Decoupling of Tensor factors in Cross Product and Braided Tensor Product Algebras

We briefly review and illustrate our procedure to 'decouple' by transformation of generators: either a Hopf algebra $H$ from a $H$-module algebra $A_1$ in their cross-product $A_1 >\triangleleft H$; or two (or more) $H$-module algebras $A_1,A_2$. These transformations are based on the existence of an algebra map $A_1 >\triangleleft H\to A_1$.Comment: Latex file,7 pages.Contribution to the Proceedings of the "International Colloquium on Group Theoretical Methods in Physics" (Group24), Paris, July 200

New approach to Hermitian q-differential operators on R_q^N

We report on our recent breakthrough in the costructionfor q>0 of Hermitean and "tractable" differential operators out of the U_qso(N)-covariant differential calculus on the noncommutative manifolds R_q^N (the socalled "quantum Euclidean spaces").Comment: Latex file, 11 page

Can QFT on Moyal-Weyl spaces look as on commutative ones?

We sketch a natural affirmative answer to the question based on a joint work [11] with J. Wess. There we argue that a proper enforcement of the "twisted Poincare'" covariance makes any differences $(x-y)^\mu$ of coordinates of two copies of the Moyal-Weyl deformation of Minkowski space like undeformed. Then QFT in an operator approach becomes compatible with (minimally adapted) Wightman axioms and time-ordered perturbation theory, and physically equivalent to ordinary QFT, as observables involve only coordinate differences.Comment: Talk given at the 21st Nishinomiya-Yukawa Memorial Symposium on Theoretical Physics "Noncommutative Geometry and Spacetime in Physics", Nishinomiya-Kyoto, Nov. 200

On very short and intense laser-plasma interactions

We briefly report on some results regarding the impact of very short and intense laser pulses on a cold, low-density plasma initially at rest, and the consequent acceleration of plasma electrons to relativistic energies. Locally and for short times the pulse can be described by a transverse plane electromagnetic travelling-wave and the motion of the electrons by a purely Magneto-Fluido-Dynamical (MFD) model with a very simple dependence on the transverse electromagnetic potential, while the ions can be regarded as at rest; the Lorentz-Maxwell and continuity equations are reduced to the Hamilton equations of a Hamiltonian system with 1 degree of freedom, in the case of a plasma with constant initial density, or a collection of such systems otherwise. We can thus describe both the well-known "wakefield" behind the pulse and the recently predicted "slingshot effect", i.e. the backward expulsion of high energy electrons just after the laser pulse has hit the surface of the plasma.Comment: Latex file, 15 pages, 6 figure

The q-Euclidean algebra Uq(eN)U_q(e^N) and the corresponding q-Euclidean lattice

We review the Euclidean Hopf algebra $U_q(e^N)$ dual of Fun(\rn_q^N\lcross SO_{q^{-1}}(N)) and describe its fundamental Hilbert space representations \cite{fioeu}, which turn out to be rather simple "lattice-regularized" versions of the classical ones, in the sense that the spectra of squared momentum components are discrete and the corresponding eigenfunctions normalizable.These representations can be regarded as describing a quantum system consisting of one free particle on the quantum Euclidean space. A suitable notion of classical limit is introduced, so that we recover the classical continuous spectra and generalized (non-normalizable) eigenfunctions in that limit.Comment: 19pages, latex. transmission error correcte

Travelling waves and a fruitful `time' reparametrization in relativistic electrodynamics

