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Gaetano Fiore

Università degli Studi di Napoli "Federico II", Dipartimento di Matematica ed Applicazioni, Faculty Member

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Papers by Gaetano Fiore

Modules Over the Noncommutative Torus and Elliptic Curves

by Davide Franco and Gaetano Fiore

Letters in Mathematical Physics, 2014

Using the Weil-Brezin-Zak transform of solid state physics, we describe line bundles over ellipti... more Using the Weil-Brezin-Zak transform of solid state physics, we describe line bundles over elliptic curves in terms of Weyl operators. We then discuss the connection with finitely-generated projective modules over the algebra A θ of the noncommutative torus. We show that such A θ -modules have a natural interpretation as Moyal deformations of vector bundles over an elliptic curve E τ , under the condition that the deformation parameter θ and the modular parameter τ satisfy a non-trivial relation. We then conclude with some remarks about formal deformations of vector bundles on the torus and twists based on the Lie algebra of the 3-dimensional Heisenberg group. arXiv:1307.6802v1 [math.OA]

Existence and Uniqueness for Some 3rd Order Dissipative Problems with Various Boundary Conditions

by Gaetano Fiore and Monica De Angelis

Acta Applicandae Mathematicae, 2012

ABSTRACT

SOq(N, - Harmonic Oscillator on the N-Dim Real Quantum Euclidean Space

Ijmpa, 1992

The Schroedinger equation for the [w.r.t. SOq(N, R)] isotropic harmonic oscillator on the N-dimen... more The Schroedinger equation for the [w.r.t. SOq(N, R)] isotropic harmonic oscillator on the N-dimensional real quantum Euclidean space RqN is algebraically solved. The Hamiltonian and the other operators are built in terms of the deformed coordinates and derivatives characterizing this noncommutative geometrical space and its two differential calculi. The spectrum and the eigenfunctions (the q-deformed Hermite functions) are studied.

Leibniz rules and reality conditions

Eur Phys J C, 2000

An analysis is made of reality conditions within the context of non-commutative geometry. We show... more An analysis is made of reality conditions within the context of non-commutative geometry. We show that if a covariant derivative satisfies a given left Leibniz rule then a right Leibniz rule fixes the reality condition for the covariant derivative itself. We show also that the map which determines the right Leibniz rule must satisfy the braid equation if the extension of the covariant derivative to tensor products is to satisfy the reality condition.

Realization ofU q (so(N)) within the differential algebra onR q N

Commun Math Phys, 1995

We realize the Hopf algebra $U_{q^{-1}}(so(N))$ as an algebra of differential operators on the qu... more We realize the Hopf algebra $U_{q^{-1}}(so(N))$ as an algebra of differential operators on the quantum Euclidean space ${\bf R}_q^N$. The generators are suitable q-deformed analogs of the angular momentum components on ordinary ${\bf R}^N$. The algebra $Fun({\bf R}_q^N)$ of functions on ${\bf R}_q^N$ splits into a direct sum of irreducible vector representations of $U_{q^{-1}}(so(N))$; the latter are explicitly constructed as highest weight representations.

format_quoteU q -1 (so(N )) is realized efficiently through differential operator relations in q-deformed space.format_quote

On plane-wave relativistic electrodynamics in plasmas and in vacuum

Journal of Physics A: Mathematical and Theoretical, 2014

We reconsider the exact microscopic equations (differential, and equivalent integral) ruling a re... more We reconsider the exact microscopic equations (differential, and equivalent integral) ruling a relativistic cold plasma after the plane-wave Ansatz, without customary approximations. We show that in the Eulerian description the motion of a very diluted plasma initially at rest and excited by an arbitrary transverse plane electromagnetic travelling-wave has a disarmingly simple and explicit expression in terms of the transverse electromagnetic potential. As a by-product, we obtain also a simple and explicit expression for the general motion of a charged test particle under the action of an electromagnetic wave of the same kind. For a non-zero density plasma the above motion is a good approximation of the real one as long as the back-reaction of the charges on the electromagnetic field can be neglected, i.e. for a time lapse decreasing with the plasma density.

