Matrix Bases for Star Products: a Review

Lizzi, Fedele; Vitale, Patrizia

doi:10.3842/sigma.2014.086

Cited by 31 publications

(47 citation statements)

References 89 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the latter case the quartic interaction term reduces to the square of φ * φ which can in turn be analyzed in terms of the Fourier transform (6.39). Another possibility would be to look for a suitable matrix basis adapted to the quantization-dequantization scheme considered here (see [28] for details). We shall come back to this issue elsewhere.…”

Section: Discussionmentioning

confidence: 99%

Noncommutative ℝ d $$ {\mathrm{\mathbb{R}}}^d $$ via closed star product

Kupriyanov

Vitale

2015

J. High Energ. Phys.

View full text Add to dashboard Cite

Abstract:We consider linear star products on R d of Lie algebra type. First we derive the closed formula for the polydifferential representation of the corresponding Lie algebra generators. Using this representation we define the Weyl star product on the dual of the Lie algebra. Then we construct a gauge operator relating the Weyl star product with the one which is closed with respect to some trace functional, Tr (f g) = Tr (f · g). We introduce the derivative operator on the algebra of the closed star product and show that the corresponding Leibniz rule holds true up to a total derivative. As a particular example we study the space R 3 θ with su(2) type noncommutativity and show that in this case the closed star product is the one obtained from the Duflo quantization map. As a result a Laplacian can be defined such that its commutative limit reproduces the ordinary commutative one. The deformed Leibniz rule is applied to scalar field theory to derive conservation laws and the corresponding noncommutative currents.

show abstract

Section: Discussionmentioning

confidence: 99%

Noncommutative ℝ d $$ {\mathrm{\mathbb{R}}}^d $$ via closed star product

Kupriyanov

Vitale

2015

J. High Energ. Phys.

View full text Add to dashboard Cite

show abstract

“…The first one is a further analysis of the Lorentzian analogue of the R 3 λ , that is to obtain the algebra of its functions and its differential structure. The second would be an effort to write the action in a specific matrix basis and study its behaviour using perturbation theory [66]. The third direction is the upgrade of the present noncommutative three-dimensional gravity model to a four-dimensional one, modifying the procedure of the first.…”

Section: Discussionmentioning

confidence: 99%

Gravity as a Gauge Theory on Three-Dimensional Noncommutative spaces

Jurman¹,

Manolakos²,

Manousselis³

et al. 2018

Proceedings of Corfu Summer Institute 2017 "Schools and Workshops on Elementary Particle Physics and Gravity" — PoS(CORFU2017)

View full text Add to dashboard Cite

We plan to translate the successful description of three-dimensional gravity as a gauge theory in the noncommutative framework, making use of the covariant coordinates. We consider two specific three-dimensional fuzzy spaces based on SU(2) and SU(1,1), which carry appropriate symmetry groups. These are the groups we are going to gauge in order to result with the transformations of the gauge fields (dreibein, spin connection and two extra Maxwell fields due to noncommutativity), their corresponding curvatures and eventually determine the action and the equations of motion. Finally, we verify their connection to three-dimensional gravity.

show abstract

“…Now we consider the two-qubit-state tomogram (19) for the given density matrix. To obtain the components, we consider the transformed density matrix…”

Section: Tomograms Of the Two-qubit Statementioning

confidence: 99%

“…The 16 components of the Wigner function (22) of the two-qubit state (19) in the AA orÂA Employing the explicit forms of the Wigner-function components (23) and basis matricesÂA (2) α , after some algebra, we obtain the reconstruction formulâ…”

Section: For One Qubit the Matrices A α Readmentioning

confidence: 99%