Guy Rideau

On the representations of quantum oscillator algebra

It is shown that for q<1, the quantum oscillator algebra has a supplementary family of representations inequivalent to the usual q-Fock representation, with no counterpart at the limit q=1. They are used to build representations of SUq(1,1) and E(2) in Schwingers way.

Classification of the quantum deformations of the superalgebra gl(1|1)

We present a classification of the possible quantum deformations of the supergroup and its Lie superalgebra . In each case, the (super)commutation relations and the Hopf structures are explicitly computed. For each R-matrix, one finds two inequivalent co-products whether one chooses an unbraided or a braided framework while the corresponding structures are isomorphic as algebras. In the braided case, one recovers the classical algebra for suitable limits of the deformation parameters but this is no longer true in the unbraided case.

On the realization of Landau gauge

Starting with the determination of the two-points function in the Landau gauge given by Wightman and Strocchi, we build a more economic realization of this gauge, following the basic principles they have proposed.

Stochastic processes and the evolution of quantum observables

The classical results of stochastic calculus are extended to the equations giving the evolution of quantum observables in terms of their Weyl symbols, when the free Hamiltonian in h0(p) + qh1(p) or p2/2m) + (mω2/2)q2 and the interaction potential is the Fourier transform of a bounded measure. The arising stochastic processes are purely jump processes.

On the extensions of mass-zero representations of the poincaré group

Abstract It is shown that the extension of two representations of the Poincare group with mass zero and definite helicities which occur in theoretical physics (two component neutrino) do not fall in to the framework of Hilbert extension theory. To get extensions useful in physics, the representation spaces are restricted according to a criterion of simplicity. Then it is proved that there exists a non-trivial extension only if the difference of the helicities is one in modulus, or if the helicities are both equal to zero. As desired, the two component neutrino fits well into this scheme.

An analytic nonlinear representation of the poincaré group: II. the case of helicities ±1/2

We construct an analytic truly nonlinear representation of the Poincaré group having as its linear part the mass zero, helicity -1/2(+1/2) unitary representation.

The cohomology of the classical and quantum Weyl CCR in curved spaces

An analysis is presented of the cohomological underpinnings for the Weyl group of the canonical commutation relations on manifolds of constant negative curvature. Several uniqueness results are obtained leading from purely classical considerations to the group associated with the systems of imprimitivity of the orthodox approach to quantum mechanics.

An analytic nonlinear representation of the Poincaré group

We prove in a constructive way the existence of an analytic nonlinear representation of the Poincaré group in a Banach space, the linear part of which is the massless representation with helicity +1 (or-1). Furthermore, this nonlinear representation is shown to be analytically unwquivalent to any unitary linear representation.

R-matrix method for Heisenberg quantum groups

TheR-matrix method is systematically applied to get several Heisenberg quantum groups depending on two or three parameters. It turns out that the associatedR-matrices have to verify a weaker form of the QYBE. Only for particular cases of quantum groups, we can imposeR to be a solution of the QYBE. The corresponding quantum Heisenberg Lie algebras are obtained by duality.

Quantum fields and poisson processes: Interaction of a cut-off boson field with a quantum particle

The solution of the Schrödinger equation for a boson field interacting with a quantum particle is written as an expectation on a Poisson process counting the variations of the boson-occupation numbers for each momentum. An energy cut-off is needed for the expectation to be meaningful.