Travelling-Wave Solutions of a Modified Sine-Gordon Equation Used in Superconductivity

Waves and Stability in Continuous Media, 2008

... f^ j Half-array of kinks (p+ or antikinks&lt; p~, with (p±(x,^):=fJ-)&#x27;,&#x27... more ... f^ j Half-array of kinks (p+ or antikinks&lt; p~, with (p±(x,^):=fJ-)&#x27;,&#x27;J, A*] 7r, onZy i/7&lt; l and for any jj,£] 0,/2 (7)[. i^ 1* 1 respectively have velocity v=±v (/j,) and fulfill lim (p±{x, t)= 9, lim [&lt;^±(x^;/i)-^&#x27;±x,^//) 0+(15) z»^ foo x»±oo for a suitable (p,: iz ... Cremonese, Roma, 1956 ...

$Research paper thumbnail of New approach to Hermitian 𝑞-differential operators on ℝ^{ℕ}_{𝕢}$

New approach to Hermitian 𝑞-differential operators on ℝ^{ℕ}_{𝕢}

Contemporary Mathematics, 2005

Decoupling of Translations from Homogeneous Transformations in Inhomogeneous Quantum Groups

Quantum Theory and Symmetries, 2002

We briefly report on our result that, if there exists a realization of a Hopf algebra H in a H-mo... more We briefly report on our result that, if there exists a realization of a Hopf algebra H in a H-module algebra A, then their cross-product is equal to the product of A itself with a subalgebra isomorphic to H and commuting with A. We illustrate its application to the Euclidean quantum groups in N ≥ 3 dimensions.

format_quoteDemonstrates cross-product equality of a Hopf algebra and a module algebra, applicable to Euclidean quantum groups in N ≥ 3 dimensions.format_quote

On q-Deformations of Clifford Algebras

Clifford Algebras and their Applications in Mathematical Physics, 2000

Several Clifford algebras that are covariant under the action of a Lie algebra g can be deformed ... more Several Clifford algebras that are covariant under the action of a Lie algebra g can be deformed in a way consistent with the deformation of U g into a quantum group (or into a triangular Hopf algebra) U q g , i.e. so as to remain covariant under the action of U q g . In this report, after recalling these facts, we review our results regarding the formal realization of the elements of such "q-deformed" Clifford algebras as "functions" (polynomials) in the generators of the undeformed ones; in particular, the intruiging interplay between the original and the q-deformed symmetry. Finally, we briefly illustrate their dramatic consequences on the representation theories of the original and of the q-deformed Clifford algebra, and mention how these results could turn out to be useful in quantum physics.

Drinfel'd Twist and q-Deforming Maps for Lie Group Covariant Heisenberg Algebrae

Rmp, 2000

Any deformation of a Weyl or Clifford algebra can be realized through a change of generators in t... more Any deformation of a Weyl or Clifford algebra can be realized through a change of generators in the undeformed algebra. q-deformations of Weyl or Clifford algebrae that were covariant under the action of a simple Lie algebra g are characterized by their being covariant under the action of the quantum group Uhg, q:=eh. We present a systematic procedure for determining all possible corresponding changes of generators, together with the corresponding realizations of the Uhg-action. The intriguing relation between g-invariants and Uhg-invariants suggests that these changes of generators might be employed to simplify the dynamics of some g-covariant quantum physical systems.

