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We introduce a finite dimensional matrix model approximation to the algebra of functions on a disc based on noncommutative geometry. The algebra is a subalgebra of the one characterizing the noncommutative plane with a * product and depends on two parameters N and θ. It is composed of functions which decay exponentially outside a disc. In the limit in which… (More)

While there has been growing interest for noncommutative spaces in recent times, most examples have been based on the simplest noncommutative algebra: [x i , x j ] = iθ ij. Here we present new classes of (non-formal) deformed products associated to linear Lie algebras of the kind [x i , x j ] = ic k ij x k. For all possible three-dimensional cases, we… (More)

- Paolo Aschieri, Fedele Lizzi, Patrizia Vitale
- 2007

We discuss the effects that a noncommutative geometry induced by a Drinfeld twist has on physical theories. We systematically deform all products and symmetries of the theory. We discuss noncommutative classical mechanics, in particular its deformed Poisson bracket and hence time evolution and symmetries. The twisting is then extended to classical fields,… (More)

The usual description of 2 + 1 dimensional Einstein gravity as a Chern-Simons (CS) theory is extended to a one parameter family of descriptions of 2 + 1 Einstein gravity. This is done by replacing the Poincaré gauge group symmetry by a q−deformed Poincaré gauge group symmetry, with the former recovered when q → 1. As a result, we obtain a one parameter… (More)

- J. M. Gracia-Bond́ıa, Fedele Lizzi, F. Ruiz Ruiz, Patrizia Vitale
- 2006

We prove that the Moyal product is covariant under linear affine spacetime transformations. From the covariance law, by introducing an (x, Θ)-space where the spacetime coordinates and the noncommutativity matrix components are on the same footing, we obtain a noncommutative representation of the affine algebra, its generators being differential operators in… (More)

Quantum Groups can be constructed by applying the quantization by deformation procedure to Lie groups endowed with a suitable Poisson bracket. Here we try to develop an understanding of these structures by investigating dynamical systems which are associated with this bracket. We look at SU (2) and SU (1, 1), as submanifolds of a 4–dimensional phase space… (More)

Gravitational fields invariant for a 2-dimensional Lie algebra of Killing fields [X, Y ] = Y , with Y of light type, are analyzed. The conditions for them to represent gravitational waves are verified and the definition of energy and polarization is addressed; realistic generating sources are described.

The recently proposed probability representation of quantum mechanics is generalized to quantum field theory. We introduce a probability distribution functional for field configurations and find an evolution equation for such a distribution. The connection to the time-dependent generating functional of Green's functions is elucidated and the classical limit… (More)

- Olga V. Man’ko, Vladimir I. Man’ko, Giuseppe Marmo, Patrizia Vitale
- 2006

Quantization of classical systems using the star-product of symbols of observables is discussed. In the star-product scheme an analysis of dual structures is performed and a physical interpretation is proposed. At the Lie algebra level duality is shown to be connected to double Lie algebras. The analysis is specified to quantum tomography. The classical… (More)

The fuzzy disc is a matrix approximation of the functions on a disc which preserves rotational symmetry. In this paper we introduce a basis for the algebra of functions on the fuzzy disc in terms of the eigenfunctions of a properly defined fuzzy Laplacian. In the commutative limit they tend to the eigenfunctions of the ordinary Laplacian on the disc, i.e.… (More)