Papers by Patrizia Vitale
Journal of High Energy Physics
We propose a field theoretical model defined on non-commutative space-time with non-constant non-... more We propose a field theoretical model defined on non-commutative space-time with non-constant non-commutativity parameter Θ(x), which satisfies two main requirements: it is gauge invariant and reproduces in the commutative limit, Θ → 0, the standard U(1) gauge theory. We work in the slowly varying field approximation where higher derivatives terms in the star commutator are neglected and the latter is approximated by the Poisson bracket, −i[f, g] ≈ {f, g}. We derive an explicit expression for both the NC deformation of Abelian gauge transformations which close the algebra [δ f , δ g ]A = δ {f,g} A, and the NC field strength F , covariant under these transformations, δ f F = {F , f }. NC Chern-Simons equations are equivalent to the requirement that the NC field strength, F , should vanish identically. Such equations are non-Lagrangian. The NC deformation of Yang-Mills theory is obtained from the gauge invariant action, S = F 2. As guiding example, the case of su(2)-like non-commutativity, corresponding to rotationally invariant NC space, is worked out in detail.
Journal of Physics Communications
Proceedings of Corfu Summer Institute on Elementary Particles and Physics - Workshop on Non Commutative Field Theory and Gravity — PoS(CNCFG2010)
We present an introduction to Group Field Theory models, motivating them on the basis of their re... more We present an introduction to Group Field Theory models, motivating them on the basis of their relationship with discretized BF models of gravity. We derive the Feynmann rules and compute quantum corrections in the coherent states basis.
Journal of High Energy Physics
The Principal Chiral Model (PCM) defined on the group manifold of SU(2) is here investigated with... more The Principal Chiral Model (PCM) defined on the group manifold of SU(2) is here investigated with the aim of getting a further deepening of its relation with Generalized Geometry and Doubled Geometry. A one-parameter family of equivalent Hamiltonian descriptions is analysed, and cast into the form of Born geometries. Then O(3, 3) duality transformations of the target phase space are performed and we show that the resulting dual models are defined on the group SB(2, ℂ) which is the Poisson-Lie dual of SU(2) in the Iwasawa decomposition of the Drinfel’d double SL(2, ℂ). A parent action with doubled degrees of freedom and configuration space SL(2, ℂ) is then defined that reduces to either one of the dually related models, once suitable constraints are implemented.
Physical Review D
We give a Lagrangian description of an electric charge in a field sourced by a continuous magneti... more We give a Lagrangian description of an electric charge in a field sourced by a continuous magnetic monopole distribution. The description is made possible thanks to a doubling of the configuration space. The Legendre transform of the nonrelativistic Lagrangian agrees with the Hamiltonian description given recently by Kupriyanov and Szabo[1]. The covariant relativistic version of the Lagrangian is shown to introduce a new gauge symmetry, in addition to standard reparametrizations. The generalization of the system to open strings coupled to a magnetic monopole distribution is also given, as well as the generalization to particles in a non-Abelian gauge field which does not satisfy Bianchi identities in some region of the space-time.
International Journal of Geometric Methods in Modern Physics
After reviewing Gribov ambiguity of non-Abelian gauge theories, a phenomenon related to the topol... more After reviewing Gribov ambiguity of non-Abelian gauge theories, a phenomenon related to the topology of the bundle of gauge connections, we show that there is a similar feature for noncommutative QED over Moyal space, despite the structure group being Abelian, and we exhibit an infinite number of solutions for the equation of Gribov copies. This is a genuine effect of noncommutative geometry which disappears when the noncommutative parameter vanishes.
Journal of High Energy Physics
A simple mechanical system, the three-dimensional isotropic rigid rotator, is here investigated a... more A simple mechanical system, the three-dimensional isotropic rigid rotator, is here investigated as a 0+1 field theory, aiming at further investigating the relation between Generalized/Double Geometry on the one hand and Doubled World-Sheet Formalism/Double Field Theory, on the other hand. The model is defined over the group manifold of SU(2) and a dual model is introduced having the Poisson-Lie dual of SU(2) as configuration space. A generalized action with configuration space SL(2, C), i.e. the Drinfel'd double of the group SU(2), is then defined: it reduces to the original action of the rotator or to its dual, once constraints are implemented. The new action contains twice as many variables as the original. Moreover its geometric structures can be understood in terms of Generalized Geometry.
