Ph. Combe

On the quantization of spin systems and Fermi systems

It is shown that spin operators and Fermi operators can be interpreted as the Weyl quantization of some functions on a ’’classical phase space’’ which is a compact group. Moreover the transition from quantum spin to Fermi operators is an isomorphism of the ’’classical phase space’’ preserving the Haar measure.

Feynman path integral and Poisson processes with piecewise classical paths

We prove the existence of a Feynman integral formula for gentle perturbations of the harmonic oscillator. This result is extended to Bose relativistic theory.

On real Hilbertian info-manifolds

The methods of differential geometry applied to probability and statistics open a new domain of investigation of the statistical manifold or information geometry. This framework provides a geometrical description of statistical quantities and leads to a new approach to complex statistical problems. A peculiar feature is that statistical manifolds are naturally associated with a family of affine-metric geometries. In active fields such as information and communication theories, the finite dimensional statistical manifolds approach is not completely satisfactory. In this short note we discuss some possible constructions of a real Hilbertian geometry.

Feynman Formula and Poisson Processes for Gentle Perturbations

Let H = H0 + V be a Hamiltonian, the solution of the corresponding Schrodinger equation is expected to be given by the Feynman path integral [1]

Stochastic jump processes associated with Dirac equation

\psi (X,T) = \int\limits_{\Gamma } {{e^{{ - i{S_{0}}(X,\gamma )}}} - i\int\limits_{o}^{T} {V(X - \gamma (t))dt} } \psi (X - \gamma (o))d\gamma

Geometrical methods in learning theory

where S0 is the free classical action associated with a path γ ∈ Γ, V is the potential and dγ is expected to be a measure.

Statistical Manifolds, Self-Parallel Curves and Learning Processes

The problem of the frustration on graphs is investigated in terms of Cramerlike systems on a hypercube. This approach provides a linear algorithmic method to characterise the non-frustrated edge configurations. The case of complete graphs is given as example.

Probabilistic Expression for the Solution of the Dirac Equation in Fourier Space

We study the stochastic jump processes associated with the Dirac equation where the space derivatives are replaced by discrete approximations.