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 Tensor fields associated with integrable systems of chiral

2019 ◽  
Vol 21 (4) ◽  
pp. 405-412
Author(s):  
Alexander V. Balandin
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Necessary Conditions ◽  
Differential Forms ◽  
Type Systems ◽  
Lax Representation ◽  
Killing Tensor ◽  
Tensor Fields ◽  
Linear Differential ◽  
Special Case

This article describes necessary conditions for chiral-type systems to admit Lax representation with values in simple compact Lie algebras. These conditions state that there exists a covariant constant tensor field with an additional property. It is proposed to construct in an invariant way some covariant tensor fields using the Lax representation of the system under consideration. These fields are constructed by taking linear differential forms with values in the Lie algebra that are constructed using the Lax representation of the system and substituting them into an arbitrary Ad-invariant form on the Lie algebra. The paper proves that such tensor fields are Killing tensor fields or covariant constant fields. The discovered necessary conditions for the existence of the Lax representation are obtained using a special case of such tensor fields associated with the Killing metric of the Lie 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.

Z-Graded Extensions of Poisson Brackets

1997 ◽  
Vol 09 (01) ◽  
pp. 1-27 ◽  
Author(s):  
Janusz Grabowski
Keyword(s):  
Poisson Bracket ◽  
Jacobi Identity ◽  
Differential Forms ◽  
Poisson Manifold ◽  
Exterior Derivative ◽  
Tensor Fields ◽  
Lie Bracket ◽  
Nijenhuis Bracket ◽  
Hamiltonian Map ◽  
Vector Valued

A Z-graded Lie bracket { , }P on the exterior algebra Ω(M) of differential forms, which is an extension of the Poisson bracket of functions on a Poisson manifold (M,P), is found. This bracket is simultaneously graded skew-symmetric and satisfies the graded Jacobi identity. It is a kind of an 'integral' of the Koszul–Schouten bracket [ , ]P of differential forms in the sense that the exterior derivative is a bracket homomorphism: [dμ, dν]P=d{μ, ν}P. A naturally defined generalized Hamiltonian map is proved to be a homomorphism between { , }P and the Frölicher–Nijenhuis bracket of vector valued forms. Also relations of this graded Poisson bracket to the Schouten–Nijenhuis bracket and an extension of { , }P to a graded bracket on certain multivector fields, being an 'integral' of the Schouten–Nijenhuis bracket, are studied. All these constructions are generalized to tensor fields associated with an arbitrary Lie algebroid.

Vertical Chern Type Classes on Complex Finsler Bundles

2011 ◽  
Vol 57 (2) ◽  
pp. 377-386
Author(s):  
Cristian Ida
Keyword(s):  
Differential Forms ◽  
Linear Connection ◽  
Curvature Form ◽  
Cohomology Groups ◽  
Finsler Manifolds ◽  
Invariance Properties ◽  
Type Classes

Vertical Chern Type Classes on Complex Finsler BundlesIn the present paper, we define vertical Chern type classes on complex Finsler bundles, as an extension of thev-cohomology groups theory on complex Finsler manifolds. These classes are introduced in a classical way by using closed differential forms with respect to the conjugated vertical differential in terms of the vertical curvature form of Chern-Finsler linear connection. Also, some invariance properties of these classes are studied.

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 CLIFFORD VALUED DIFFERENTIAL FORMS, AND SOME ISSUES IN GRAVITATION, ELECTROMAGNETISM AND "UNIFIED" THEORIES

2004 ◽  
Vol 13 (09) ◽  
pp. 1879-1915 ◽  
Author(s):  
W. A. RODRIGUES ◽  
E. CAPELAS DE OLIVEIRA
Keyword(s):  
Lie Algebra ◽  
Differential Forms ◽  
Linear Connection ◽  
Gravitational Theory ◽  
Momentum Conservation ◽  
Unified Field Theory ◽  
Mathematical Methods ◽  
Unified Field ◽  
Theory Of Gravitation ◽  
Local Trivialization

In this paper we show how to describe the general theory of a linear metric compatible connection with the theory of Clifford valued differential forms. This is done by realizing that for each spacetime point the Lie algebra of Clifford bivectors is isomorphic to the Lie algebra of [Formula: see text]. In that way the pullback of the linear connection under a local trivialization of the bundle (i.e., a choice of gauge) is represented by a Clifford valued 1-form. That observation makes it possible to realize immediately that Einstein's gravitational theory can be formulated in a way which is similar to a [Formula: see text] gauge theory. Such a theory is compared with other interesting mathematical formulations of Einstein's theory, and particularly with a supposedly "unified" field theory of gravitation and electromagnetism proposed by M. Sachs. We show that his identification of Maxwell equations within his formalism is not a valid one. Also, taking profit of the mathematical methods introduced in the paper we investigate a very polemical issue in Einstein gravitational theory, namely the problem of the 'energy–momentum' conservation. We show that many statements appearing in the literature are confusing or even wrong.

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