp. 1550019 ◽  
Author(s):  
Domagoj Kovačević ◽  
Stjepan Meljanac ◽  
Andjelo Samsarov ◽  
Zoran Škoda
Keyword(s):  
Minkowski Space ◽  
Star Product ◽  
Plane Waves ◽  
Space Time ◽  
Star Products ◽  
Systematic Construction ◽  
Minkowski Space Time ◽  
Noncommutative Plane

General realizations, star products and plane waves for κ-Minkowski space–time are considered. Systematic construction of general Hermitian realization is presented, with special emphasis on noncommutative plane waves and Hermitian star product. Few examples are elaborated and possible physical applications are mentioned.

MOYAL STAR PRODUCT OF μ-HOLOMORPHIC j-DIFFERENTIALS

2008 ◽  
Vol 05 (03) ◽  
pp. 363-373
Author(s):  
M. KACHKACHI
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Quantum Effect ◽  
Star Product ◽  
Complex Structure ◽  
Complex Structures ◽  
Star Products ◽  
First Order ◽  
Conformal Fields ◽  
Conformal Covariance

It was shown in [1], only for scalar conformal fields, that the Moyal–Weyl star product can introduce the quantum effect as the phase factor to the ordinary product. In this paper we show that, even on the same complex structure, the Moyal–Weyl star product of two j-differentials (conformal fields of weights (j, 0)) does not vanish but it generates the quantum effect at the first order of its perturbative series. More generally, we get the explicit expression of the Moyal–Weyl star product of j-differentials defined on any complex structure of a bi-dimensional Riemann surface Σ. We show that the star product of two j-differentials is not a j-differential and does not preserve the conformal covariance character. This can shed some light on the Moyal–Weyl deformation quantization procedure connection's with the deformation of complex structures on a Riemann surface. Hence, the situation might relate the star products to the Moduli and Teichmüller spaces of Riemann surfaces.

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.

scholarly journals ACTIONS, CHARGES AND OFF-SHELL FIELDS IN THE UNFOLDED DYNAMICS APPROACH

2006 ◽  
Vol 03 (01) ◽  
pp. 37-80 ◽  
Author(s):  
M. A. VASILIEV
Keyword(s):  
Dynamical System ◽  
Closed Form ◽  
Minkowski Space ◽  
Star Product ◽  
Einstein Equations ◽  
Yang Mills ◽  
Conserved Charges ◽  
Zero Curvature ◽  
Product Algebra

Within unfolded dynamics approach, we represent actions and conserved charges as elements of cohomology of the L∞ algebra underlying the unfolded formulation of a given dynamical system. The unfolded off-shell constraints for symmetric fields of all spins in Minkowski space are shown to have the form of zero curvature and covariant constancy conditions for 1-forms and 0-forms taking values in an appropriate star product algebra. Unfolded formulation of Yang–Mills and Einstein equations is presented in a closed form.

scholarly journals Wick rotations in deformation quantization

2021 ◽  
pp. 2150035
Author(s):  
Philipp Schmitt ◽  
Matthias Schötz
Keyword(s):  
Analytic Functions ◽  
Star Product ◽  
Holomorphic Extension ◽  
Star Products ◽  
Formal Deformation ◽  
Space Reduction ◽  
Function Algebras ◽  
Special Cases ◽  
Complex Projective ◽  
Holomorphic Extensions

We study formal and non-formal deformation quantizations of a family of manifolds that can be obtained by phase space reduction from [Formula: see text] with the Wick star product in arbitrary signature. Two special cases of such manifolds are the complex projective space [Formula: see text] and the complex hyperbolic disc [Formula: see text]. We generalize several older results to this setting: The construction of formal star products and their explicit description by bidifferential operators, the existence of a convergent subalgebra of “polynomial” functions, and its completion to an algebra of certain analytic functions that allow an easy characterization via their holomorphic extensions. Moreover, we find an isomorphism between the non-formal deformation quantizations for different signatures, linking, e.g., the star products on [Formula: see text] and [Formula: see text]. More precisely, we describe an isomorphism between the (polynomial or analytic) function algebras that is compatible with Poisson brackets and the convergent star products. This isomorphism is essentially given by Wick rotation, i.e. holomorphic extension of analytic functions and restriction to a new domain. It is not compatible with the [Formula: see text]-involution of pointwise complex conjugation.

scholarly journals Traces for Star Products on the Dual of a Lie Algebra

2003 ◽  
Vol 15 (05) ◽  
pp. 425-445 ◽  
Author(s):  
Pierre Bieliavsky ◽  
Simone Gutt ◽  
Martin Bordemann ◽  
Stefan Waldmann
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Star Product ◽  
Coadjoint Orbits ◽  
Star Products ◽  
Compact Lie Group ◽  
Regular Orbit ◽  
Elementary Argument

In this paper, we describe all traces for the BCH star-product on the dual of a Lie algebra. First we show by an elementary argument that the BCH as well as the Kontsevich star-product are strongly closed if and only if the Lie algebra is unimodular. In a next step we show that the traces of the BCH star-product are given by the ad-invariant functionals. Particular examples are the integration over coadjoint orbits. We show that for a compact Lie group and a regular orbit one can even achieve that this integration becomes a positive trace functional. In this case we explicitly describe the corresponding GNS representation. Finally we discuss how invariant deformations on a group can be used to induce deformations of spaces where the group acts on.

scholarly journals Green’s functions for translation invariant star products

2015 ◽  
Vol 30 (36) ◽  
pp. 1550194
Author(s):  
Fedele Lizzi ◽  
Manolo Rivera ◽  
Patrizia Vitale
Keyword(s):  
Field Theory ◽  
Scalar Field ◽  
Commutation Relation ◽  
Green's Functions ◽  
Green’S Functions ◽  
Star Product ◽  
Canonical Commutation Relation ◽  
Star Products ◽  
Translation Invariant ◽  
The One

We calculate the Green’s functions for a scalar field theory with quartic interactions for which the fields are multiplied with a generic translation invariant star product. Our analysis involves both non-commutative products, for which there is the canonical commutation relation among coordinates, and nonlocal commutative products. We give explicit expressions for the one-loop corrections to the two- and four-point functions. We find that the phenomenon of ultraviolet/infrared mixing is always a consequence of the presence of non-commuting variables. The commutative part of the product does not have the mixing.

Star Products and Certain Star Product Functions

2013 ◽  
pp. 39-45
Author(s):  
Mari Iida ◽  
Chishu Tsukamoto ◽  
Akira Yoshioka
Keyword(s):  
Star Product ◽  
Star Products
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