Patrizia Vitale

Twisting all the way: from Classical Mechanics to Quantum Fields

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, and then to the main interest of this work: quantum fields. This leads to a geometric formulation of quantization on noncommutative space-time, i.e., we establish a noncommutative correspondence principle from *-Poisson brackets to * commutators. In particular commutation relations among creation and annihilation operators are deduced.

The Fuzzy Disc

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 the size of the matrices goes to infinity and the noncommutativity parameter goes to zero the disc becomes sharper. We introduce a laplacian defined on the whole algebra and calculate its eigenvalues. We also calculate the two-points correlation function for a free massless theory (Greens function). In both cases the agreement with the exact result on the disc is very good already for relatively small matrices. This opens up the possibility for the study of field theories on the disc with nonperturbative methods. The model contains edge states, a fact studied in a similar matrix model independently introduced by Balachandran, Gupta and Kurkcuoglu.

Twisted noncommutative field theory with the Wick-Voros and Moyal products

We present a comparison of the noncommutative field theories built using two different star products: Moyal and Wick-Voros (or normally ordered). For the latter we discuss both the classical and the quantum field theory in the quartic potential case and c

Star products, duality and double Lie algebras

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 tomographic Poisson bracket is found.

Phase space distributions and a duality symmetry for star products

Abstract A duality property for star products is exhibited. In view of it, known star-product schemes, like the Weyl–Wigner–Moyal formalism, the Husimi and the Glauber–Sudarshan maps are revisited. The tomographic map, which has been recently described as yet another star product scheme, is considered. It yields a noncommutative algebra of operator symbols which are positive definite probability distributions. Through the duality symmetry a new noncommutative algebra of operator symbols is found, equipped with a new star product. The kernel of the star product is established in explicit form and examples are considered.

A GENERALIZATION OF THE JORDAN-SCHWINGER MAP: THE CLASSICAL VERSION AND ITS q DEFORMATION

For all three-dimensional Lie algebras the construction of generators in terms of functions on four-dimensional real phase space is given with a realization of the Lie product in terms of Poisson brackets. This is the classical Jordan-Schwinger map, which is also given for the deformed algebras , ℰq(2) and ℋq(1). The algebra is discussed in the same context.

The beat of a fuzzy drum: fuzzy Bessel functions for the disc

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. Bessel functions of the first kind, thus deserving the name of fuzzy Bessel functions.

Twisted conformal symmetry in noncommutative two dimensional quantum field theory

By twisting the commutation relations between creation and annihilation operators, we show that quantum conformal invariance can be implemented in the 2-d Moyal plane. This is an explicit realization of an infinite dimensional symmetry as a quantum algebra.

ON THE ⋆-PRODUCT QUANTIZATION AND THE DUFLO MAP IN THREE DIMENSIONS

We analyze the ⋆-product induced on ℱ(ℝ3) by a suitable reduction of the Moyal product defined on ℱ(ℝ4). This is obtained through the identification ℝ3≃𝔤*, with 𝔤 a three-dimensional Lie algebra. We consider the 𝔰𝔲(2) case, exhibit a matrix basis and realize the algebra of functions on 𝔰𝔲(2)* in such a basis. The relation to the Duflo map is discussed. As an application to quantum mechanics we compute the spectrum of the hydrogen atom.

Noncommutative field theories on

A bstractWe consider the noncommutative space