We simplify the nonlinear equations of motion of charged particles in an external electromagnetic field that is the sum of a plane travelling wave F_t(ct-z) and a static part F_s(x,y,z): by adopting the light-like coordinate ct-z instead of time t as an independent variable in the Action, Lagrangian and Hamiltonian, and deriving the new Euler-Lagrange and Hamilton equations accordingly, we make the unknown z(t) disappear from the argument of F_t. We study and solve first the single particle equations in few significant cases of extreme accelerations. In particular we obtain a rigorous formulation of a Lawson-Woodward-type (no-final-acceleration) theorem and a compact derivation of cyclotron autoresonance, beside new solutions in the presence of uniform F_s. We then extend our method to plasmas in hydrodynamic conditions and apply it to plane problems: the system of partial differential equations may be partially solved and sometimes even completely reduced to a family of decoupled systems of ordinary ones; this occurs e.g. with the impact of the travelling wave on a vacuum-plasma interface (what may produce the slingshot effect). Since Fourier analysis plays no role in our general framework, the method can be applied to all kind of travelling waves, ranging from almost monochromatic to socalled "impulses", which contain few, one or even no complete cycle.Comment: Latex file, 35 pages, 6 figures. Final version to appear in J. Phys. A: Math. Theo

Noncommutative spaces with twisted symmetries and second quantization

In a minimalistic view, the use of noncommutative coordinates can be seen just as a way to better express non-local interactions of a special kind: 1-particle solutions (wavefunctions) of the equation of motion in the presence of an external field may look simpler as functions of noncommutative coordinates. It turns out that also the wave-mechanical description of a system of n such bosons/fermions and its second quantization is simplified if we translate them in terms of their deformed counterparts. The latter are obtained by a general twist-induced *-deformation procedure which deforms in a coordinated way not just the spacetime algebra, but the larger algebra generated by any number n of copies of the spacetime coordinates and by the particle creation and annihilation operators. On the deformed algebra the action of the original spacetime transformations looks twisted. In a non-conservative view, we thus obtain a twisted covariant framework for QFT on the corresponding noncommutative spacetime consistent with quantum mechanical axioms and Bose-Fermi statistics. One distinguishing feature is that the field commutation relations remain of the type "field (anti)commutator=a distribution". We illustrate the results by choosing as examples interacting non-relativistic and free relativistic QFT on Moyal space(time)s.Comment: Latex file 16 pages. Talk given at the conference "Noncommutative Structures in Mathematics and Physics" (Satellite Conference to the 5th European Congress of Mathematics), Brussels 22-26/7/2008. Appeared in the Proceedings, Ed. S. Caenepeel, J. Fuchs, S. Gutt, C. Schweigert, A. Stolin, F. Van Oystaeyen, Royal Flemish Academy of Belgium for Sciences and Arts, brussels, 2010, pp. 163-17

The SOq(N,R)SO_q(N,{\bf R})-Symmetric Harmonic Oscillator on the Quantum Euclidean Space RqN{\bf R}_q^N and its Hilbert Space Structure

We show that the isotropic harmonic oscillator in the ordinary euclidean space ${\bf R}^N$ ($N\ge 3$) admits a natural q-deformation into a new quantum mechanical model having a q-deformed symmetry (in the sense of quantum groups), $SO_q(N,{\bf R})$. The q-deformation is the consequence of replacing $R^N$ by ${\bf R}^N_q$ (the corresponding quantum space). This provides an example of quantum mechanics on a noncommutative geometrical space. To reach the goal, we also have to deal with a sensible definition of integration over ${\bf R}^N_q$, which we use for the definition of the scalar product of states.Comment: 55 pages, tex, to appear in the Int. J Mod. Phys. A, 1993. Revised Feb. 199

Embedding q-deformed Heisenberg Algebras into Undeformed Ones

Any deformation of a Weyl or Clifford algebra can be realized through some change of generators in the undeformed algebra. Here we briefly describe and motivate our systematic procedure for constructing all such changes of generators for those particular deformations where the original algebra is covariant undersome Lie group and the deformed algebra is covariant under the corresponding quantum group.Comment: LaTex2e file, 8 pages, no figure. To appear in Rep. Math Phy

The Euclidean Hopf algebra Uq(eN)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)) 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 $q\rightarrow 1$.Comment: 67 pages, 1 figures. Revised version: Format changed, typos amended, some citations adde