On the relation between quantum mechanics with a magnetic field on R^n and on a torus T^n

We show in elementary terms the equivalence in a general gauge of a U(1)-gauge theory of a scalar... more We show in elementary terms the equivalence in a general gauge of a U(1)-gauge theory of a scalar charged particle on a torus mathbb{T}n=mathbb{R}n/\varLambda to the analogous theory on ℝ n constrained by quasiperiodicity under translations in the lattice Λ. The latter theory provides a global description of the former: the quasiperiodic wavefunctions ψ defined on ℝ n play the role of sections of the associated hermitean line bundle E on mathbb{T}n, since also E admits a global description as a quotient. The components of the covariant derivatives corresponding to a constant (necessarily integral) magnetic field B= dA generate a Lie algebra g Q and together with the periodic functions the algebra of observables mathcal {O}Q. The non-abelian part of g Q is a Heisenberg Lie algebra with the electric charge operator Q as the central generator; the corresponding Lie group G Q acts on the Hilbert space as the translation group up to phase factors. Also the space of sections of E is mapped into itself by g∈ G Q . We identify the socalled magnetic translation group as a subgroup of the observables' group Y Q . We determine the unitary irreducible representations of mathcal{O}Q,YQ corresponding to integer charges and for each of them an associated orthonormal basis explicitly in configuration space. We also clarify how in the n=2 m case a holomorphic structure and Theta functions arise on the associated complex torus. These results apply equally well to the physics of charged scalar particles on ℝ n and on mathbb{T}n in the presence of periodic magnetic field B and scalar potential. They are also necessary preliminary steps for the application to these theories of the deformation procedure induced by Drinfel'd twists.

On plane waves in diluted relativistic cold plasmas

Xvii International Conference on Waves and Stability in Continuous Media, May 15, 2013

We consider the exact microscopic equations ruling a relativistic cold plasma after the plane-wav... more We consider the exact microscopic equations ruling a relativistic cold plasma after the plane-wave Ansatz. We show that the motion of a very diluted plasma initially at rest and excited by a given purely forward (or backward) transverse plane wave has a disarmingly simple and explicit expression in terms of the transverse electromagnetic potential A ⊥ (x − ct), valid for all profiles and intensities of the latter. We also mention the relevance of these results for understanding the acceleration mechanisms of charged particles (in particular, electrons) by nowadays available very short and intense laser pulses.

Frame Formalism for the N-Dimensional Quantum Euclidean Spaces

Ijmpb, 2000

We sketch our application1 of a non-commutative version of the Cartan "moving-frame" formalism to... more We sketch our application1 of a non-commutative version of the Cartan "moving-frame" formalism to the quantum Euclidean space RNqthe space which is covariant under the action of the quantum group SOq(N). For each of the two covariant differential calculi over RNq based on the R-matrix formalism, we summarize our construction of a frame, the dual inner derivations, a metric and two torsion-free almost metric compatible covariant derivatives with a vanishing curvature. To obtain these results we have developed a technique which fully exploits the quantum group covariance of RNq. We first find a frame in the larger algebra Ω * ( RNq)⋊ Uq so (N). Then we define homomorphisms from RNq⋊ Uq± so (N) to RNq which we use to project this frame in Ω * ( RNq).

On the hermiticity of q-differential operators and forms on the quantum Euclidean spaces R_q^N

We show that the complicated ⋆-structure characterizing for positive q the U q so(N )-covariant d... more We show that the complicated ⋆-structure characterizing for positive q the U q so(N )-covariant differential calculus on the non-commutative manifold R N q boils down to similarity transformations involving the ribbon element of a central extension of U q so(N ) and its formal square rootṽ. Subspaces of the spaces of functions and of p-forms on R N q are made into Hilbert spaces by introducing non-conventional "weights" in the integrals defining the corresponding scalar products, namely suitable positive-definite qpseudodifferential operatorsṽ ′±1 realizing the action ofṽ ±1 ; this serves to make the partial q-derivatives antihermitean and the exterior coderivative equal to the hermitean conjugate of the exterior derivative, as usual. There is a residual freedom in the choice of the weight m(r) along the 'radial coordinate' r. Unless we choose a constant m, then the square-integrables functions/forms mustfulfill an additional condition, namely their analytic continuations to thecomplex r plane can have poles only on the sites of some special lattice.Among the functions naturally selected by this condition there are q-special functions with 'quantized' free parameters.