Journal of High Energy Physics
We consider linear star products on R d of Lie algebra type. First we derive the closed formula f... more 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.
The symmetry properties under permutation of tomograms representing the states of a system of ide... more The symmetry properties under permutation of tomograms representing the states of a system of identical particles are studied. Starting from the action of the permutation group on the density matrix we define its action on the tomographic probability distribution. Explicit calculations are performed in the case of the two-dimensional harmonic oscillator.
Eprint Arxiv 0807 1492, Jul 9, 2008
We study the noncommutative scalar field theory in the presence of the Wick-Voros product (or nor... more We study the noncommutative scalar field theory in the presence of the Wick-Voros product (or normally ordered product), a variant of the more studied Moyal product. We discuss both the classical and the quantum field theory in the quartic potential case, and calculate the Green's functions up to one loop, for the two and four points cases.
Physical Review D Particles and Fields, Apr 27, 2006
We prove that the Moyal product is covariant under linear affine spacetime transformations. From ... more 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 (x, Θ)-space. As a particular case, the Weyl Lie algebra is studied and known results for Weyl invariant noncommutative field theories are rederived in a nutshell. We also show that this covariance cannot be extended to spacetime transformations generated by differential operators whose coefficients are polynomials of order larger than one. We compare our approach with the twist-deformed enveloping algebra description of spacetime transformations.
Journal of High Energy Physics, 2016
It is shown that in the noncommutative version of QED (NCQED) Gribov copies induced by the noncom... more It is shown that in the noncommutative version of QED (NCQED) Gribov copies induced by the noncommutativity of space-time do appear in the Landau gauge. This is a genuine effect of noncommutative geometry which disappears when the noncommutative parameter vanishes. On the basis of existing applications of the Gribov-Zwanziger propagator in NCQED to deal with the UV/IR mixing problem, we argue that the two problems may have a common origin and possibly a common solution.
We discuss the effects that a noncommutative geometry induced by a Drinfeld twist has on physical... more 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 spacetime, i.e. we establish a noncommutative correspondence principle from ⋆-Poisson brackets to ⋆-commutators. In particular commutation relations among creation and annihilation operators are deduced.
Physical Review A
In the probability representation of quantum mechanics, quantum states are represented by a class... more In the probability representation of quantum mechanics, quantum states are represented by a classical probability distribution, the marginal distribution function (MDF), whose time dependence is governed by a classical evolution equation. We find and explicitly solve, for a wide class of Hamiltonians, new equations for the Green's function of such an equation, the so-called classical propagator. We elucidate the connection of the classical propagator to the quantum propagator for the density matrix and to the Green's function of the Schrödinger equation. Within the new description of quantum mechanics we give a definition of coherence solely in terms of properties of the MDF and we test the new definition recovering well known results. As an application, the forced parametric oscillator is considered. Its classical and quantum propagator are found, together with the MDF for coherent and Fock states.
Modern Physics Letters A, 2015
We calculate the Green’s functions for a scalar field theory with quartic interactions for which ... more 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.
Using the tools of q--differential calculus and quantum Lie algebras associated to quantum groups... more Using the tools of q--differential calculus and quantum Lie algebras associated to quantum groups, we find a one--parameter family of q-gauge theories associated to the quantum group $ISO_q(3,1)$. Although the gauge fields, that is the spin--connection and the vierbeins are non--commuting objects depending on a deformation parameter, $q$, it is possible to construct out of them a metric theory which is insensitive to the deformation. The Christoffel symbols and the Riemann tensor are ordinary commuting objects. Hence it is argued that torsionless Einstein's General Relativity is the common invariant sector of a one--parameter family of deformed gauge theories.
Journal of High Energy Physics, 2015
A duality property for star products is exhibited. In view of it, known star- product schemes, li... more 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 and their dual partners elucidated. The tomographic map, which has been recently described as yet another star product scheme, is con- sidered. It yields a noncommutative algebra of operator symbols which
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Papers by Patrizia Vitale