Renormalization group and continuum limit in Quantum Mechanics

1 ∆t ). Furthermore, this nondifferentiability of the trajectories is the source of the Itô calculus [2] in Quantum Mechanics. One expects that the velocity dependent interactions become more important for short time processes. Our goal is to see whether the concept of running coupling constant is meaningful in Quantum Mechanics and it shows an enhancement for velocity dependent interactions at high energy or short time. Quantum Mechanics can formally be considered as a lattice regulated Quantum Field Theory in 0+1 dimension with lattice spacing ∆t. One finds that the transition amplitudes are not necessarily ultraviolet finite when velocity dependent interactions are present. The power counting argument shows that the vertex ( ∆t ) x is renormalizable, i.e. its contributions do not become more and more ultraviolet singular in the higher orders of the perturbation expansion for d ≤ 2+s [3]. We shall find that some of the ultraviolet singularities predicted by the simple power counting argument are curiously enough cancel in the simple perturbation expansion and are only present when the expansion is made around non-static trajectories. This raises the question about the equivalence of the operator and the path integral formalism. These questions will be considered here in the framework of the perturbation expansion. The renormalization of Quantum Mechanics has already been discussed in the presence of singular potential in [4]. Our motivation to look for the running coupling constant is different, the goal is now to understand the time dependence of the transition amplitude