Existence, uniqueness and stability for a class of third order dissipative problems depending on time

We prove new results regarding the existence, uniqueness, (eventual) boundedness, (total) stabili... more We prove new results regarding the existence, uniqueness, (eventual) boundedness, (total) stability and attractivity of the solutions of a class of initial-boundary-value problems characterized by a quasi-linear third order equation which may contain time-dependent coefficients. The class includes equations arising in Superconductor Theory and in the Theory of Viscoelastic Materials. In the proof we use a Liapunov functional V depending on two parameters, which we adapt to the characteristics of the problem.

On the Consequences of Twisted Poincaré Symmetry Upon QFT on Moyal Noncommutative Spaces

Quantum Field Theory and Beyond, 2008

We explore some general consequences of a consistent formulation of relativistic quantum field th... more

Some aspects of the quantization of p -forms and reducible gauge theories

Nuovo Cimento a Nucl Part F, 1991

The quantization of the p-form in D dimensions is carried out according to both an extension of t... more The quantization of the p-form in D dimensions is carried out according to both an extension of the BRST formalism and a method due to A. S. Schwarz. The latter yields a suggestive interpretation of the role of nonphysical fields in the quantization of reducible gauge theories. The introduction of gauge conditions of a different type (namely self-dual) modifies the ghost spectrum but does not affect the partition function. PACS 11.15. -Gauge field theories. PACS 03.70. -Theories of quantized fields. 1. -Many examples of reducible gauge theories such as supergravity and string theories have arisen in the past decade. The term ,,reducible~ is currently understood to mean that the generators of the local gauge algebra which naturally appear in the formulation of the theory are linearly dependent on each other. Using a fairly condensed notation, (r being the set of the fields, S(r the classical action, R~ and ~ the gauge generators and the gauge parameters, respectively, gauge invariance implies ~S R~ (1) ~ ~ = O. (*) The authors of this paper have agreed to not receive the proofs for correction. 56 -I1 Nuovo Cimento A 899 ~00 L. CAPPIELLO and G. FIORE

format_quoteTwo simplified quantization methods for reducible gauge theories provide clearer insights into nonphysical fields' roles in the quantization process.format_quote

Braided Chains of q-Deformed Heisenberg Algebrae

Given M copies of a q-deformed Weyl or Clifford algebra in the defining representation of a quant... more Given M copies of a q-deformed Weyl or Clifford algebra in the defining representation of a quantum group $G_q$, we determine a prescription to embed them into a unique, inclusive $G_q$-covariant algebra. The different copies are "coupled" to each other and are naturally ordered into a "chain". In the case $G_q=SL_q(N)$ a modified prescription yields an inclusive algebra which is even explicitly $SL_q(M) X SL_q(N)$-covariant, where $SL_q(M)$ is a symmetry relating the different copies. By the introduction of these inclusive algebrae we significantly enlarge the class of $G_q$-covariant deformed Weyl/Clifford algebrae available for physical applications.

The Euclidean Hopf algebra $U_q(e^N)$ and its fundamental Hilbert space representations

We construct the Euclidean Hopf algebra $U_q(e^N)$ dual of $Fun(\rn_q^N\lcross SO_{q^{-1}}(N))$ b... more We construct the Euclidean Hopf algebra $U_q(e^N)$ dual of $Fun(\rn_q^N\lcross SO_{q^{-1}}(N))$ by realizing it as a subalgebra of the differential algebra $\DFR$ on the quantum Euclidean space $\rn_q^N$; in fact, we extend our previous realization \cite{fio4} of $U_{q^{-1}}(so(N))$ within $\DFR$ through the introduction of q-derivatives as generators of q-translations. The fundamental Hilbert space representations of $U_q(e^N)$ turn out to be of highest weight type and rather simple `` lattice-regularized '' versions of the classical ones. The vectors of a basis of the singlet (i.e. zero-spin) irrep can be realized as normalizable functions on $\rn_q^N$, going to distributions in the limit $q\rightarrow 1$.

format_quoteFundamental irreducible representations of U_q(e_N) are established, relevant for quantum physics applications.format_quote