Gaetano Fiore

The Euclidean Hopf algebra Uq(eN) and its fundamental Hilbert‐space representations

The Euclidean Hopf algebra Uq(eN) dual of Fun(RNq■SOq−1(N)) is constructed by realizing it as a subalgebra of the differential algebra Diff(RNq) on the quantum Euclidean space RNq; in fact, the previous realization [G. Fiore, Commun. Math. Phys. 169, 475–500 (1995)] of Uq−1(so(N)) is extended within Diff(RNq) through the introduction of q derivatives as generators of q translations. The fundamental Hilbert‐space representations of Uq(eN) 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 RNq, going to distributions in the limit q → 1.

Deforming maps for Lie group covariant creation and annihilation operators

Any deformation of a Weyl or Clifford algebra A can be realized through a “deforming map,” i.e., a formal change of generators in A. This is true in particular if A is covariant under a Lie algebra g and its deformation is induced by some triangular deformation Uhg of the Hopf algebra Ug. We propose a systematic method to construct all the corresponding deforming maps, together with the corresponding realizations of the action of Uhg. The method is then generalized and explicitly applied to the case that Uhg is the quantum group Uhsl(2). A preliminary study of the status of deforming maps at the representation level shows in particular that “deformed” Fock representations induced by a compact Uhg can be interpreted as standard “undeformed” Fock representations describing particles with ordinary Bose or Fermi statistics.

Unbraiding the braided tensor product

We show that the braided tensor product algebra A1⊗_A2 of two module algebras A1,A2 of a quasitriangular Hopf algebra H is isomorphic to the ordinary tensor product A1⊗A2, provided there exists a realization of H within A1. In other words, under this assumption we construct a transformation of generators which decouples A1,A2 (i.e., makes them commuting). We apply the theorem to the braided tensor product algebras of two or more quantum group covariant quantum spaces, deformed Heisenberg algebras and q-deformed fuzzy spheres.

The geometry of the quantum Euclidean space

Abstract A detailed study is made of the noncommutative geometry of R 3 q , the quantum space covariant under the quantum group SO q (3). For each of its two SO q (3)-covariant differential calculi we find its metric, the corresponding frame and two torsion-free covariant derivatives that are metric compatible up to a conformal factor and both which yield a vanishing linear curvature. A discussion is given of various ways of imposing reality conditions. The delicate issue of the commutative limit is discussed at the formal algebraic level. Two rather different ways of taking the limit are suggested, yielding S 2 × R and R 3 , respectively, as the limit Riemannian manifolds.

Existence and uniqueness of solutions of a class of third order dissipative problems with various boundary conditions describing the Josephson effect

Abstract We prove existence and uniqueness of solutions of a large class of initial–boundary-value problems characterized by a quasi-linear third order equation (the third order term being dissipative) on a finite space interval with Dirichlet, Neumann or pseudoperiodic boundary conditions. The class includes equations arising in superconductor theory, in the form of various modified sine–Gordon equation describing the Josephson effect, and in the theory of viscoelastic materials.

On the decoupling of the homogeneous and inhomogeneous parts in inhomogeneous quantum groups

We show that, if there exists a realization of a Hopf algebra H in a H-module algebra , then one can split their cross-product into the tensor product algebra of itself with a subalgebra isomorphic to H and commuting with . This result applies in particular to the algebra underlying inhomogeneous quantum groups like the Euclidean groups, which are obtained as cross-products of the quantum Euclidean spaces qN with the quantum groups of rotation Uq so(N) of qN, for which it has no classical analogue.

DRINFEL'D TWIST AND q-DEFORMING MAPS FOR LIE GROUP COVARIANT HEISENBERG ALGEBRAE

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.

The slingshot effect: a possible new laser-driven high energy acceleration mechanism for electrons

We show that under appropriate conditions the impact of a very short and intense laser pulse onto a plasma causes the expulsion of surface electrons with high energy in the direction opposite to the one of the propagations of the pulse. This is due to the combined effects of the ponderomotive force and the huge longitudinal field arising from charge separation (“slingshot effect”). The effect should also be present with other states of matter, provided the pulse is sufficiently intense to locally cause complete ionization. An experimental test seems to be feasible and, if confirmed, would provide a new extraction and acceleration mechanism for electrons, alternative to traditional radio-frequency-based or laser-wake-field ones.

Class of invariants for the two-dimensional time-dependent Landau problem and harmonic oscillator in a magnetic field

We consider an isotropic two-dimensional harmonic oscillator with arbitrarily time-dependent mass M(t) and frequency Ω(t) in an arbitrarily time-dependent magnetic field B(t). We determine two commuting invariant observables (in the sense of Lewis and Riesenfeld) L, I in terms of some solution of an auxiliary ordinary differential equation and an orthonormal basis of the Hilbert space consisting of joint eigenvectors φλ of L, I. We then determine time-dependent phases αλ(t) such that the ψλ(t)=eiαλϕλ are solutions of the time-dependent Schrodinger equation and make up an orthonormal basis of the Hilbert space. These results apply, in particular, to a two-dimensional Landau problem with time-dependent M, B, which is obtained from the above just by setting Ω(t) ≡ 0. By a mere redefinition of the parameters, these results can be applied also to the analogous models on the canonical non-commutative plane.

q-quaternions and q-deformed su(2) instantons

We construct (anti-)instanton solutions of a would-be q-deformed su(2) Yang-Mills theory on the quantum Euclidean space Rq4 [the SOq(4)-covariant noncommutative space] by reinterpreting the function algebra on the latter as a q-quaternion bialgebra. Since the (anti-)self-duality equations are covariant under the quantum group of deformed rotations, translations, and scale change, by applying the latter we can generate new solutions from the one centered at the origin and with unit size. We also construct multi-instanton solutions. As they depend on noncommuting parameters playing the roles of “sizes” and “coordinates of the centers” of the instantons, this indicates that the moduli space of a complete theory should be a noncommutative manifold. Similarly, gauge transformations should be allowed to depend on additional noncommutative parameters.