J. Bertrand

Coherent states and the role of the affine group in the quantum mechanics of the Morse potential

The coherent states of the Morse potential that have been obtained earlier from supersymmetric quantum mechanics, are shown to be connected with the representations of the affine group of the real line and some of its extensions. This relation is similar to the one between the Heisenberg-Weyl group and the coherent states of the harmonic oscillator. The states that minimize the uncertainty product of the generators of the affine Lie algebra are shown to contain all the coherent states of the Morse oscillator plus the intelligent states of the Morse Hamiltonians with different shape parameter s. The representations of the central extension of the affine group denoted by G0 and its further extension will be shown to define the phase space relevant to the problem by choosing an appropriate orbit of the coadjoint representation of . This allows one to construct a generalized Wigner function on this phase space, which is again essentially in the same relation with the affine group, as the ordinary Wigner function with the Heisenberg-Weyl group.

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.

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.

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.

Characterization of SU(1, 1) coherent states in terms of affine group wavelets

The Perelomov coherent states of SU(1, 1) are labelled by elements of the quotient of SU(1, 1) by the compact subgroup. Taking advantage of the fact that this quotient is isomorphic to the affine group of the real line, we are able to parametrize the coherent states by elements of that group or equivalently by points in the half-plane. Such a formulation permits to find new properties of the SU(1, 1) coherent states and to relate them to affine wavelets.

Invariance quantum groups of the deformed oscillator algebra

A differential calculus is set up on a deformation of the oscillator algebra. It is uniquely determined by the requirement of invariance under a seven-dimensional quantum group. The quantum space and its associated differential calculus are also shown to be invariant under a nine generator quantum group containing the previous one.

Non-uniqueness in writing Schrödinger kernel as a functional integral

We consider a path integral in phase space involving a linear functional of the classical Hamiltonian and find the Schrödinger equation of which it is a propagator. Conversely, to any quantum Hamiltonian, we associate a whole family of functionals and hence of expressions of the same Schrödinger kernel; all this is carried out independently of any correspondence principle.

Quantum fields and poisson processes II: Interaction of boson-boson and boson-fermion fields with a cut-off

Quantum field evolutions are written as expectation values with respect to Poisson processes in two simple models: interaction of two boson fields (with conservation of the number of particles in one field) and interaction of a boson with a fermion field. The introduction of a cut-off ensures that the expectation values are well-defined.

Fock spaces for mass-zero helicity-one particles described by an antisymmetrical two-tensor

Using mainly Poincaré covariance arguments, we find all the zero-mass ‘particles’ (in Wigners sense) that can be described by a skew-symmetric two-tensor on the light cone and we construct the corresponding free field theories, when possible.