Quantization of a parametric Hamiltonian formalism

Ph. Droz-Vincent

Mark S. Hillery

The process of n-photon down conversion is considered both with a quantized pump and in the parametric approximation. In the parametric approximation it is formally found that the photon number diverges at a finite time for n>2. This divergence limits the times for which the parametric approximation is useful. The divergence does not occur if the pump is quantized. Furthermore, it is shown that the Hamiltonian with a quantized pump is essentially self-adjoint. This means that the questions that have been raised about the existence of the time-development operator in the parametric approximation do not arise when the pump is quantized.

Sergio Giardino

Abstract A generalization of classical mechanics is obtained from a complex parametrization of the phase space. The formalism supports complex Hamiltonian functions describing non-conservative classical mechanical systems. A quantization scheme that is general enough to incorporate non-stationary physical processes is also achieved.

Paweł Błasiak, A. Horzela

Adnan Hamoui, André Lichnerowicz

The quantization of a dynamical system with time-dependent constraints or time-dependent Hamiltonians presents particular difficulties not only from a technical, but also from a theoretical point of view. We give here an approach to the evolution of such a quantum system in terms of star products on symplectic or canonical manifolds. These star products are particular deformations of the usual product of functions, deformations that start with the Poisson bracket. The introduction of the star products corresponds to a generalization of the Weyl–Wigner quantization on a flat phase space. This approach leads to the introduction of ‘‘two times’’ denoted t and τ, playing different roles; the time t plays the role of a geometrical time corresponding to the control, while the other, τ, which parametrizes the evolution, plays the role of a dynamical time. The classical as well as the quantum results point out the role played by the variable (t+τ). This reflects the coherence of the formalism.